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Hierarchical multiscale modeling for flows in fractured media using generalized multiscale finite element method

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Abstract

In this paper, we develop a multiscale finite element method for solving flows in fractured media. Our approach is based on generalized multiscale finite element method (GMsFEM), where we represent the fracture effects on a coarse grid via multiscale basis functions. These multiscale basis functions are constructed in the offline stage via local spectral problems following GMsFEM. To represent the fractures on the fine grid, we consider two approaches (1) discrete fracture model (DFM) (2) embedded fracture model (EFM) and their combination. In DFM, the fractures are resolved via the fine grid, while in EFM the fracture and the fine grid block interaction is represented as a source term. In the proposed multiscale method, additional multiscale basis functions are used to represent the long fractures, while short-size fractures are collectively represented by a single basis functions. The procedure is automatically done via local spectral problems. In this regard, our approach shares common concepts with several approaches proposed in the literature as we discuss. We would like to emphasize that our goal is not to compare DFM with EFM, but rather to develop GMsFEM framework which uses these (DFM or EFM) fine-grid discretization techniques. Numerical results are presented, where we demonstrate how one can adaptively add basis functions in the regions of interest based on error indicators. We also discuss the use of randomized snapshots (Calo et al. Randomized oversampling for generalized multiscale finite element methods, 2014), which reduces the offline computational cost.

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Acknowledgments

Efendievs work is partially supported by the US 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. Yao’s work is partially supported by the National Basic Research Program of China (973 Program) (Grant No. 2011CB201004), the National Natural Science Foundation of China (Grant No. 51234007), the Outstanding Doctoral Dissertation Training program of China university of petroleum (Grant No. LW120201A).

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Authors and Affiliations

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

    Yalchin Efendiev

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

    Yalchin Efendiev

  3. Chevron ETC, Houston, TX, 77002, USA

    Seong Lee

  4. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Guanglian Li

  5. School of Petroleum Engineering, China University of Petroleum (East China), Qingdao, 266555, Shandong, China

    Jun Yao & Na Zhang

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  1. Yalchin Efendiev
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  2. Seong Lee
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  3. Guanglian Li
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  4. Jun Yao
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  5. Na Zhang
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Correspondence to Yalchin Efendiev.

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Efendiev, Y., Lee, S., Li, G. et al. Hierarchical multiscale modeling for flows in fractured media using generalized multiscale finite element method. Int J Geomath 6, 141–162 (2015). https://doi.org/10.1007/s13137-015-0075-7

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  • DOI: https://doi.org/10.1007/s13137-015-0075-7

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