Computational Geosciences

, Volume 15, Issue 3, pp 399–419 | Cite as

Multidimensional upstream weighting for multiphase transport in porous media

  • Jeremy Edward Kozdon
  • Bradley T. Mallison
  • Margot G. Gerritsen
Original Paper


Truly multidimensional methods for hyperbolic equations use flow-based information to determine the computational stencil, as opposed to applying one-dimensional methods dimension by dimension. By doing this, the numerical errors are less correlated with the underlying computational grid. This can be important for reducing bias in flow problems that are inherently unstable at simulation scale, such as in certain porous media problems. In this work, a monotone, multi-D framework for multiphase flow and transport in porous media is developed. A local coupling of the fluxes is introduced through the use of interaction regions, resulting in a compact stencil. A relaxed volume formulation of the coupled hyperbolic–elliptic system is used that allows for nonzero residuals in the pressure equation to be handled robustly. This formulation ensures nonnegative masses and saturations (volume fractions) that sum to one (Acs et al., SPE J 25(4):543–553, 1985). Though the focus of the paper is on immiscible flow, an extension of the methods to a class of more general scalar hyperbolic equations is also presented. Several test problems demonstrate that the truly multi-D schemes reduce biasing due to the computational grid.


Multi-D transport Porous media Phase-based upwinding Two phase Interaction regions Monotonicity Hyperbolic equations Finite volume Finite difference Immiscible Partially miscible 


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Jeremy Edward Kozdon
    • 1
  • Bradley T. Mallison
    • 2
  • Margot G. Gerritsen
    • 3
  1. 1.Geophysics DepartmentStanford UniversityStanfordUSA
  2. 2.Reservoir and Production EngineeringChevron Energy Technology CompanySan RamonUSA
  3. 3.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

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