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Analysis of generalized Forchheimer flows of compressible fluids in porous media

J. Math. Phys. 50, 103102 (2009); doi:10.1063/1.3204977

Published 16 October 2009

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Eugenio Aulisa, Lidia Bloshanskaya, Luan Hoang, and Akif Ibragimov
Department of Mathematics and Statistics, Texas Tech University, P.O. Box 41042, Lubbock, Texas 79409-1042, USA
This work is focused on the analysis of nonlinear flows of slightly compressible fluids in porous media not adequately described by Darcy's law. We study a class of generalized nonlinear momentum equations which covers all three well-known Forchheimer equations, the so-called two-term, power, and three-term laws. The generalized Forchheimer equation is inverted to a nonlinear Darcy equation with implicit permeability tensor depending on the pressure gradient. This results in a degenerate parabolic equation for the pressure. Two classes of boundary conditions are considered, given pressure and given total flux. In both cases they are allowed to be unbounded in time. The uniqueness, Lyapunov and asymptotic stabilities, and other long-time dynamical features of the corresponding initial boundary value problems are analyzed. The results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters. Some numerical simulations are also included. ©2009 American Institute of Physics
History: Received 29 May 2009; accepted 23 July 2009; published 16 October 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/103102/1
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0022-2488 (print)   1089-7658 (online)
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REFERENCES (42)
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  1. Abodun, M. A., “Mathematical Model for Darcy-Forchheimer Flow with application to well performance analysis,” Master thesis, Texas Tech University, 2007.
  2. Ai, L. and Vafai, K., “A coupling model for macromolecule transport in a stenosed arterial wall,” Int. J. Heat Mass Transfer 49, 1568 (2006).
  3. Aulisia, E., Ibragimov, A. I., Valkó, P. P., and Walton, J. R., “A new method for evaluating the productivity index for nonlinear flows,” SPEJ, in press (2009).
  4. Aulisa, E., Ibragimov, A. I., Valkó, P. P., and Walton, J. R., Proceedings of COMSOL Users Conference 2006, Boston, 2006 (unpublished).
  5. Aulisa, E., Cakmak, A., Ibragimov, A., and Solynin, A., “Variational principle and steady state invariants for non-linear hydrodynamic interactions in porous media,” Dyn. Contin. Discrete Impulsive Syst.: Ser. A Suppl, Adv. Dyn. Syst. 14(S2), 148 (2007).
  6. Aulisa, E., Ibragimov, A., and Toda, M., “Geometric framework for modeling nonlinear flows in porous media and its applications in engineering,” Nonlinear Anal.: Real World Appl., in press (2009).
  7. Aulisa, E., Ibragimov, A. I., Valkó, P. P., and Walton, J. R., “Mathematical framework of the well productivity index for fast Forchheimer (non-Darcy) flow in porous media,” Math. Models Meth. Appl. Sci. 19(8), 1241 (2009).
  8. Balhoff, M. T. and Wheeler, M. F., “A predictive pore-scale model for non-Darcy flow in anisotropic media,” SPE, published online (2007).
  9. Balhoff, M. T., Mikelic, A., andWheeler, M. F., “Polynomial filtration laws for low Reynolds number flows through porous media,” Transp. Porous Media, in press (2009).
  10. Bear, J., Dynamics of Fluids in Porous Media (Dover, New York, 1972).
  11. Blick, E. F., Enga, P. N., and Lin, P. C., “Theoretical stability analysis of flowing oil wells and gas-lift wells,” SPE Prod. Eng. 3, 508 (1988).
  12. Chadam, J. and Qin, Y., “Spatial decay estimates for flow in a porous medium,” SIAM J. Math. Anal. 28, 808 (1997).
  13. Dake, L. P., Fundamental in Reservoir Engineering (Elsevier, Amsterdam, 1978).
  14. DiBenedetto, E., Degenerate Parabolic Equations (Springer, New York, 1993).
  15. Douglas, J., Jr., Paes-Leme, P. J., and Giorgi, T., Generalized Forchheimer Flow in Porous Media (Masson, Paris, 1993), Vol. 29, pp. 99–111.
  16. Evans, L. C., Partial Differential Equations (American Mathematical Society, Providence, 1998).
  17. Ewing, E., Lazarov, R., Lyons, S., and Papavassiliou, D., “Numerical well model for non Darcy flow,” Comput. Geosci. 3, 185 (1999).
  18. Franchi, F. and Straughan, B., “Continuous dependence and decay for the Forchheimer equations,” Proc. R. Soc. London, Ser. A 459, 3195 (2003).
  19. Forchheimer, P., “Wasserbewegung durch Boden Zeit,” Z. Ver. Dtsch. Ing. 45, 1782 (1901).
  20. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, 2nd ed. (Springer-Verlag, Berlin, 1983).
  21. Ibragimov, A. I., Khalmanova, D., Valkó, P. P., and Walton, J. R., “On a mathematical model of the productivity index of a well from reservoir engineering,” SIAM J. Appl. Math. 65, 1952 (2005).
  22. Li, D. and Engler, T. W., “Literature review on correlations of the non-Darcy coefficient,” SPEJ, published online (2001).
  23. Mazya, V. G., Sobolev Spaces (Springer-Verlag, Berlin, 1985) (Translated from the Russian by T. O. Shaposhnikova, Springer Series in Soviet Mathematics).
  24. Miskimins, J. L., Lopez-Hernandez, H. D., and Barree, R. D., “Non-Darcy flow in hydraulic fractures: Does it really matter?,” SPEJ, published online (2005).
  25. Muskat, M., The Flow of Homogeneous Fluids Through Porous Media (McGraw-Hill, New York, 1937).
  26. Park, E.-J., “Mixed finite element methods for generalized Forchheimer flow in porous media,” Numer. Methods Partial Differ. Equ. 21(2), 213 (2005).
  27. Payne, L. E. and Straughan, B., “Convergence and continuous dependence for the Brinkman-Forchheimer equations,” Stud. Appl. Math. 102, 419 (1999).
  28. Payne, L. E., Song, J. C., and Straughan, B., “Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity,” Proc. R. Soc. London, Ser. A 455, 2173 (1999).
  29. Payne, L. E. and Song, J. C., “Spatial decay estimates for the Brinkman and Darcy flows in a semi-infinite cylinder,” Continuum Mech. Thermodyn. 9, 175 (1997).
  30. Payne, L. E. and Straughan, B., “Stability in the initial-time geometry problem for the Brinkman and Darcy equations of flow in porous media,” J. Math. Pures Appl. 75, 225 (1996).
  31. Polubarinova-Kochina, P., Theory of Ground Water Movement (Princeton University Press, Princeton, 1962) (translated from the Russian by J. M. Roger De Wiest).
  32. Ruth, D. and Ma, H., “On the derivation of the Forchheimer equation by means of the averaging theorem,” Transp. Porous Media 7, 255 (1992).
  33. Quarteroni, A., Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 (unpublished).
  34. Raghavan, R., Well Test Analysis (Prentice Hall, New York, 1993).
  35. Rajagopal, K. R., “On a hierarchy of approximate models for flows of incompressible fluids through porous solids,” Math. Models Meth. Appl. Sci. 17, 215 (2007).
  36. Rajagopal, K. R. and Tao, L., Mechanics of Mixtures (World Scientific, Singapore, 1995).
  37. Sanchez-Palencia, E., Non-Homogeneous Media and Vibration Theory, Lecture Notes in Physics (Springer-Verlag, Berlin, 1980).
  38. Straughan, B., Stability and Wave Motion in Porous Media, Applied Mathematical Sciences Vol. 165 (Springer, New York, 2008).
  39. Tavera, C. A. P., Kazemi, H., and Ozkan, E., “Combine effect of non-Darcy flow and formation damage on gas well performance of dual-porosity and dual permeability reservoirs,” SPEJ 9(9), 543–552 (2006).
  40. Wang, B. and Lin, S., “Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation,” Math. Methods Appl. Sci. 31, 1479 (2008).
  41. Wheeler, M. F. and Peszynska, M., “Computational engineering and science methodologies for modeling and simulation of subsurface applications,” Adv. Water Resour. 25, 1147 (2002).
  42. Whitaker, S., “The Forchheimer equation: A theoretical development,” Transp. Porous Media 25, 27 (1996).

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