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Fractal Analytical Solutions for Nonlinear Two-Phase Flow in Discontinuous Shale Gas Reservoir

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  • Xiaoji Shang

    (State Key Laboratory of Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China
    Key Laboratory of Deep Earth Science and Engineering (Sichuan University), Ministry of Education, Chengdu 610065, China
    YunLong Lake Laboratory of Deep Underground Science and Engineering, Xuzhou 221116, China)

  • Zhizhen Zhang

    (State Key Laboratory of Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China
    Key Laboratory for Urban Underground Engineering of the Education Ministry, Beijing Jiaotong University, Beijing 100044, China)

  • Zetian Zhang

    (Key Laboratory of Deep Earth Science and Engineering (Sichuan University), Ministry of Education, Chengdu 610065, China)

  • J. G. Wang

    (State Key Laboratory of Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China)

  • Yuejin Zhou

    (State Key Laboratory of Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China)

  • Weihao Yang

    (State Key Laboratory of Geomechanics and Deep Underground Engineering, School of Mechanics and Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China)

Abstract

The paths of a two-phase flow are usually non-linear and discontinuous in the production of shale gas development. To research the influence mechanism between shale gas and water, several integer two-phase flow models have been studied but few analytical solutions have been obtained on shale gas and water pressure. This study first developed a local fractional mathematical model for gas and water two-phase flow in shale gas production. The model thus created considers the effects of capillary pressure, the fractal dimension of the flow pipe, and the discontinuity of the flow path. Second, the local fractional traveling wave method and variational iteration method were applied to this model for the development of iterative analytical solutions. Both shale gas and water pressure were analytically derived. Third, the depressurization process of the shale gas and water was analyzed, and a parametric study was conducted to explore the impacts of fractional dimension, entry capillary pressure, and travel wave velocity on shale gas pressure. Finally, our conclusions are drawn, based on the results of these studies.

Suggested Citation

  • Xiaoji Shang & Zhizhen Zhang & Zetian Zhang & J. G. Wang & Yuejin Zhou & Weihao Yang, 2022. "Fractal Analytical Solutions for Nonlinear Two-Phase Flow in Discontinuous Shale Gas Reservoir," Mathematics, MDPI, vol. 10(22), pages 1-14, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:22:p:4227-:d:970675
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    References listed on IDEAS

    as
    1. Dossan Baigereyev & Nurlana Alimbekova & Abdumauvlen Berdyshev & Muratkan Madiyarov, 2021. "Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media," Mathematics, MDPI, vol. 9(18), pages 1-25, September.
    2. Suran Wang & Linsong Cheng & Yongchao Xue & Shijun Huang & Yonghui Wu & Pin Jia & Zheng Sun, 2018. "A Semi-Analytical Method for Simulating Two-Phase Flow Performance of Horizontal Volatile Oil Wells in Fractured Carbonate Reservoirs," Energies, MDPI, vol. 11(10), pages 1-21, October.
    3. Xiao-Jun Yang & Dumitru Baleanu & J. A. Tenreiro Machado, 2013. "Systems of Navier-Stokes Equations on Cantor Sets," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-8, June.
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