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The effects of non-inverse-square-root stress singularity on some crack propagation criteria in hydraulic fracturing

非倒平方根应力奇异性对水力压裂某些裂纹扩展准则的影响

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Abstract

Numerous crack propagation criteria have been employed in numerical investigations of hydraulic fracturing. Some criteria rely on linear elastic fracture mechanics, which assume the presence of an inverse-square-root stress singularity and an elliptical crack profile. However, it has been observed that hydraulic fracturing exhibits a weaker stress singularity, which is governed by a dimensionless parameter known as apparent fracture toughness. This parameter represents the energy required to fracture the solid compared to the energy needed to overcome fluid friction. Abnormal stress singularity was found for the viscosity dominated regime where the apparent fracture toughness approaches zero from the positive side. Variations in results can occur when different criteria are employed, primarily due to the breakdown of the inverse-square-root stress singularity. In this paper, a mathematical model utilizing a straight slit crack was employed to investigate these discrepancies. Four commonly used criteria were implemented in the same mathematical model, and high-precision methods based on modified Chebyshev polynomials and implicit algorithms were employed for calculations. The transient crack dimensions and wellbore pressure were found to diverge for these criteria. Notably, the CTOD, CDM, and FPZ criteria were found to be challenging to apply in the viscosity-dominated regime where the apparent fracture toughness was less than 1.0. The underlying mechanisms were discussed based on crack tip asymptotics and crack opening behavior. This study provides valuable insights for selecting appropriate crack propagation criteria in hydraulic fracturing.

摘要

水力压裂的数值研究可采用不同的裂纹扩展准则, 其中一些准则依赖于线弹性断裂力学的假设, 即存在倒平方根应力奇异性和 椭圆裂纹轮廓. 然而, 水力压裂的裂纹尖端可能表现出更弱的应力奇异性, 这种现象由称为表观断裂韧性的无量纲参数控制, 该参数表 示使固体破裂所需的能量与克服流体摩擦所需的能量的比. 在流体黏度占主导的情况下, 表观断裂韧性从正数一侧趋近于零, 应力奇 异性阶次降低. 当采用一些裂纹扩展准则计算时, 结果可能会发生变化, 主要是由于倒平方根应力奇异性发生改变, 本文采用平直裂纹 的数学模型来研究这种现象. 在相同的数学模型中分别采用4种常用裂纹扩展准则, 使用基于修正切比雪夫多项式和隐式时间积分的 高精度方法进行计算. 发现瞬态裂缝尺寸和井筒压力在使用不同裂纹扩展准则时存在差异. 值得注意的是, 当表观断裂韧性小于1.0时, 裂纹尖端张开位移准则、连续介质损伤模型和断裂过程区准则在流体黏度占主导的情况下的计算模型很难收敛, 本文基于裂纹尖端 渐近性质讨论了其中隐含的机制. 本文研究为水力压裂中裂缝扩展准则的选择提供了理论基础.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11902213).

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Authors and Affiliations

  1. College of Mechanical and Vehicle Engineering, Taiyuan University of Technology, Taiyuan, 030024, China

    Wenhao Shen  (沈文豪) & Xuan Wu  (吴璇)

  2. Institute of Applied Mechanics, Taiyuan University of Technology, Taiyuan, 030024, China

    Wenhao Shen  (沈文豪) & Xuan Wu  (吴璇)

  3. State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190, China

    Wenhao Shen  (沈文豪) & Xianfu Huang  (黄先富)

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  1. Wenhao Shen  (沈文豪)
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Contributions

Author contributions Wenhao Shen designed the research, built the mathematical model and the numerical calculation procedure, and wrote the first draft of the manuscript. Xuan Wu conducted the numerical calculations and processed the data. Xianfu Huang and Xuan Wu helped organize the manuscript, and revised and edited the final version.

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Correspondence to Wenhao Shen  (沈文豪).

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Shen, W., Wu, X. & Huang, X. The effects of non-inverse-square-root stress singularity on some crack propagation criteria in hydraulic fracturing. Acta Mech. Sin. 40, 423246 (2024). https://doi.org/10.1007/s10409-023-23246-x

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