Advertisement

Transport in Porous Media

, Volume 32, Issue 1, pp 75–95 | Cite as

A Least-Squares Mixed Scheme for the Simulation of Two-Phase Flow in Porous Media on Unstructured Grids

  • Deepankar Biswas
  • Graham F. Carey
Article

Abstract

A least-squares mixed formulation is developed for simulation of two-phase flow in porous media. Such problems arise in petroleum applications and ground-water flow. An adaptive strategy based on the element residual as an error indicator is developed in conjunction with unstructured remeshing and tested for the two-phase flow of oil and water. An element-by-element conjugate-gradient scheme (EBE-CG) is compared to a band solution algorithm.

least squares finite elements adaptive mesh conjugate-gradient scheme 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arbogast, T. and Wheeler, M. F.: 1995, A characteristic-mixed finite element method for advection dominated problems, SIAM J. Num. Anal. 31(4), 982–999.Google Scholar
  2. Aziz, K. and Settari, A.: 1979, Petroleum Reservoir Simulation, Applied Science, London, p. 476.Google Scholar
  3. Barragy, E. and Carey, G. F.: 1988, A parallel element-by-element solution scheme, Int. J. Num. Meth. Eng. 26, 2367–2382.Google Scholar
  4. Biswas, D.: 1997, Least-squares finite element method for hydrocarbon transport in porous media, Ph.D. dissertation, The Univ. of Texas, Austin.Google Scholar
  5. Biswas, D. and Carey, G. F.: 1997, An adaptive least-squares finite element method to predict areal compositional variations in large hydrocarbon reservoirs, SPEJ (in press).Google Scholar
  6. Carey, G. F., Barragy, E., McLay, R. and Sharma, M.: 1987, Element-by-element vector and parallel schemes, Specialty Conf. on Computer Science and Engineering, Dallas, TX, April 13-14.Google Scholar
  7. Carey, G. F.: 1997, Computational Grids: Generation, Adaptation and Solution Strategies, Taylor and Francis, Washington D.C.Google Scholar
  8. Carey, G. F. and Plover, T.: 1983, Variable upwinding and adaptive mesh refinement in convection-diffusion, Int. J. Num. Meth. Eng. 19, 341–353.Google Scholar
  9. Cavendish, J. C., Price, H. S. and Varga, R. S.: 1969, Galerkin methods for the numerical solution of boundary value problems, J. Soc. Pet. Eng. 9, 204–220.Google Scholar
  10. Chiang, C., Bedient, P. and Wheeler, M. F.: 1989, A modified method of characteristics technique and mixed finite element method for simulation of ground-water solute transport, Water Resour. Res. 25, 1541–1549.Google Scholar
  11. Culham, W. E. and Varga, R. S.: 1971, Numerical methods for time-dependent, nonlinear boundary value problems, Soc. Pet. Eng. J. 374–388.Google Scholar
  12. Dalen, V.: 1976, Immiscible flow by finite elements, paper presented at the Intl. Conference on Finite Elements in Water Resources, Princeton U., Princeton, NJ, July 1976.Google Scholar
  13. Ewing, R. E. and Heinemann, R. F.: 1984, Mixed finite element approximation of phase velocities in compositional reservoir simulation, Comp. Meth. Appl. Mech. Eng. 47, 161–175.Google Scholar
  14. Huyakorn, P. S. and Pinder, G. F.: 1983, Computational Methods in Subsurface Flow, Academic Press, New York.Google Scholar
  15. Huyakorn, P. S. and Pinder, G. F.: 1978, A new finite element technique for the solution of two-phase flow through porous media, Adv. Water Resour. 1(5), 285–298.Google Scholar
  16. Javendel, I. and Witherspoon, P. A.: 1968, Application of the finite element method to transient flow in porous media, Soc. Pet. Eng. J. 241–252.Google Scholar
  17. Jiang, B. N. and Carey, G. F.: 1988, A stable least-squares finite element method for nonlinear hyperbolic problems, Int. J. Num. Meth. Fl. 8, 933–942.Google Scholar
  18. Lake, L. W., Pope, G. A., Carey, G. F. and Sepehrnoori, K.: 1984, Isothermal, multiphase, multicomponent fluid flow in permeable media, In Situ 8(1), 1–40.Google Scholar
  19. Langsrud, O.: 1970, Simulation of two-phase flow by finite element method, Paper SPE 5725 presented at the SPE-AIME Symp. on Numerical Simulation of reservoir Performance, Los Angeles, Feb, 19-20.Google Scholar
  20. Moissis, D., Miller, C. and Wheeler, M. F.: 1993, Simulation of miscible viscous fingering using modified method of characteristics: effects of gravity and heterogeneity, SPE Reservoir Adv. Tech. Series 1, 62–72.Google Scholar
  21. Pehlivanov, A. I. and Carey, G. F.: 1993, Convergence analysis of least-squares mixed finite elements, Computing 51, 11–123.Google Scholar
  22. Pehlivanov, A. I. and Carey, G. F.: 1994, Error estimates for least-squares mixed finite elements, Math. Modeling Num. Anal. 28(5), 499–516.Google Scholar
  23. Price, H. S., Cavendish, J. C. and Varga, R. S.: 1968, Numerical methods of higher-order accuracy for diffusion-convection equations, Soc. Pet. Eng. J. 293–303.Google Scholar
  24. Shen, Y. and Carey, G. F.: 1989, Convergence studies of least-squares finite elements for first-order systems, Comm. Appl. Num. Meth. 5, 427–434.Google Scholar
  25. Spivak, A., Price, H. S. and Settari, A.: 1997, Solution of the equations for multidimensional, two-phase, immiscible flow by variational methods, J. Soc. Pet. Eng. 26–41.Google Scholar
  26. West, W. J., Garvin, W. W. and Sheldon, J. W.: 1954, Solution of the equations of unsteady state two-phase flow in oil reservoirs, Trans. AIME 201, 217–229.Google Scholar
  27. Young, L. C.: 1984, A study of spatial approximation for simulating fluid displacements in petroleum reservoirs, Comp. Meth. Appl. Mech. Eng. 47, 3–46.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Deepankar Biswas
    • 1
  • Graham F. Carey
    • 1
  1. 1.University of Texas at AustinAustinU.S.A.

Personalised recommendations