Transport in Porous Media

, Volume 71, Issue 3, pp 345–377 | Cite as

A variable relaxation scheme for multiphase, multicomponent flow

Article

Abstract

We propose a variable relaxation scheme for the simulation of 1D, two-phase, multicomponent flow in porous media. For these strongly nonlinear systems, traditional high order upwind schemes are impractical: Riemann solutions are not directly available when the phase behavior is complex, and the systems are weakly hyperbolic at isolated points. Relaxation schemes avoid the dependency on the eigenstructure and nonlinear Riemann solvers by approximating the original system with a strongly hyperbolic linear system. We exploit the known information about the eigenvalues to construct first order and second order variable relaxation schemes with much reduced numerical diffusion as compared to the standard relaxation formulations. The proposed second order variable relaxation scheme is competitive in accuracy and efficiency with a third order component-wise ENO reconstruction, and performs at least as well as second order component-wise TVD schemes.

Keywords

Multicomponent multiphase flows Miscible displacement Weak hyperbolicity Relaxation schemes Gas injection 

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References

  1. Anderson W.K., Thomas J.L., Van Leer B. (1986). Comparison of finite volume flux volume flux vector splittings for the Euler equations. AIAA J. 24: 1453–1460 CrossRefGoogle Scholar
  2. Aregba-Driollet D., Natalini R. (1996). Convergence of relaxation schemes for conservation laws. Appl. Anal. 61: 163–193 CrossRefGoogle Scholar
  3. Banda M.K. (2005). Variants of relaxed schemes and two-dimensional gas dynamics. J. Comp. Appl. Math. 175: 41–62 CrossRefGoogle Scholar
  4. Batycky R.P., Blunt M., Thiele M.R. (1997). A 3D field-scale streamline based reservoir simulator. Soc. Petrol. Eng. Res. Eng. 12: 246–254 Google Scholar
  5. Chalabi A. (1999). Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms. Math. Comput. 68: 955–970 CrossRefGoogle Scholar
  6. Chapman, S., Cowling, T.G.: The Mathematical Theory of Nonuniform Gases. Cambridge Univ. Press (1970)Google Scholar
  7. Delis A.I., Katsounis Th. (2005). Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods. Appl. Math. Mod. 29(8): 754–783 CrossRefGoogle Scholar
  8. Dindoruk, B.: Analytical theory of multiphase, multicomponent displacement in porous media. PhD thesis. Stanford University (1982)Google Scholar
  9. Gerritsen M., Durlofsky L.J. (2005). Modeling fluid flow in oil reservoirs. Annu. Rev. Fluid Mech. 37: 211–238 CrossRefGoogle Scholar
  10. Harten A., Engquist B., Osher S., Chakravarthy S. (1987). Uniformly high order accurate essentially non-oscillatory schemes III. J. Comp. Phys. 71: 231–303 CrossRefGoogle Scholar
  11. Hundsdorfer, W., Verwer, J.G.: Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Comp. Math (2003)Google Scholar
  12. Jessen K., Wang Y., Ermakov P., Zhu J., Orr F.M. Jr. (2001). Fast, approximate solutions for 1D multicomponent gas injection problems. Soc. Petrol. Eng. J. 6(4): 442–451 Google Scholar
  13. Jessen, K., Orr, F.M. Jr.: Compositional streamline simulation. Presented at the SPE Annu. Tech. Conf. Ex., San Antonio, Texas (2002)Google Scholar
  14. Jessen K., Stenby E.H., Orr F.M. Jr. (2004). Interplay of phase behavior and numerical dispersion in finite difference compositional simulation. Soc. Pet. Eng. J. 9(2): 193–201 Google Scholar
  15. Jiang G.S., Levy D., Lin C.T., Osher S., Tadmor E. (1998). High-resolution nonoscillatory central schemes with nonstaggered grids for hyperbolic conservation laws. SIAM J. Num. Anal. 35(6): 2147–2168 CrossRefGoogle Scholar
  16. Jin S. (1995). Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comp. Phys. 122: 51–67 CrossRefGoogle Scholar
  17. Jin S., Xin Z.P. (1995). The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48: 235–277 CrossRefGoogle Scholar
  18. Jin S., Xin Z.P. (1998). Numerical passage from systems of conservation laws to Hamilton-Jacobi equation and a relaxation scheme. SIAM J. Num. Anal. 35: 2385–2404 CrossRefGoogle Scholar
  19. Juanes R., Patzek T.W. (2004). Relative permeabilities for strictly hyperbolic models of three-phase flow in porous media. Trans. Porous Med. 57: 125–152 CrossRefGoogle Scholar
  20. Kurganov A., Tadmor E. (2000). New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comp. Phys. 160: 241–282 CrossRefGoogle Scholar
  21. Kurganov A., Noelle S., Petrova G. (2001). Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equations. SIAM J. Sci. Comp. 23(3): 707–740 CrossRefGoogle Scholar
  22. Lake L.W. (1989). Enhanced Oil Recovery. Prentice-Hall, Englewood Cliffs, NJ Google Scholar
  23. LeVeque R.J., Pelanti M. (2001). A class of approximate Riemann solvers and their relation to relaxation schemes. J. Comp. Phys. 172: 572–591 CrossRefGoogle Scholar
  24. LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge texts Appl. Math (2002)Google Scholar
  25. Liu T.P. (1987). Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108: 153–175 CrossRefGoogle Scholar
  26. Lohrenz J., Bray B.G., Clark C.R. (1964). The viscosity of pure substances in dense liquid and gaseous phases. J Petrol. Technol. 1: 1171–1176 Google Scholar
  27. Mallison, B.T., Gerritsen, M.G., Jessen, K., Orr, F.M. Jr.: High order upwind schemes for twophase multicomponent flows. Presented at the Soc. Petrol. Eng. Res. Sim. Symp., Houston, Texas (2003)Google Scholar
  28. Mallison, B.T.: Streamline-based simulation of two-phase, multicomponent flow in porous media. PhD thesis. Stanford Univ (2004)Google Scholar
  29. Nessyahu H., Tadmor E. (1990). Non-oscillatory central differencing for hyperbolic conservation laws. J. Comp. Phys. 87(2): 408–463 CrossRefGoogle Scholar
  30. Orr F.M. Jr., Dindoruk B., Johns R.T. (1995). Theory of multicomponent gas/oil displacements. Ind. Eng. Chem. Res. 34: 2661–2669 CrossRefGoogle Scholar
  31. Orr, F.M. Jr.: Theory of Gas Injection Processes. Stanford Univ (2005)Google Scholar
  32. Qiu J., Shu C.W. (2002). On the construction, comparison and local characteristic decomposition for high order central WENO schemes. J. Comp. Phys. 183(1): 187–209 CrossRefGoogle Scholar
  33. Sweby P.K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 21: 995–1011 CrossRefGoogle Scholar
  34. Tadmor E., Tang T. (2001). Pointwise error estimates for relaxation approximations to conservation laws. SIAM J. Math. Anal. 32: 870–886 CrossRefGoogle Scholar
  35. van Leer B (1979). Towards the ultimate conservative difference schemes V: A second order sequel to Godunov’s method. J. Comp. Phys. 32: 101–136 CrossRefGoogle Scholar
  36. Younis, R., Gerritsen, M.: Multiscale process coupling by adaptive fractional stepping: An in-situ combustion model. Presented at the Soc. Petrol. Eng. Symp. on Improved Oil Recovery, Tulsa, Oklahoma (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  • Shalini B. Krishnamurthy
    • 1
  • Margot G. Gerritsen
    • 2
  1. 1.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

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