Computational Geosciences

, Volume 6, Issue 3–4, pp 453–481 | Cite as

Implementation of a Locally Conservative Numerical Subgrid Upscaling Scheme for Two-Phase Darcy Flow

  • Todd Arbogast


We present a locally mass conservative scheme for the approximation of two-phase flow in a porous medium that allows us to obtain detailed fine scale solutions on relatively coarse meshes. The permeability is assumed to be resolvable on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We define a two-scale mixed finite element space and resulting method, and describe in detail the solution algorithm. It involves a coarse scale operator coupled to a subgrid scale operator localized in space to each coarse grid element. An influence function (numerical Greens function) technique allows us to solve these subgrid scale problems independently of the coarse grid approximation. The coarse grid problem is modified to take into account the subgrid scale solution and solved as a large linear system of equations posed over a coarse grid. Finally, the coarse scale solution is corrected on the subgrid scale, providing a fine grid representation of the solution. Numerical examples are presented, which show that near-well behavior and even extremely heterogeneous permeability barriers and streaks are upscaled well by the technique.

heterogeneity numerical Greens functions porous media subgrid upscaling 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. Amaziane, A. Bourgeat and J. Koebbe, Numerical simulation and homogenization of two-phase flow in heterogeneous porous media, Transport Porous Media 9 (1991) 519-547.Google Scholar
  2. [2]
    T. Arbogast, The existence of weak solutions to single-porosity and simple dual-porosity models of two-phase incompressible flow, J. Nonlinear Anal. 19 (1992) 1009-1031.Google Scholar
  3. [3]
    T. Arbogast, Numerical subgrid upscaling of two-phase flow in porous media, in: Numerical Treat-ment of Multiphase Flows in Porous Media, eds. Z. Chen et al., Lecture Notes in Physics, Vol. 552 (Springer, Berlin, 2000) pp. 35-49.Google Scholar
  4. [4]
    T. Arbogast and S. Bryant, Efficient forward modeling for DNAPL site evaluation and remediation, in: Computational Methods in Water Resources, Vol. XIII, eds. Bentley et al. (Rotterdam, 2000) pp. 161-166.Google Scholar
  5. [5]
    T. Arbogast and S. Bryant, Numerical subgrid upscaling for waterflood simulations, SPE 66375, in: Proc. of the 16th SPE Symposium on Reservoir Simulation, Houston, TX, 11-14 February 2001.Google Scholar
  6. [6]
    T. Arbogast, S.E. Minkoff and P.T. Keenan, An operator-based approach to upscaling the pressure equation, in: Computational Methods in Water Resources, Vol. XII(1), eds. V.N. Burganos et al. (Southampton, UK, 1998) pp. 405-412.Google Scholar
  7. [7]
    T. Arbogast, M.F. Wheeler and N.-Y. Zhang, A nonlinear mixed finite element method for a degenerate parabolic equation arising in flow in porous media, SIAM J. Numer. Anal. 33 (1996) 1669-1687.Google Scholar
  8. [8]
    D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, post-processing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985) 7-32.Google Scholar
  9. [9]
    F. Brezzi, Interacting with the subgrid world, in: Proc. of the Dundee Conference, 1999.Google Scholar
  10. [10]
    F. Brezzi, J. Douglas, Jr., R. Duràn and M. Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987) 237-250.Google Scholar
  11. [11]
    F. Brezzi, J. Douglas, Jr. and L.D. Marini, Two families of mixed elements for second order elliptic problems, Numer. Math. 47 (1985) 217-235.Google Scholar
  12. [12]
    F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods (Springer, New York, 1991).Google Scholar
  13. [13]
    G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation (Elsevier Science, New York, 1986).Google Scholar
  14. [14]
    Z. Chen, Large-scale averaging analysis of single phase flow in fractured reservoirs, SIAM J. Appl. Math. 54 (1994) 641-659.Google Scholar
  15. [15]
    R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in: First Internat. Symposium on Domain Decomposition Methods for Partial Differential Equations, eds. R. Glowinski et al. (SIAM, Philadelphia, 1988) pp. 144-172.Google Scholar
  16. [16]
    U. Hornung, ed., Homogenization and Porous Media, Interdisciplinary Applied Mathematics Series (Springer, New York, 1997) to appear.Google Scholar
  17. [17]
    T.Y. Hou and X.H. Wu, A multiscale finite element method for elliptic problems in composite mate-rials and porous media, J. Comput. Phys. 134 (1997) 169-189.Google Scholar
  18. [18]
    T.J.R. Hughes, Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1995) 387-401.Google Scholar
  19. [19]
    T.J.R. Hughes, G.R. Feijóo, L. Mazzei and J.-B. Quincy, The variational multiscale method-a para-digm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998) 3-24.Google Scholar
  20. [20]
    L.W. Lake, Enhanced Oil Recovery (Prentice-Hall, Englewood Cliffs, NJ, 1989).Google Scholar
  21. [21]
    J.T. Oden and K.S. Vemaganti, Adaptive hierarchical modeling of heterogeneous structures, Phys. D: Nonlinear Phenomena 133 (1999) 404-415.Google Scholar
  22. [22]
    J.T. Oden and K.S. Vemaganti, Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms, J. Comput. Phys. 164 (2000) 22-47.Google Scholar
  23. [23]
    D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation (Elsevier, Amsterdam, 1977).Google Scholar
  24. [24]
    D.W. Peaceman, Interpretation of well-block pressures in numerical reservoir simulation, Soc. Petrol. Engrg. J. (1978) 183-194.Google Scholar
  25. [25]
    D.W. Peaceman, Interpretation of well-block pressures in numerical reservoir simulation with non-square grid blocks and anisotropic permeability, Soc. Petrol. Engrg. J. (1983) 531-543.Google Scholar
  26. [26]
    M. Peszynska, M.F. Wheeler and I. Yotov, Mortar upscaling for multiphase flow in porous media (2001) submitted.Google Scholar
  27. [27]
    M. Quintard and S. Whitaker, Two-phase flow in heterogeneous porous media: The method of large-scale averaging, Transport Porous Media 3 (1988) 357-413.Google Scholar
  28. [28]
    R.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in: Mathematical Aspects of Finite Element Methods, Lecture Notes in Mathematics, Vol. 606 (Springer, New York, 1977) pp. 292-315.Google Scholar
  29. [29]
    R.K. Romeu and B. Noetinger, Calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media, Water Resourc. Res. 31 (1995) 943-959.Google Scholar
  30. [30]
    J.M. Thomas, Sur l'analyse numerique des methodes d'elements finis hybrides et mixtes, Ph.D. thesis, Sciences Mathematiques, l'Université Pierre et Marie Curie (1977).Google Scholar
  31. [31]
    J. Xu, The method of subspace corrections, J. Comput. Appl. Math. 128 (2001) 335-362.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Todd Arbogast
    • 1
  1. 1.Department of Mathematics, C1200The University of Texas at AustinAustinUSA

Personalised recommendations