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A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media

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Abstract

A block-centered finite difference method is introduced to solve an initial and boundary value problem for a nonlinear parabolic equation to model the slightly compressible flow in porous media, in which the velocity–pressure relation is described by Darcy–Forchheimer’s Law. The method can be thought as the lowest order Raviart–Thomas mixed element method with proper quadrature formulation. By using the method the velocity and pressure can be approximated simultaneously. We established the second-order error estimates for pressure and velocity in proper discrete norms on non-uniform rectangular grid. No time-step restriction is needed for the error estimates. The numerical experiments using the scheme show that the convergence rates of the method are in agreement with the theoretical analysis.

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Acknowledgements

The authors thank the anonymous referees for their constructive comments, suggestions, careful checking of the manuscript and listing papers about this problem published recently, which lead to improvements of the presentation. This work is supported by the National Natural Science Foundation of China Grant Nos. 11671233,91330106,11301307; the Foundation of Shandong Province Outstanding Young Scientist Award No. BS2013NJ002.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Hongxing Rui

  2. School of Information Science and Engineering, Shandong Agricultural University, Taian, 271018, China

    Hao Pan

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  1. Hongxing Rui
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  2. Hao Pan
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Correspondence to Hongxing Rui.

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Rui, H., Pan, H. A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media. J Sci Comput 73, 70–92 (2017). https://doi.org/10.1007/s10915-017-0406-y

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