Computational Geosciences

, Volume 19, Issue 1, pp 99–114 | Cite as

Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media

  • Patrick Henning
  • Mario Ohlberger
  • Ben Schweizer


In this contribution, we present the first formulation of a heterogeneous multiscale method for an incompressible immiscible two-phase flow system with degenerate permeabilities. The method is in a general formulation, which includes oversampling. We do not specify the discretization of the derived macroscopic equation, but we give two examples of possible realizations, suggesting a finite element solver for the fine scale and a vertex-centered finite volume method for the effective coarse scale equations. Assuming periodicity, we show that the method is equivalent to a discretization of the homogenized equation. We provide an a posteriori estimate for the error between the homogenized solutions of the pressure and saturation equations and the corresponding HMM approximations. The error estimate is based on the results recently achieved as reported by Cancès et al. (Math. Comp. 83(285):153–188, 2014). An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow.


Adaptivity HMM Multiscale problem Two-phase flow Porous media 

Mathematics Subject Classification (2010)

76S05 35B27 65G99 65N08 65N30 


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Patrick Henning
    • 1
  • Mario Ohlberger
    • 1
  • Ben Schweizer
    • 2
  1. 1.Institut für Numerische und Angewandte MathematikWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

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