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A finite difference approximation of reduced coupled model for slightly compressible Forchheimer fractures in Karst aquifer system

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Abstract

A finite difference method is proposed for solving the compressible reduced coupled model, in which the flow is governed by Forchheimer’s law in the fracture and Darcy’s law in the surrounding porous media. By using the averaging technique, the fracture is reduced to a lower dimensional interface and a more complicated transmission condition is derived on the fracture-interface. Different degrees of freedom are located on both sides of fracture-interface in order to capture the jump of velocity and pressure. Second-order error estimates in discrete norms are derived on nonuniform staggered grids for both pressure and velocity. The proposed scheme can also be extended to nonmatching spatial and temporal grids without loss of accuracy. Numerical experiments are performed to demonstrate the efficiency and accuracy of the numerical method. It is shown that the parameter ξ has little influence on the fluid flow, and the permeability tensor of fracture has a significant impact on the flow rate in both the surrounding porous and fracture-interface.

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Funding

The work is supported by the National Natural Science Foundation of China Grant No. 11771367, the Shandong Provincial Natural Science Foundation No. ZR2019MA049, Shandong Province Higher Educational Science and Technology Program No. J16LI05 and The Hong Kong RGC General Research Fund, Grant No. 15302518.

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Authors and Affiliations

  1. School of Mathematics and Statistics Science, Ludong University, Yantai, 264025, China

    Wei Liu

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

    Jintao Cui

  3. School of Mathematics and Information Sciences, Yantai University, Yantai, 264005, China

    Zhifeng Wang

Authors
  1. Wei Liu
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  2. Jintao Cui
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  3. Zhifeng Wang
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Correspondence to Wei Liu.

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Liu, W., Cui, J. & Wang, Z. A finite difference approximation of reduced coupled model for slightly compressible Forchheimer fractures in Karst aquifer system. Numer Algor 84, 133–163 (2020). https://doi.org/10.1007/s11075-019-00749-z

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  • DOI: https://doi.org/10.1007/s11075-019-00749-z

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