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Role of Fractal Effect in Predicting Crack Initiation Angle and Its Application in Hydraulic Fracturing

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Abstract

In hydraulic fracturing, a prominent issue is to predict the crack initiation angle, especially for the directional fracturing process. However, the influence of the geometric irregularity of the hydraulic fracture on its initiation angle is unclear. In this study, the geometric irregularity of fractures is represented by fractal dimension, three fractal fracture criteria are derived to judge the crack initiation angle, and then their predicted values are compared to the reported experimental results. The result shows that the energy-based fractal criterion prediction is the most accurate. Furthermore, the energy-based fractal criterion is applied to evaluate the crack initiation angle in hydraulic fracturing. The predicted initiation angle of hydraulic fractures considered the crack fractal effect is more consistent with the experimental data. When the crack roughness is ignored, the predicted crack initiation angles are smaller than the experimental values, with 36–50% deviations. According to the parametric sensitivity analysis results, the initiation angle of hydraulic fractures increases with the crack fractal dimension. Moreover, the effect of the crack fractal dimension on the initiation angle of hydraulic fractures can be amplified with the increase of in situ stress difference or decrease of the Biot coefficient.

Highlights

  • Three fractal fracture criteria are derived to judge the crack initiation angle.

  • The energy-based fractal fracture criterion is applied to evaluate the crack initiation angle of hydraulic fracturing.

  • The prediction deviation of the hydraulic fracture initiation angle can be effectively reduced by considering the crack fractal dimension.

  • The hydraulic fracture initiation angle increases with the crack fractal dimension and the increasing magnitude is affected by the in situ stress and Biot coefficient.

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Abbreviations

DHF:

Directional hydraulic fracturing

CPF:

Complex stress potential function

FMTS:

Fractal maximum tangential stress

FMTSN:

Fractal maximum tangential strain

FDSED:

Fractal distortional strain energy density

\(\sigma_{\theta }\) :

Tangential stress

\(\tau_{r\theta }\) :

Tangential stress

\(\varepsilon_{\theta }\) :

Tangential strain

\(\varepsilon_{\theta T}\) :

Tangential strain by considering T-stress

\(G\) :

Energy release rate

\(U\) :

Strain energy density

\(U_{d}\) :

Distortional strain energy density

\(U_{v}\) :

Dilatational strain energy density

\(r_{b}\) :

Plastic radius

\(r_{bcr}\) :

Critical plastic radius

\(I_{p}\) :

Second stress invariant

\(R_{p}\) :

Non-dimensional plastic radius

\(\sigma_{ij}\) :

Crack-tip stress

\(r\) :

Radial polar coordinate

\(\theta\) :

Circumpolar coordinate

\(D\) :

Fractal dimension

\(\sigma\) :

Remote stress

\(\alpha\) :

Stress singularity exponent

\(Z_{N}\) :

Complex stress potential function

\(z\) :

Complex variable

\(\sigma_{H}\) :

Maximum horizontal in situ stress

\(\sigma_{h}\) :

Minimum horizontal in situ stress

\(P\) :

Fluid injection pressure

\(\rho_{\rho }\) :

Dimensionless plastic radius a hydraulic crack tip

\(\left( {\sigma_{xx} } \right)_{{{\text{II}}}}^{f} ,\left( {\sigma_{yy} } \right)_{{{\text{II}}}}^{f} ,\left( {\tau_{xy} } \right)_{{{\text{II}}}}^{f}\) :

Stress components at a mode II fractal crack tip

\({\text{Re}} ,{\text{Im}}\) :

Real part and imaginary part of complex variable

\(\sigma_{x} ,\sigma_{y} ,\tau_{xy}\) :

Crack-tip stress components in Cartesian coordinates

\(K_{N}^{f}\) :

Intensity factors of fractal cracks

\(K_{{\text{I}}}^{f}\) :

Intensity factors of mode I fractal cracks

\(K_{{{\text{II}}}}^{f}\) :

Intensity factors of mode II fractal cracks

\(\beta\) :

Crack inclination angle

\(E\) :

Elastic modulus

\(\mu\) :

Poisson’s ratio

\(\Delta \theta\) :

Crack initiation deviation angle

\(p\) :

Pore pressure

\(\sigma_{n}\) :

Normal stress on the crack surface

\(\tau_{n}\) :

Shear stress on the crack surface

\(C_{n}\) :

Compression transmitting factor

\(C_{s}\) :

Shear transmitting factor

\(k_{n}\) :

Normal stiffness of the crack surface

\(k_{s}\) :

Shear stiffness of the crack surface

\(a\) :

Nominal length of fractal cracks

\(\alpha_{1}\) :

Biot coefficient

\(f\) :

Crack friction coefficient

\(\tau_{n}^{eff}\) :

Effective shear stress on the crack surface

\(\delta\) :

Box side length

\(N\left( \delta \right)\) :

Box counts

\(\chi \left( D \right)\) :

A parameter related to fractal dimension

\(\sigma_{e}\) :

Yield strength of rock under uniaxial tension

\(r_{\rho }\) :

Plastic radius at a hydraulic crack tip

\(\left( {\sigma_{xx} } \right)_{\rm I}^{f} ,\left( {\sigma_{yy} } \right)_{\rm I}^{f} ,\left( {\tau_{xy} } \right)_{\rm I}^{f}\) :

Stress components at a mode I fractal crack tip

\(\sigma_{x}^{f} ,\sigma_{y}^{f} ,\sigma_{z}^{f} ,\tau_{xy}^{f}\) :

Stress components at a mixed-mode I-II fractal crack tip

\(\sigma_{r}^{f} ,\sigma_{\theta }^{f} ,\tau_{\theta r}^{f}\) :

Fractal crack-tip stress components in polar coordinates

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Acknowledgements

This work was financially supported by the Natural Science Foundation of China (No. 12002270), the Key Research and Development project of Shaanxi Province (No. 2022SF-197) and the China Postdoctoral Science Foundation (Nos. 2020M673451, 2020M683686XB, 2021T140553, 2021M692600).

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Authors and Affiliations

  1. State Key Laboratory of Eco-Hydraulics in Northwest Arid Region, Xi’an University of Technology, Xi’an, 710048, China

    Xin Liang, Peng Hou, Yi Xue, Jia Liu & Faning Dang

  2. School of Civil Engineering and Architecture, Xi’an University of Technology, Xi’an, 710048, China

    Xin Liang, Yi Xue & Jia Liu

  3. IPPH, Purdue University, West Lafayette, IN, 47907, USA

    Peng Hou

  4. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, 221116, Jiangsu Province, China

    Yanan Gao & Feng Gao

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  1. Xin Liang
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Correspondence to Peng Hou.

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Liang, X., Hou, P., Xue, Y. et al. Role of Fractal Effect in Predicting Crack Initiation Angle and Its Application in Hydraulic Fracturing. Rock Mech Rock Eng 55, 5491–5512 (2022). https://doi.org/10.1007/s00603-022-02940-6

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