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A Study of the Non-Linear Seepage Problem in Porous Media via the Homotopy Analysis Method

Author

Listed:
  • Xiangcheng You

    (State Key Laboratory of Petroleum Resources and Prospecting, College of Petroleum Engineering, China University of Petroleum-Beijing, Beijing 102249, China)

  • Shiyuan Li

    (State Key Laboratory of Petroleum Resources and Prospecting, College of Petroleum Engineering, China University of Petroleum-Beijing, Beijing 102249, China)

  • Lei Kang

    (State Key Laboratory of Petroleum Resources and Prospecting, College of Petroleum Engineering, China University of Petroleum-Beijing, Beijing 102249, China)

  • Li Cheng

    (State Key Laboratory of Petroleum Resources and Prospecting, College of Petroleum Engineering, China University of Petroleum-Beijing, Beijing 102249, China)

Abstract

A non-Darcy flow with moving boundary conditions in a low-permeability reservoir was solved using the homotopy analysis method (HAM), which was converted into a fixed-boundary mathematical model via similarity transformation. Approximate analytical solutions based on the HAM are guaranteed to be more accurate than exact analytical solutions, with relative errors between 0.0089% and 2.64%. When λ = 0, the pressure drop of the Darcy seepage model could be instantaneously transmitted to infinity. When λ > 0, the pressure drop curve of the non-Darcy seepage model exhibited the characteristics of tight support, which was clearly different from the Darcy seepage model’s formation pressure distribution curve. According to the results of the HAM, a moving boundary is more influenced by threshold pressure gradients with a longer time. When the threshold pressure gradients were smaller, the moving boundaries move more quickly and are more sensitive to external influences. One-dimensional, low-permeability porous media with a non-Darcy flow with moving boundary conditions can be reduced to a Darcy seepage model if the threshold pressure gradient values tend to zero.

Suggested Citation

  • Xiangcheng You & Shiyuan Li & Lei Kang & Li Cheng, 2023. "A Study of the Non-Linear Seepage Problem in Porous Media via the Homotopy Analysis Method," Energies, MDPI, vol. 16(5), pages 1-13, February.
    • Handle: RePEc:gam:jeners:v:16:y:2023:i:5:p:2175-:d:1078744
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    References listed on IDEAS

    as
    1. Al-Qudah, Alaa & Odibat, Zaid & Shawagfeh, Nabil, 2022. "A linearization-based computational algorithm of homotopy analysis method for nonlinear reaction–diffusion systems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 194(C), pages 505-522.
    2. Zhou, Yang & Zhang, Li-ying & Wang, Tao, 2021. "Analytical solution for one-dimensional non-Darcy flow with bilinear relation in porous medium caused by line source," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    3. Wei, Qing & Zhou, Hongwei & Yang, Shuai, 2020. "Non-Darcy flow models in porous media via Atangana-Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    4. Sardanyés, Josep & Rodrigues, Carla & Januário, Cristina & Martins, Nuno & Gil-Gómez, Gabriel & Duarte, Jorge, 2015. "Activation of effector immune cells promotes tumor stochastic extinction: A homotopy analysis approach," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 484-495.
    5. Jun Yao & Wenchao Liu & Zhangxin Chen, 2013. "Numerical Solution of a Moving Boundary Problem of One-Dimensional Flow in Semi-Infinite Long Porous Media with Threshold Pressure Gradient," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-7, December.
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