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Theoretical and numerical models of rock wing crack subjected to hydraulic pressure and far-field stresses

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Abstract

The evolution laws of stress intensity factor (SIF) at the crack tip subjected to hydraulic pressure have still not elucidated clearly. This article attempts to study the evolution laws of SIF at the wing crack tip subjected to hydraulic pressure and far-field stresses, theoretically and numerically, based on the previous proposed wing crack models without considering hydraulic pressure. The numerical model of wing crack subjected to hydraulic pressure and far-field stresses is proposed by ANSYS based on finite element model (FEM). Research results show that the curves of the dimensionless SIF at the wing crack tip versus equivalent crack propagation length are in three major types: D type, DR type, and R type. The D type curve exhibits a steady propagation behavior of wing crack; however, the DR type and R type curves exhibit unsteady propagation behavior. The D type curve gradually transfers to the DR and R type curves with increasing hydraulic pressure. On the whole, the tendency of theoretical model curves is in agreement with that of numerical simulation curves. The average SSRs of HN, S, B, LK, W, and Z model solutions to SIF at the wing crack tip are 0.0079, 0.0348, 0.0099, 0.0127, 0.0077, and 0.0068, respectively. So the average SSRs of the Z and S model solutions are the lowest and highest among all theoretical model solutions. The Z model solution to SIF at the wing crack tip subjected to the combined action of hydraulic pressure and far-field stresses can be considered an optimal solution due to the lowest average SSR. The study further enhances the understanding of the mechanical behavior of hydraulic fracturing in rock mass engineering.

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Abbreviations

a :

Half of crack length

β:

Main crack inclination angle

l :

Wing crack propagation length

l*:

The equivalent crack length, l* = 0.27a

τ eff :

Effective shear stress applied on main crack face

\( {\sigma}_n^{\hbox{'}} \) :

Normal stress applied on wing crack

μ :

The friction coefficient of the main crack surface

θ :

The orientation of straight wing crack against main crack

σ1 and σ3 :

The maximum and minimum principal stress, respectively

l eq :

Equivalent wing crack length

K I :

The mode I SIF at the tip of wing crack

p :

Hydraulic pressure

α:

The ratio of connected area to total area of crack

τ peff :

Effective shear stress applied on main crack face considering hydraulic pressure

\( {\sigma}_{pn}^{\hbox{'}} \) :

Normal stress applied on wing crack considering hydraulic pressure

w(r):

Opening displacement near a crack

Δw(r):

Relative opening displacements

r,θ :

Radius vector and polar angle in a local cylindrical coordinate system, respectively

G and ν:

Shear modulus and Poisson’s ratio of material, respectively

KΙ(r) and KΙΙ(r):

Mode Ι and II SIFs near the wing crack tip, respectively

λ :

Lateral pressure coefficient

‾K I :

The dimensionless SIF

L :

Equivalent crack propagation length

K IC :

Fracture toughness of rock

SSR:

Sum of squared residual

SIF:

Stress intensity factor

FEM:

Finite element model

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Funding

This research is supported by the National Natural Science Foundation of China (51774131, 51774107, 51274097)

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Authors and Affiliations

  1. Hunan Provincial Key Laboratory of Safe Mining Techniques of Coal Mines, Work Safety Key Laboratory on Prevention and Control of Gas and Roof Disasters for Southern Coal Mines, Hunan University of Science and Technology, Xiangtan, 411201, China

    Yanlin Zhao, Qiang Liu, Jian Liao & Liming Tang

  2. School of Civil Engineering, Hefei University of Technology, Hefei, 230009, China

    Yixian Wang

Authors
  1. Yanlin Zhao
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  2. Qiang Liu
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  3. Jian Liao
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  4. Yixian Wang
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  5. Liming Tang
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Corresponding author

Correspondence to Yanlin Zhao.

Additional information

Responsible Editor: Murat Karakus

Highlights

(1) The revised wing crack models subjected to hydraulic pressure and far-field stresses are proposed, based on previous proposed wing crack models without considering hydraulic pressure.

(2) The curves of the dimensionless SIF at the wing crack tip versus equivalent crack propagation length are in three major types: D type, DR type, and R type. The D type curve exhibits a steady propagation behavior of wing crack; however, the DR type and R type curves exhibit unsteady propagation behavior.

(3) The Z model solution to SIF at the wing crack tip subjected to the combined action of hydraulic pressure and far-field stresses can be considered an optimal solution.

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Zhao, Y., Liu, Q., Liao, J. et al. Theoretical and numerical models of rock wing crack subjected to hydraulic pressure and far-field stresses. Arab J Geosci 13, 926 (2020). https://doi.org/10.1007/s12517-020-05957-9

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  • DOI: https://doi.org/10.1007/s12517-020-05957-9

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