Finite Volume Approximations for Incompressible Miscible Displacement Problems in Porous Media with Modified Method of Characteristics

  • Sarvesh Kumar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8236)


The incompressible miscible displacement problem in porous media is modeled by a coupled system of two nonlinear partial differential equations, the pressure-velocity equation and the concentration equation. The pressure-velocity is elliptic type and the concentration equations is convection dominated diffusion type. It is known that miscible displacement problems follow the natural law of conservation and finite volume methods are conservative. Hence, in this paper, we present a mixed finite volume element method (FVEM) for the approximation of the pressure-velocity equation. Since concentration equation is convection dominated diffusion type and most of the numerical methods suffer from the grid orientation effect and modified method of characteristics(MMOC) minimizes the grid orientation effect. Therefore, for the approximation of the concentration equation we apply a standard FVEM combined MMOC. A priori error estimates are derived for velocity, pressure and concentration. Numerical results are presented to substantiate the validity of the theoretical results.


modified method of characteristics mixed methods finite volume element methods miscible displacement problems error estimates numerical experiments 


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  1. 1.
    Kumar, S., Nataraj, N., Pani, A.K.: Finite volume element method for second order hyperbolic equations. Int. J. Numerical Analysis and Modeling 5, 132–151 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Kumar, S.: A mixed and discontinuous Galerkin finite volume element method for incompressible miscible displacement problems in porous media. Numer. Methods Partial Differential Equations 28, 1354–1381 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Douglas Jr., J., Ewing, R.E., Wheeler, M.F.: A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media. RAIRO Anal. Numér. 17, 249–265 (1983)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chadam, J., Peirce, A., Ortoleva, P.: Stability of reactive flows in porous media: coupled porosity and viscosity changes. SIAM J. Appl. Math. 51, 684–692 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ewing, R.E., Russell, T.F., Wheeler, M.F.: Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics. Comput. Meth. Appl. Mech. Engrg. 47, 73–92 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Chou, S.H., Kwak, D.Y., Vassilevski, P.: Mixed covolume methods for elliptic problems on triangular grids. SIAM J. Numer. Anal. 35, 1850–1861 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43, 195–219 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Duran, R.G.: On the approximation of miscible displacement in porous media by a method characteristics combined with a mixed method. SIAM J. Numer. Anal. 14, 989–1001 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Wang, H., Liang, D., Ewing, R.E., Lyons, S.L., Qin, G.: An approximation to miscible fluid flows in porous media with point sources and sinks by an Eulerian-Lagrangian localized adjoint method and mixed finite element methods. SIAM J. Sci. Comput. 22, 561–581 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, Z., Ewing, R.E.: Mathematical analysis for reservoir models. SIAM J. Numer. Anal. 30, 431–453 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Feng, X.: On existence and uniqueness results for a coupled system modeling miscible displacement in porous media. J. Math. Anal. Appl. 194, 883–910 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Russell, T.F.: An incompletely iterated characteristic finite element method for a miscible displacement problem. Ph.DThesis, University of Chicago, Illinois (1980)Google Scholar
  13. 13.
    Kumar, S., Yadav, S.: Modified Method of Characteristics Combined with Finite Volume Element Methods for Incompressible Miscible Displacement Problems in Porous Media (To be Communicated)Google Scholar

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sarvesh Kumar
    • 1
  1. 1.Department of MathematicsIndian Institute of Space Science and TechnologyThiruvananthapuramIndia

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