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A generalized multiscale finite element method for elastic wave propagation in fractured media

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Abstract

In this paper, we consider elastic wave propagation in fractured media applying a linear-slip model to represent the effects of fractures on the wavefield. Fractured media, typically, are highly heterogeneous due to multiple length scales. Direct numerical simulations for wave propagation in highly heterogeneous fractured media can be computationally expensive and require some type of model reduction. We develop a multiscale model reduction technique that captures the complex nature of the media (heterogeneities and fractures) in the coarse scale system. The proposed method is based on the generalized multiscale finite element method, where the multiscale basis functions are constructed to capture the fine-scale information of the heterogeneous, fractured media and effectively reduce the degrees of freedom. These multiscale basis functions are coupled via the interior penalty discontinuous Galerkin method, which provides a block-diagonal mass matrix. The latter is needed for fast computation in an explicit time discretization, which is used in our simulations. Numerical results are presented to show the performance of the presented multiscale method for fractured media. We consider several cases where fractured media contain fractures of multiple lengths. Our numerical results show that the proposed reduced-order models can provide accurate approximations for the fine-scale solution.

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Acknowledgments

The research of Eric Chung is supported by Hong Kong RGC General Research Fund (Project 400411). YE’s work is partially supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project.

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Authors and Affiliations

  1. Department of Mathematics, The Chinese University of Hong Kong (CUHK), Sha Tin, Hong Kong SAR

    Eric T. Chung

  2. Department of Mathematics, Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX, USA

    Yalchin Efendiev

  3. Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Yalchin Efendiev

  4. Department of Geology and Geophysics, Texas A&M University, College Station, TX, 77843, USA

    Richard L. Gibson Jr.

  5. Department of Computational Technologies, Institute of Mathematics and Informatics, North-Eastern Federal University, Yakutsk, Republic of Sakha (Yakutia), 677980, Russia

    Maria Vasilyeva

  6. Institute for Scientific Computation, Texas A&M University, College Station, TX, 77843-3368, USA

    Maria Vasilyeva

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  1. Eric T. Chung
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  2. Yalchin Efendiev
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  4. Maria Vasilyeva
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Correspondence to Yalchin Efendiev.

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Chung, E.T., Efendiev, Y., Gibson, R.L. et al. A generalized multiscale finite element method for elastic wave propagation in fractured media. Int J Geomath 7, 163–182 (2016). https://doi.org/10.1007/s13137-016-0081-4

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