Computational Geosciences

, Volume 6, Issue 1, pp 73–100 | Cite as

Mortar Upscaling for Multiphase Flow in Porous Media

  • Małgorzata Peszyńska
  • Mary F. Wheeler
  • Ivan Yotov


In mortar space upscaling methods, a reservoir is decomposed into a series of subdomains (blocks) in which independently constructed numerical grids and possibly different physical models and discretization techniques can be employed in each block. Physically meaningful matching conditions are imposed on block interfaces in a numerically stable and accurate way using mortar finite element spaces. Coarse mortar grids and fine subdomain grids provide two-scale approximations. In the resulting effective solution flow is computed in subdomains on the fine scale while fluxes are matched on the coarse scale. In addition the flexibility to vary adaptively the number of interface degrees of freedom leads to more accurate multiscale approximations. This methodology has been implemented in the Center for Subsurface Modeling's multiphysics multiblock simulator IPARS (Integrated Parallel Accurate reservoir Simulator). Computational experiments demonstrate that this approach is scalable in parallel and it can be applied to non-matching grids across the interface, multinumerics and multiphysics models, and mortar adaptivity. Moreover unlike most upscaling approaches the underlying systems can be treated fully implicitly.

fully implicit mixed finite element method mortar spaces multiblock multiphase flow multiphysics porous media upscaling 


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  1. [1]
    L. An, J. Glimm, D. Sharp and Q. Zhang, Scale up of flow in porous media, in: Mathematical Modeling of Flow through Porous Media, eds. A.P. Bourgeat, C. Carasso, S. Luckhaus and A. Mikelić (World Scientific, Singapore,1995) pp. 26–44.Google Scholar
  2. [2]
    T. Arbogast, Numerical subgrid upscaling of two-phase flow in porous media, Technical Report 99-30, TICAM, University of Texas at Austin (1999).Google Scholar
  3. [3]
    T. Arbogast, L. Cowsar, M.F. Wheeler and I. Yotov, Mixed finite element methods on non-matching multiblock grids, SIAM J. Numer. Anal. 37(4) (2000) 1295–1331.Google Scholar
  4. [4]
    T. Arbogast, C. N. Dawson, P.T. Keenan, M.F. Wheeler and I. Yotov, Enhanced cell-centered finite differences for elliptic equations on general geometry, SIAM J. Sci. Comput. 19(1998) 404–425.Google Scholar
  5. [5]
    T. Arbogast, S. Minkoff and P. Keenan, An operator-based approach to upscaling the pressure equation, in: Computational Methods in Water Resources XII, eds. V.N. Burganos et al. (Computational Mechanics Publications, Southampton, 1998) pp. 405–412.Google Scholar
  6. [6]
    T. Arbogast, M.F. Wheeler and I. Yotov, Mixed finite elements for elliptic problems with tensor coef-ficients as cell-centered finite differences, SIAM J. Numer. Anal. 34(1997) 828–852.Google Scholar
  7. [7]
    K. Aziz and A. Settari, Petroleum Reservoir Simulation (Applied Science, 1979).Google Scholar
  8. [8]
    R.P. Batycky, M.J. Blunt and M.R. Thiele, A 3d multi-phase streamline simulator with gravity and changing well conditions, in: 17th Internat.Energy Agency Coll.Project on Enhanced Oil Recovery, Sydney, Australia, 29 September–2 October 1996.Google Scholar
  9. [9]
    F. Ben Belgacem, The mortar finite element method with Lagrange multipliers, Numer. Math. 84(2) (1999) 173–197.Google Scholar
  10. [10]
    F. Ben Belgacem, The mixed mortar finite element method for the incompressible Stokes problem: Convergence analysis, SIAM J. Numer. Anal. 37(4) (2000) 1085–1100.Google Scholar
  11. [11]
    L. Bergamaschi, S. Mantica and G. Manzini, A mixed finite element-finite volume formulation of the blackoil model, SIAM J. Sci. Comput. 20(3) (1998) 970–997.Google Scholar
  12. [12]
    C. Bernardi, Y. Maday and A.T. Patera, A new nonconforming approach to domain decomposition: The mortar element method, in: Nonlinear Partial Differential Equations and Their Applications,eds. H. Brezis and J.L. Lions (Longman Scientific & Technical, UK, 1994).Google Scholar
  13. [13]
    A. Bourgeat, Homogenized behavior of two-phase flows in naturally fractured reservoirs with uniform fractures distribution, Comput. Methods Appl. Mech. Engrg. 47(1984) 205–216.Google Scholar
  14. [14]
    F. Brezzi, L.P. Franca, T.J.R. Hughes and A. Russo, b =_g, Comput. Methods Appl. Mech. Engrg. 145(3/4) (1997) 329–339.Google Scholar
  15. [15]
    M.A. Christie, M. Mansfield, P.R. King, J.W. Barker and I.D. Culverwell, A renormalization-based upscaling technique for WAG floods in heterogeneous reservoirs, in: Expanded Abstracts (Society of Petroleum Engineers, 1995) pp. 353–361.Google Scholar
  16. [16]
    K.H. Coats, L.K. Thomas and R.G. Pierson, Compositional and black oil reservoir simulator, in: 13th SPE Symposium on Reservoir Simulation, San Antonio, TX, 12–15 February 1995.Google Scholar
  17. [17]
    L.C. Cowsar, J. Mandel and M.F. Wheeler, Balancing domain decomposition for mixed finite elements, Math. Comp. 64(211)(1995) 989–1015.Google Scholar
  18. [18]
    C.N. Dawson, H. Klie, M.F. Wheeler and C. Woodward, A parallel, implicit, cell-centered method for two-phase flow with a preconditioned Newton–Krylov solver, Comput. Geosci. 1(1997) 215–249.Google Scholar
  19. [19]
    J. Douglas, R.E. Ewing and M.F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, R.A.I.R.O. Analyse Numerique 17(1983) 249–265.Google Scholar
  20. [20]
    L.J. Durlofsky, Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water Resources Res. 27(5) (1991) 699–708.Google Scholar
  21. [21]
    M.J. Economides and A.D. Hill, Petroleum Production Systems (Prentice-Hall, Englewood Cliffs, NJ, 1994).Google Scholar
  22. [22]
    H.C. Edwards, A parallel multilevel-preconditioned GMRES solver for multiphase flow models in the implicit parallel accurate reservoir simulator, Technical Report 98-04, TICAM, University of Texas at Austin (1998).Google Scholar
  23. [23]
    S.C. Eigenstat and H.F. Walker, Globally convergent inexact Newton method, SIAM J. Sci. Optim. 4 (1994) 393–422.Google Scholar
  24. [24]
    R. Glowinski and M.F. Wheeler, Domain decomposition and mixed finite element methods for elliptic problems, in: Domain Decomposition Methods for Partial Differential Equations (SIAM, Philadelphia, PA, 1988) pp. 144–172.Google Scholar
  25. [25]
    T.Y. Hou and X. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys134(1) (1997) 169–189.Google Scholar
  26. [26]
    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(1–4) (1995) 387–401.Google Scholar
  27. [27]
    C.T. Kelley, Iterative Methods for Linear and Nonlinear Equations (SIAM, Philadelphia, PA, 1995).Google Scholar
  28. [28]
    H. Klie, Krylov-secant methods for solving large scale systems of coupled nonlinear parabolic equations, Ph.D. thesis, Rice University, Houston, TX (1996).Google Scholar
  29. [29]
    S. Lacroix, Y. Vassilevski and M.F. Wheeler, Iterative solvers of the implicit parallel accurate reservoir simulator (IPARS), to appear in Numer. Linear Algebra Appl.Google Scholar
  30. [30]
    L.W. Lake, Enhanced Oil Recovery (Prentice-Hall, Englewood Cliffs, NJ, 1989).Google Scholar
  31. [31]
    Q. Lu, A parallel multi-block multi-physics approach for multi-phase flow in porous media, Ph.D. thesis, The University of Texas at Austin (2000).Google Scholar
  32. [32]
    Q. Lu, M. Peszynska and M.F. Wheeler, A parallel multi-block black-oil model in multi-model im-plementation, in: 2001 SPE Reservoir Simulation Symposium, Houston, TX, 2001, SPE 66359.Google Scholar
  33. [33]
    Q. Lu, M. Peszynska, M.F. Wheeler and I. Yotov, Multiphysics and multinumerics couplings for multiphase flow in porous media, in preparation.Google Scholar
  34. [34]
    C.C. Mattax and R.L. Dalton, Reservoir simulation, in: SPE Monograph Series, Vol. 13 (Richardson, Texas, 1990).Google Scholar
  35. [35]
    N. Moes, J.T. Oden and K. Vemaganti, A two-scale strategy and a posteriori error estimation for modeling heterogeneous structures, in: On New Advances in Adaptive Computational Methods in Mechanics (Elsevier, Amsterdam, 1998).Google Scholar
  36. [36]
    M. Parashar, J.A. Wheeler, J.C. Browne, G. Pope, K. Wang and P. Wang, A new generation EOS compositional reservoir simulator: Part II – framework and multiprocessing, in: 1997 SPE Reservoir Simulation Symposium, Houston, TX, 1997, SPE 37977.Google Scholar
  37. [37]
    M. Parashar and I. Yotov, An environment for parallel multi-block, multi-resolution reservoir simulations, in: ISCA 11th Internat.Conf.on Parallel and Distributed Computing System, September 1998, pp. 230–235.Google Scholar
  38. [38]
    D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation,1sted.(ElsevierScientfic, Amsterdam, 1977).Google Scholar
  39. [39]
    D.W. Peaceman, Interpretation of well-block pressure in numerical reservior simulation with non-square grid blocks and anisotropic permeability, Trans. AIME 275(1983) 10–22.Google Scholar
  40. [40]
    G. Pencheva and I. Yotov, Balancing domain decomposition for porous media flow in multiblock domains, in: Summer Research Conf.on Fluid Flow and Transport in Porous Media: Mathematical and Numerical Treatment (Amer. Math. Soc., Providence, RI, 2001) to appear.Google Scholar
  41. [41]
    M. Peszynska, Advanced techniques and algorithms for reservoir simulation, III: Multiphysics cou-pling for two phase flow in degenerate conditions, in: IMA Volumes on Resource Recovery (submitted August 2000).Google Scholar
  42. [42]
    M. Peszynska, E. Jenkins and M.F. Wheeler, Boundary conditions for fully implicit two-phase flow model, submitted.Google Scholar
  43. [43]
    M. Peszynska, Q. Lu and M.F. Wheeler, Coupling different numerical algorithms for two phase fluid flow, in: MAFELAP Proc.of Mathematics of Finite Elements and Applications, ed. J.R. Whiteman, Brunel University, Uxbridge, UK, 1999, pp. 205–214.Google Scholar
  44. [44]
    M. Peszynska, Q. Lu and M.F. Wheeler, Multiphysics coupling of codes, in: Computational Methods in Water Resources, eds. L.R. Bentley, J.F. Sykes, C.A. Brebbia, W.G. Gray and G.F. Pinder (A.A. Balkema, 2000) pp. 175–182.Google Scholar
  45. [45]
    D.K. Ponting, B.A. Foster, P.F. Naccache, M.O. Nicholas, R.K. Pollard, J. Rae, D. Banks and Walsh S.K., An efficient fully implicit simulator, in: European Offshore Petroleum Conference and Exihibition, 1980.Google Scholar
  46. [46]
    R.A. Raviart and J.M. Thomas, A mixed finite element method for 2nd order elliptic problems, in: Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, Vol. 606 (Springer, New York, 1977) pp. 292–315.Google Scholar
  47. [47]
    M. Snir, S. Otto, S. Huss-Lederman, D. Walker and J. Dongarra, MPI: The Complete Reference (MIT Press, Cambridge, MA, 1996).Google Scholar
  48. [48]
    J.A. Trangenstein and J.B. Bell, Mathematical structure of the black-oil model for petroleum reservoir simulation, SIAM J. Appl. Math. 49(3) (1989) 749–783.Google Scholar
  49. [49]
    P. Wang, I. Yotov, M. Wheeler, T. Arbogast, C. Dawson, M. Parashar and K. Sephernoori, A new generation EOS compositional reservoir simulator: Part I – formulation and discretization, in: 1997 SPE Reservoir Simulation Symposium, Houston, TX, 1997, SPE 37979.Google Scholar
  50. [50]
    M.F. Wheeler, Advanced techniques and algorithms for reservoir simulation, II: The multiblock approach in the integrated parallel accurate reservoir simulator (IPARS), in: IMA Volume on Resource Recovery (submitted August 2000).Google Scholar
  51. [51]
    M.F. Wheeler, T. Arbogast, S. Bryant, J. Eaton, Q. Lu, M. Peszynska and I. Yotov, A parallel multi-block/ multidomain approach for reservoir simulation, in: 1999 SPE Symposium on Reservoir Simulation, Houston, TX, 1999, SPE 51884.Google Scholar
  52. [52]
    M.F. Wheeler, M. Peszynska, X. Gai and O. El-Domeiri, Modeling subsurface flow on pc cluster, in: High Performance Computing, ed. A. Tentner (SCS, 2000) pp. 318–323.Google Scholar
  53. [53]
    M.F. Wheeler, J.A. Wheeler and M. Peszynska, A distributed computing portal for coupling multi-physics and multiple domains in porous media, in: Computational Methods in Water Resources, eds. L.R. Bentley, J.F. Sykes, C.A. Brebbia, W.G. Gray and G.F. Pinder (A.A. Balkema, 2000) pp. 167–174.Google Scholar
  54. [54]
    M.F. Wheeler and I. Yotov, Physical and computational domain decompositions for modeling sub-surface flows, in: Tenth Internat.Conf.on Domain Decomposition Methods, ed. J. Mandel et al., Contemporary Mathematics, Vol. 218(Amer. Math. Soc., Providence, RI, 1998) pp. 217–228.Google Scholar
  55. [55]
    I. Yotov, Mixed finite element methods for flow in porous media, Ph.D. thesis, Rice University, Houston,TX (1996), TR96-09, Department of Comp. Appl. Math., Rice University and TICAM Report 96-23, University of Texas at Austin.Google Scholar
  56. [56]
    I. Yotov, A mixed finite element discretization on non-matching multiblock grids for a degenerate parabolic equation arising in porous media flow, East–West J. Numer. MAth. 5(1997) 211–230.Google Scholar
  57. [57]
    I. Yotov, Interface solvers and preconditioners of domain decomposition type for multiphase flow in multiblock porous media, in: Advances in Computation: Theory and Practice,eds. P.Minev, Y.Lin and Y.S. Wong, Vol. 7(Nova Science, 2001) pp. 157–167.Google Scholar
  58. [58]
    T.I. Zohdi, J.T. Oden and G.J. Rodin, Hierarchical modeling of heterogeneous bodies, Comput. Methods Appl. Mech. Engrg. 138(1996) 273–298.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Małgorzata Peszyńska
    • 1
  • Mary F. Wheeler
    • 1
  • Ivan Yotov
    • 2
  1. 1.Texas Institute for Computational and Applied MathematicsUniversity of TexasAustinUSA
  2. 2.Department of MathematicsUniversity of PittsburgPittsburghUSA

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