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Weak Galerkin Finite Element Methods for the Simulation of Single-Phase Flow in Fractured Porous Media

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Abstract

This paper presents a numerical simulation of the flow in fractured porous media, which can be efficiently described by the reduced model consisting of the bulk problem in the porous matrix and the fracture problem in the fracture together with physically consistent coupling conditions. The reduced model is solved by using two types of weak Galerkin finite element methods (on simplex mesh and polygonal mesh, respectively) for the bulk problem coupled with the cell centered finite volume method for the fracture problem. We prove that the algebraic system arising in the discrete formulation is symmetric and positive-definite, which leads to the well-posedness of the discrete problem. Error estimates for the pressure in suitable norms are established. A series of numerical experiments demonstrate the accuracy and robustness of our method with respect to the shape of the grid and the permeability of the fractures.

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Acknowledgements

The authors would like to thank the reviewer and editor for their valuable comments and suggestions which lead to an improvement of the paper. The authors wish to express sincere thanks to the authors in [15] who proposed the benchmarks and published the grid and result files in the form of a Git repository. This gives us the chance to present most of the numerical experiments in this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, People’s Republic of China

    Gang Wang, Yinnian He & Jinjin Yang

Authors
  1. Gang Wang
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  2. Yinnian He
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  3. Jinjin Yang
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Corresponding author

Correspondence to Yinnian He.

Additional information

The work was supported by the Major Research and Development Program of China under Grant No. 2016YFB0200901 and the National Natural Science Foundation of China under Grant No. 11771348.

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Wang, G., He, Y. & Yang, J. Weak Galerkin Finite Element Methods for the Simulation of Single-Phase Flow in Fractured Porous Media. J Sci Comput 76, 1274–1300 (2018). https://doi.org/10.1007/s10915-018-0673-2

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  • DOI: https://doi.org/10.1007/s10915-018-0673-2

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