Multiscale Computation of Fluid Flow in Heterogeneous Media

  • Thomas Y. Hou

Abstract

There are many interesting physical problems that have multiscale solutions. These problems range from composite materials to wave propagation in random media, flow and transport through heterogeneous porous media, and turbulent flow. Computing these multiple scale solutions accurately presents a major challenge due to the wide range of scales in the solution. It is very expensive to resolve all the small scale features on a fine grid by direct num-erical simulations. A natural question is if it is possible to develop a multiscale computational method that captures the effect of small scales on the large scales using a coarse grid, but does not require resolving all the small scale features. Such multiscale method can offer significant computational savings.

Keywords

Permeability Porosity Convection Propa Shale 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    T.Y. Hou and X. Wu, “A multiscale finite element method for elliptic problems in composite materials and porous media,” J. Comput. Phys., 134, 169–189, 1997.MATHCrossRefMathSciNetADSGoogle Scholar
  2. [2]
    T.Y. Hou, X. Wu, and Z. Cai, “Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,” Math. Comput., 68, 913–943, 1999.MATHCrossRefMathSciNetADSGoogle Scholar
  3. [3]
    Y.R. Efendiev, T.Y. Hou, and X. Wu, “Convergence of a nonconforming multiscale finite element method,” SIAM J. Numer. Anal., 37, 888–910, 2000b.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    Z. Chen and T. Hou, “A mixed finite element method for elliptic problems with rapidly oscillating coefficients,” Math. Comput., 72, 541–576, 2002.CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    I. Babuska, G. Caloz, and E. Osborn, “Special finite element methods for a class of second order elliptic problems with rough coefficients,” SIAM J. Numer. Anal., 31, 945–981, 1994.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    F. Brezzi and A. Russo, “Choosing bubbles for advection-diffusion problems,” Math. Models Methods Appl. Sci., 4, 571–587, 1994.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    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, 387–401, 1995.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, 1st edn., North-Holland, Amsterdam, 1978.MATHGoogle Scholar
  9. [9]
    T.Y. Hou, “Multiscale modeling and computation of incompressible flow,” In: J.M. Hill and R. Moore (eds.), Applied Mathematics Entering the 21st Century, Invited Talks from the ICIAM 2003 Congress, SIAM, Philadelphia, pp. 177–209, 2004.Google Scholar
  10. [10]
    P. Park and T.Y. Hou, “Multiscale numerical methods for singularly-perturbed convection-diffusion equations,” Int. J. Comput. Meth., 1(1), 17–65, 2004.MATHCrossRefGoogle Scholar
  11. [11]
    L.J. Durlofsky, “Numerical calculation of equivalent grid block permeability tensors for Heterogeneous porous media,” Water Resour. Res., 27, 699–708, 1991.CrossRefADSGoogle Scholar
  12. [12]
    P. Jenny, S.H. Lee, and H. Tchelepi, “Multi-scale finite volume method for elliptic problems in subsurface flow simulation,” J. Comput. Phys., 187, 47–67, 2003.MATHCrossRefADSGoogle Scholar
  13. [13]
    P. Jenny, S.H. Lee, and H. Tchelepi, “Adaptive multi-scale finite volume method for multi-phase flow and transport in porous media,” Multiscale Model. Simul., 3, 50–64, 2005.CrossRefGoogle Scholar
  14. [14]
    L. Tartar, “Nonlocal effects induced by homogenization,” In: F. Culumbini (ed.), PDE and Calculus of Variations, Birkhäuser, Boston, pp. 925–938, 1989.Google Scholar
  15. [15]
    Y.R. Efendiev, L.J. Durlofsky, and S.H. Lee, “Modeling of subgrid effects in coarsescale simulations of transport in heterogeneous porous media,” Water Resour. Res., 36, 2031–2041, 2000a.CrossRefADSGoogle Scholar
  16. [16]
    J. Smogorinsky, “General circulation experiments with the primitive equations,” Mon. Weather Rev., 91, 99–164, 1963.CrossRefADSGoogle Scholar
  17. [17]
    M. Germano, U. Pimomelli, P. Moin, and W. Cabot, “A dynamic subgrid-scale eddy viscosity model,” Phys. Fluids A, 3, 1760–1765, 1991.MATHCrossRefADSGoogle Scholar
  18. [18]
    D.W. McLaughlin, G.C. Papanicolaou, and O. Pironneau, “Convection of microstructure and related problems,” SIAM J. Appl. Math., 45, 780–797, 1985.MATHCrossRefMathSciNetADSGoogle Scholar
  19. [19]
    T.Y. Hou, D. Yang, and K. Wang, “Homogenization of incompressible Euler equations,” J. Comput. Math., 22(2), 220–229, 2004b.MATHMathSciNetGoogle Scholar
  20. [20]
    T.Y. Hou, D. Yang, and H. Ran, “Multiscale analysis in the Lagrangian formulation for the 2-D incompressible Euler equation,” Discr. Continuous Dynam. Sys., 12, to appear, 2005.Google Scholar
  21. [21]
    T.Y. Hou, A. Westhead, and D. Yang, “Multiscale analysis and computation for two-phase flows in strongly heterogeneous porous media,” (in preparation), 2005a.Google Scholar
  22. [22]
    M. Dorobantu and B. Engquist, “Wavelet-based numerical homogenization,” SIAM J. Numer. Anal., 35, 540–559, 1998.MATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    T. Wallstrom, S. Hou, M.A. Christie, L.J. Durlofsky, and D. Sharp, “Accurate scale up of two-phase flow using renormalization and nonuniform coarsening,” Comput. Geosci., 3, 69–87, 1999.MATHCrossRefGoogle Scholar
  24. [24]
    T. Arbogast, “Numerical subgrid upscaling of two-phase flow in porous media,” In: Z. Chen (ed.), Numerical Treatment of Multiphase Flows in Porous Media, Springer, Berlin, pp. 35–49, 2000.CrossRefGoogle Scholar
  25. [25]
    A. Matache, I. Babuska, and C. Schwab, “Generalized p-FEM in homogenization,” Numer. Math., 86, 319–375, 2000.MATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    L.Q. Cao, J.Z. Cui, and D.C. Zhu, “Multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equations with rapidly oscillating coefficients over general convex domains,” SIAM J. Numer. Anal., 40, 543–577, 2002.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    W.E. and B. Engquist, “The heterogeneous multi-scale methods,” Comm. Math. Sci, 1, 87–133, 2003.MATHMathSciNetGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  • Thomas Y. Hou
    • 1
  1. 1.California Institute of TechnologyPasadenaUSA

Personalised recommendations