Henning, Patrick; Ohlberger, Mario; Schweizer, Ben Adaptive heterogeneous multiscale methods for immiscible two-phase flow in porous media. (English) Zbl 1326.76060 Comput. Geosci. 19, No. 1, 99-114 (2015). Summary: 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 C. Cancès et al. [Math. Comput. 83, No. 285, 153–188 (2014; Zbl 1430.76369)]. An a posteriori error estimate for vertex-centered finite volume discretizations of immiscible incompressible two-phase flow. Cited in 7 Documents MSC: 76M10 Finite element methods applied to problems in fluid mechanics 76S05 Flows in porous media; filtration; seepage 76T99 Multiphase and multicomponent flows 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 65N08 Finite volume methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs Keywords:adaptivity; HMM; multiscale problem; two-phase flow; porous media Citations:Zbl 1430.76369 PDFBibTeX XMLCite \textit{P. Henning} et al., Comput. Geosci. 19, No. 1, 99--114 (2015; Zbl 1326.76060) Full Text: DOI arXiv References: [1] Aarnes, J.E., Efendiev, Y.: An adaptive multiscale method for simulation of fluid flow in heterogeneous porous media. Multiscale Model. 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