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

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Abstract

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.

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Authors and Affiliations

  1. Institut für Numerische und Angewandte Mathematik, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149, Münster, Germany

    Patrick Henning & Mario Ohlberger

  2. Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, D-44227, Dortmund, Germany

    Ben Schweizer

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  1. Patrick Henning
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  2. Mario Ohlberger
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  3. Ben Schweizer
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Correspondence to Patrick Henning.

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This work was supported by the Deutsche Forschungsgemeinschaft under the contract numbers SCHW 639/3-2 and OH 98/4-2.

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Henning, P., Ohlberger, M. & Schweizer, B. Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media. Comput Geosci 19, 99–114 (2015). https://doi.org/10.1007/s10596-014-9455-6

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