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A variable relaxation scheme for multiphase, multicomponent flow

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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.

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  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

  2. Aregba-Driollet D., Natalini R. (1996). Convergence of relaxation schemes for conservation laws. Appl. Anal. 61: 163–193

  3. Banda M.K. (2005). Variants of relaxed schemes and two-dimensional gas dynamics. J. Comp. Appl. Math. 175: 41–62

  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

  5. Chalabi A. (1999). Convergence of relaxation schemes for hyperbolic conservation laws with stiff source terms. Math. Comput. 68: 955–970

  6. Chapman, S., Cowling, T.G.: The Mathematical Theory of Nonuniform Gases. Cambridge Univ. Press (1970)

  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

  8. Dindoruk, B.: Analytical theory of multiphase, multicomponent displacement in porous media. PhD thesis. Stanford University (1982)

  9. Gerritsen M., Durlofsky L.J. (2005). Modeling fluid flow in oil reservoirs. Annu. Rev. Fluid Mech. 37: 211–238

  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

  11. Hundsdorfer, W., Verwer, J.G.: Numerical solution of time-dependent advection-diffusion-reaction equations. Springer Series in Comp. Math (2003)

  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

  13. Jessen, K., Orr, F.M. Jr.: Compositional streamline simulation. Presented at the SPE Annu. Tech. Conf. Ex., San Antonio, Texas (2002)

  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

  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

  16. Jin S. (1995). Runge-Kutta methods for hyperbolic conservation laws with stiff relaxation terms. J. Comp. Phys. 122: 51–67

  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

  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

  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

  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

  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

  22. Lake L.W. (1989). Enhanced Oil Recovery. Prentice-Hall, Englewood Cliffs, NJ

  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

  24. LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge texts Appl. Math (2002)

  25. Liu T.P. (1987). Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108: 153–175

  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

  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)

  28. Mallison, B.T.: Streamline-based simulation of two-phase, multicomponent flow in porous media. PhD thesis. Stanford Univ (2004)

  29. Nessyahu H., Tadmor E. (1990). Non-oscillatory central differencing for hyperbolic conservation laws. J. Comp. Phys. 87(2): 408–463

  30. Orr F.M. Jr., Dindoruk B., Johns R.T. (1995). Theory of multicomponent gas/oil displacements. Ind. Eng. Chem. Res. 34: 2661–2669

  31. Orr, F.M. Jr.: Theory of Gas Injection Processes. Stanford Univ (2005)

  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

  33. Sweby P.K. (1984). High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Num. Anal. 21: 995–1011

  34. Tadmor E., Tang T. (2001). Pointwise error estimates for relaxation approximations to conservation laws. SIAM J. Math. Anal. 32: 870–886

  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

  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)

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Correspondence to Shalini B. Krishnamurthy.

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Krishnamurthy, S.B., Gerritsen, M.G. A variable relaxation scheme for multiphase, multicomponent flow. Transp Porous Med 71, 345–377 (2008).

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  • Multicomponent
  • multiphase flows
  • Miscible displacement
  • Weak hyperbolicity
  • Relaxation schemes
  • Gas injection