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A fully conservative block-centered finite difference method for simulating Darcy-Forchheimer compressible wormhole propagation

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Abstract

In this paper, a block-centered finite difference method is derived for the Darcy-Forchheimer compressible wormhole propagation in porous media by introducing an auxiliary flux variable to guarantee full mass conservation. Error estimates for the pressure, velocity, porosity, concentration, and auxiliary flux in different discrete norms are established rigorously and carefully on nonuniform grids. Finally, some numerical experiments are demonstrated to verify the theoretical analysis and effectiveness of the given scheme.

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References

  1. Kou, J., Sun, S., Wu, Y.: Mixed finite element-based fully conservative methods for simulating wormhole propagation. Comput. Methods Appl. Mech. Eng. 298, 279–302 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  2. Panga, M.K., Ziauddin, M., Balakotaiah, V.: Two-scale continuum model for simulation of wormholes in carbonate acidization. AIChE J 51(12), 3231–3248 (2005)

    Article  Google Scholar 

  3. Wu, Y., Salama, A., Sun, S.: Parallel simulation of wormhole propagation with the Darcy-Brinkman-Forchheimer framework. Comput. Geotech. 69, 564–577 (2015)

    Article  Google Scholar 

  4. Fredd, C.N., Fogler, H.S.: Influence of transport and reaction on wormhole formation in porous media. AIChE J 44(9), 1933–1949 (1998)

    Article  Google Scholar 

  5. Golfier, F., Zarcone, C., Bazin, B., Lenormand, R., Lasseux, D., Quintard, M.: On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium. J. Fluid Mech. 457, 213–254 (2002)

    Article  MATH  Google Scholar 

  6. Liu, M., Zhang, S., Mou, J., Zhou, F.: Wormhole propagation behavior under reservoir condition in carbonate acidizing. Transp. Porous Media 96(1), 203–220 (2013)

    Article  Google Scholar 

  7. Zhao, C., Hobbs, B., Hornby, P., Ord, A., Peng, S., Liu, L.: Theoretical and numerical analyses of chemical-dissolution front instability in fluid-saturated porous rocks. Int. J. Numer. Anal. Methods Geomech. 32(9), 1107–1130 (2008)

    Article  MATH  Google Scholar 

  8. Girault, V., Wheeler, M.: Numerical discretization of a Darcy-Forchheimer model. Numer. Math. 110(2), 161–198 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Park, E.-J.: Mixed finite element methods for generalized Forchheimer flow in porous media. Numer. Methods Partial Differential Equations 21(2), 213–228 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pan, H., Rui, H.: Mixed element method for two-dimensional Darcy-Forchheimer model. J. Sci. Comput. 52(3), 563–587 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pan, H., Rui, H.: A mixed element method for Darcy-Forchheimer incompressible miscible displacement problem. Comput. Methods Appl. Mech. Eng. 264, 1–11 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for 2-nd order elliptic problems. Springer, Berlin (1977)

    Book  Google Scholar 

  13. Arbogast, T., Wheeler, M.F., Yotov, I.: Mixed finite elements for elliptic problems with tensor coefficients as cell-centered finite differences. SIAM J. Numer. Anal. 34(2), 828–852 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Rui, H., Pan, H.: A block-centered finite difference method for the Darcy-Forchheimer model. SIAM J. Numer. Anal. 50(5), 2612–2631 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Li, X., Rui, H.: Characteristic block-centred finite difference methods for nonlinear convection-dominated diffusion equation [J]. Int. J. Comput. Math. 94(2), 386–404 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, X., Rui, H.: A two-grid block-centered finite difference method for nonlinear non-Fickian flow model. Appl. Math. Comput. 281, 300–313 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Rui, H., Pan, H.: Block-centered finite difference methods for parabolic equation with time-dependent coefficient. Jpn. J. Ind. Appl. Math. 30(3), 681–699 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, W., Cui, J., Xin, J.: A block-centered finite difference method for an unsteady asymptotic coupled model in fractured media aquifer system. J. Comput. Appl. Math. 337, 319–340 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, X., Rui, H.: Block-centered finite difference method for simulating compressible wormhole propagation. J. Sci. Comput. 74(2), 1115–1145 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, X., Rui, H.: Characteristic block-centered finite difference method for simulating incompressible wormhole propagation. Computers & Mathematics with Applications 73(10), 2171–2190 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mauran, S., Rigaud, L., Coudevylle, O.: Application of the Carman-Kozeny correlation to a high-porosity and anisotropic consolidated medium: the compressed expanded natural graphite. Transp. Porous Media 43(2), 355–376 (2001)

    Article  Google Scholar 

  22. Nédélec, J.-C.: Mixed finite elements in \(\mathbb {R}\)3. Numer. Math. 35(3), 315–341 (1980)

    Article  MathSciNet  Google Scholar 

  23. Dawson, C.N., Wheeler, M.F., Woodward, C.S.: A two-grid finite difference scheme for nonlinear parabolic equations. SIAM J. Numer. Anal. 35(2), 435–452 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Durán, R.: Superconvergence for rectangular mixed finite elements. Numer. Math. 58(1), 287–298 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  25. Diersch, H.-J.G.: FEFLOW: finite element modeling of flow, mass and heat transport in porous and fractured media. Springer Science & Business Media, Berlin (2013)

    Google Scholar 

  26. Berinde, V.: Iterative approximation of fixed points, vol. 1912. Springer, Berlin (2007)

    MATH  Google Scholar 

  27. Rui, H., Zhao, D., Pan, H.: A block-centered finite difference method for Darcy-Forchheimer model with variable Forchheimer number. Numer. Methods Partial Differential Equations 31(5), 1603–1622 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this paper. This work is supported by the National Natural Science Foundation of China Grant No. 11671233. The author X. Li thanks for the financial support from China Scholarship Council.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, Shandong, 250100, China

    Xiaoli Li & Hongxing Rui

  2. Beijing Institute for Scientific and Engineering Computing (BISEC), Beijing University of Technology, Beijing, 100124, China

    Shuangshuang Chen

Authors
  1. Xiaoli Li
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  2. Hongxing Rui
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  3. Shuangshuang Chen
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Correspondence to Hongxing Rui.

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Li, X., Rui, H. & Chen, S. A fully conservative block-centered finite difference method for simulating Darcy-Forchheimer compressible wormhole propagation. Numer Algor 82, 451–478 (2019). https://doi.org/10.1007/s11075-018-0609-9

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  • DOI: https://doi.org/10.1007/s11075-018-0609-9

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