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Towards A Posteriori Error Estimators for Realistic Problems in Incompressible Miscible Displacement

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Numerical Mathematics and Advanced Applications 2011

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Abstract

The incompressible miscible displacement problem has attracted interest in recent years as it models economically important activities such as oil recovery and groundwater flow. It is important that numerical simulations can accurately model the types of problems seen in industry. We discuss a posteriori finite element indicators for the incompressible miscible displacement problem and propose an extension to a mixed-discontinuous Galerkin scheme. Furthermore we highlight some physically realistic scenarios not covered by the existing analysis and outline the theory of weighted spaces required to address them.

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References

  1. Ammann, B., Nistor, V.: Weighted Sobolev spaces and regularity for polyhedral domains. Comp. Meth. in Appl. Mech. and Eng. 196(37–40), 3650–3659 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bartels, S., Jensen, M., Müller, R.: Discontinuous Galerkin finite element convergence for incompressible miscible displacement problems of low regularity. SIAM J. Numer. Anal. 47(5), 3720–3743 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bear, J.: Dynamics of Fluids in Porous Media. Dover, New York (1988)

    MATH  Google Scholar 

  4. Bear, J., Cheng, A.H.D.: Modeling Groundwater Flow and Contaminant Transport. Theory and Applications of Transport in Porous Media. Springer (2009)

    Google Scholar 

  5. Chapman, J.: Finite element methods for the incompressible miscible displacement problem. Ph.D. Thesis, University of Durham (2012). In preparation

    Google Scholar 

  6. Chen, Y., Liu, W.: A posteriori error estimates of mixed methods for miscible displacement problems. Int. J. Numer. Meth. Eng. 73(3), 331–343 (2008)

    Article  MATH  Google Scholar 

  7. Chen, Z., Ewing, R.: Mathematical analysis for reservoir models. SIAM J. Math. Anal. 30(2), 431–453 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cui, M.: A combined mixed and discontinuous Galerkin method for compressible miscible displacement problem in porous media. J. Comput. Appl. Math. 198(1), 19–34 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cui, M.: Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem. J. Comput. Appl. Math. 214(2), 617–636 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Douglas, J., Ewing, R.E., Wheeler, M.F.: The approximation of the pressure by a mixed method in the simulation of miscible displacement. RAIRO Anal. Numer. 17, 17–33 (1983)

    MathSciNet  MATH  Google Scholar 

  11. Douglas, J., Roberts., J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comp. 44(169), 39–52 (1985)

    Google Scholar 

  12. Evans, L.C.: Partial Differential Equations. American Mathematical Society, Rhode Island (2008)

    Google Scholar 

  13. Ewing, R.E., Wheeler, M.F.: Galerkin methods for miscible displacement problems in porous media. SIAM J. Numer. Anal. 17(3), 351–365 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jensen, M., Müller, R.: Stable Crank-Nicolson discretisation for incompressible miscible displacement problems of low regularity. In: G. Kreiss, P. Ltstedt, A. Mlqvist, M. Neytcheva (eds.) Numerical Mathematics and Advanced Applications 2009, pp. 469–477. Springer Berlin Heidelberg (2010)

    Google Scholar 

  15. Lake, L.W.: Enhanced Oil Recovery. Prentice Hall (1996)

    Google Scholar 

  16. Larson, M.G., Målqvist, A.: Goal oriented adaptivity for coupled flow and transport problems with applications in oil reservoir simulations. Comp. Meth. Appl. Mech. Eng. 196(37–40), 3546–3561 (2007)

    Article  MATH  Google Scholar 

  17. Latil, M.: Enhanced Oil Recovery. Institut Français du Pétrole Publications. Éditions Technip (1980)

    Google Scholar 

  18. Li, H.: Elliptic equations with singularities: A priori analysis and numerical approaches. PhD Thesis, The Pennsylvania State University (2008)

    Google Scholar 

  19. Li, H., Mazzucato, A., Nistor, V.: Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains. Elec. Trans. Numer. Anal. 37, 41–69 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Peaceman, D.W.: Fundementals of Numerical Reservoir Simulation. Developments in Petroleum Science. Elsevier Scientific Publishing Company (1977)

    Google Scholar 

  21. Rivière, B., Walkington, N.: Convergence of a discontinuous Galerkin method for the miscible displacement equation under low regularity. SIAM J. Numerical Analysis 49(3), 1085–1110 (2011)

    Article  MATH  Google Scholar 

  22. Sun, S., Rivière, B., Wheeler, M.F.: A combined mixed finite element and discontinuous Galerkin method for miscible displacement problem in porous media. In: Recent Progress in Computational and Applied PDEs, pp. 323–351. Kluwer/Plenum, New York (2002)

    Google Scholar 

  23. Yang, J.: A posteriori error of a discontinuous Galerkin scheme for compressible miscible displacement problems with molecular diffusion and dispersion. Int. J. Numer. Meth. Fluids 65(7), 781–797 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Durham University, Durham, UK

    J. Chapman & M. Jensen

Authors
  1. J. Chapman
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  2. M. Jensen
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Corresponding author

Correspondence to J. Chapman .

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Editors and Affiliations

  1. , Department of Mathematics, University of Leicester, Leicester, LE!1 7RH, United Kingdom

    Andrea Cangiani

  2. , Department of Mathematics, University of Leicester, Leicester, LE1 7RL, United Kingdom

    Ruslan L. Davidchack

  3. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Emmanuil Georgoulis

  4. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Alexander N. Gorban

  5. , Department of Mathematics, University of Leicester, Leicester, LE1 7RH, United Kingdom

    Jeremy Levesley

  6. , School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, United Kingdom

    Michael V. Tretyakov

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Chapman, J., Jensen, M. (2013). Towards A Posteriori Error Estimators for Realistic Problems in Incompressible Miscible Displacement. In: Cangiani, A., Davidchack, R., Georgoulis, E., Gorban, A., Levesley, J., Tretyakov, M. (eds) Numerical Mathematics and Advanced Applications 2011. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33134-3_35

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