Skip to main content
Springer Nature Link
Log in

Numerical Modeling of Degenerate Equations in Porous Media Flow

Degenerate Multiphase Flow Equations in Porous Media

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  3. Aarnes, J.E., Krogstad, S., Lie, K.-A.: Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci. 12, 297–315 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  9. Blunt, M.J., Thiele, M.R., Batycky, R.P.: A 3D field scale streamline-based reservoir simulator. SPE Reserv. Eng. 246–254 (1997)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. Chen, Z., Ewing, R.E.: Comparison of various formulation of three-phase flow in porous media. J. Comput. Phys. 132, 362–373 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  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. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics, 3rd edn. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Doughty, C., Pruess, K.: Modeling supercritical carbon dioxide injection in heterogeneous porous media. Vadose Zone J. 3, 837–847 (2004)

    Article  Google Scholar 

  23. Douglas, J. Jr., Furtado, F., Pereira, F.: On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs. Comput. Geosci. 1, 155–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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)

    Article  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  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)

  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)

    Article  Google Scholar 

  31. Fortin, M., Brezzi, F.: Mixed and hybrid finite element methods. In: Springer Series in Computational Mathematics. Springer, Berlin (1991)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  34. Gerritsen, M.G., Durlofsky, L.J.: Modeling fluid flow in oil reservoirs. Annu. Rev. Fluid Mech. 37, 211–238 (2006)

    Article  Google Scholar 

  35. Glimm, J., Sharp, D.M.: Stochastic methods for the prediction of complex multiscale phenomena. Q. Appl. Math. 56, 741–765 (1998)

    MathSciNet  MATH  Google Scholar 

  36. Glimm, J., Sharp, D.M.: Prediction and the quantification of uncertainty. Physica D 133, 142–170 (1999)

    Article  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97–127 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Isaacson, E., Marchesin, D., Plohr, B., Temple, J.B.: Multiphase flow models with singular Riemann problems. Comput. Appl. Math. 11, 147–166 (1992)

    MathSciNet  MATH  Google Scholar 

  42. Karlsen, K.H., Risebro, N.H.: Corrected operator splitting for nonlinear parabolic equations. SIAM J. Numer. Anal. 37, 980–1003 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Leverett, M.C.: Capillary behavior in porous solids. Trans. Soc. Pet. Eng. 142, 152–169 (1941)

    Google Scholar 

  47. Li, K.: More general capillary pressure and relative permeability models from fractal geometry. J. Contam. Hydrol. 111(1–4), 13–24 (2010)

    Article  Google Scholar 

  48. Marchesin, D., Plohr, B.: Wave structure in WAG recovery. Soc. Pet. Eng. J. 71314, 209–219 (2001)

    Google Scholar 

  49. Nessyahu, N., Tadmor, E.: Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  50. Oleinik, O.: Discontinuous solutions of non-linear differential equations. Transl. Am. Math. Soc. 26(2), 95–172 (1963)

    MathSciNet  Google Scholar 

  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. 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)

    Article  Google Scholar 

  53. Shi, J., Hu, C., Shu, C.-W.: A technique for treating negative weights in WENO schemes. J. Comput. Phys. 175, 108–127 (2002)

    Article  MATH  Google Scholar 

  54. Stone, H.L.: Probability model for estimating three-phase relative permeability. J. Pet. Technol. 22, 214–218 (1970)

    Google Scholar 

  55. Titareva, V.A., Toro, E.F.: Finite-volume WENO schemes for three-dimensional conservation laws. J. Comput. Phys. 201(1), 238–260 (2004)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, IMECC, University of Campinas (UNICAMP), 13.083-859, Campinas, SP, Brazil

    Eduardo Abreu

  2. Department of Mathematics, Federal University Rural of Rio de Janeiro, Rod. BR 465, Km 7, 23.890-000, Seropédica, RJ, Brazil

    Duilio Conceição

Authors
  1. Eduardo Abreu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Duilio Conceição
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Eduardo Abreu.

Additional information

E. Abreu acknowledges financial support from State of São Paulo Research Foundation grant 2011/11897-6-FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) and from grant 519.292-785/11 by Fundo de Apoio ao Ensino, Pesquisa e Extensão da Universidade Estadual de Campinas (UNICAMP/FAEPEX). D. Conceição acknowledges financial support grants 620049/2008-1 from Edital Casadinho/CNPq and 558764/2008-8 from Bolsa de Pós-doutorado PNPD/CNPq.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abreu, E., Conceição, D. Numerical Modeling of Degenerate Equations in Porous Media Flow. J Sci Comput 55, 688–717 (2013). https://doi.org/10.1007/s10915-012-9653-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-012-9653-0

Keywords