A Numerical Method for Fracture Crossing Based on Average Stress Levels


Due to the complexity of the interaction between hydraulic fractures (HFs) and natural fractures (NFs), several different interaction behaviours (arrest/cross/offset) may occur, according to hydraulic fracturing experiments. The existing analytical criteria cannot accurately describe this interaction process and predict the interaction behaviours by adopting several simple assumptions. An innovative numerical method combining strength and energy is proposed to determine the critical average stress at the most likely re-initiation position along the NF. The critical average stress for the HF crossing can be calculated by averaging the stresses around the fracture re-initiation point when the NF does not open and slip and simultaneously satisfying the energy criterion. Then, the proposed numerical method is incorporated in an extended finite element method (XFEM) scheme, in which the friction and contact of the NF are solved by a penalty function method. The numerical results suggest that the average stress can be numerically calculated and used to predict the HF crossing behaviour. The varying trends of the required maximum confining stress with respect to the coefficient of friction obtained by the proposed numerical method agree well with those predicted by the R-P and ER-P criteria. In addition, the proposed average stress method can quantitatively predict the crossing behaviour in the presence of slippage and opening after the HF terminates at the NF.

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A Ele :

Area of the element containing the intersection point

B :

Discretized gradient operators

B α :

Crack tip enrichment functions

b :

Gravitational acceleration force

D :

Elastic material property matrix

D cont :

Contact constitutive matrix

E :

Elastic modulus

H :

Heaviside enrichment function

J :

Jacobian matrix

J H :

Junction enrichment function

K :

Stiffness matrix

K :

Stress intensity factor

K c :

Critical stress intensity factor

k N :

Normal penalty parameter

k T :

Tangential penalty parameter

f cont :

Contact force vector

f p :

Fluid pressure force vector

L HF :

Length of the HF

L NF :

Length of the NF

L 0 :

Length of the fluid pressure activation zone

m :

Tangential unit vector

N :

Shape function

n :

Normal unit vector

p f :

Fracturing fluid pressure

R :

Radius of the average stress computation zone

r, θ r :

Polar coordinates at the fracture tip

T :

Tensile strength of rock

t :

External traction

u :

Displacement vector

w :

Weight values of the Gauss points

w N :

Normal fracture opening

w T :

Tangential fracture opening

Г :

Domain boundary

ε :

Strain vector

η :

Relative error

θ :

Approaching angle

μ f :

Coefficient of friction

v :

Poisson’s ratio

ρ :

Bulk density

σ :

Cauchy stress tensor

σ max :

Maximum principal compressive stress

σ min :

Minimum principal compressive stress

σ p :

Average principal stress

σ cp :

Critical average stress

φ :

Signed distance function

ψ :

Residual force vector

Ω :

Computational domain


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The authors gratefully acknowledge the support of the National Natural Science Foundation of China (Grant Nos. 51879260, 51879258, and 41572290), CAS Interdisciplinary Innovation Team (JCTD-2018-17), and Hubei Provincial Natural Science Foundation of China (Grant No. 2018CFA012).

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Appendix 1: Enrichment Functions for Displacement

Heaviside Enrichment Function

$$H({\mathbf{x}}) = \left\{ \begin{aligned} + 1\quad {\text{ if }}\varphi ({\mathbf{x}} ) { > 0} \hfill \\ - 1 \quad {\text{ if }}\varphi ({\mathbf{x}} ) { < 0,} \hfill \\ \end{aligned} \right.$$

where \(\varphi ({\mathbf{x}} ) { = }\left\| {{\mathbf{x}}^{*} - {\mathbf{x}}} \right\|{\text{sign}}\left( {({\mathbf{x}}^{*} - {\mathbf{x}}) \cdot {\mathbf{n}}_{\varGamma d} } \right)\) is the signed distance function. \({\mathbf{x}}^{*}\) is the closest projection of point \({\mathbf{x}}\) onto the crack.

Junction enrichment function (the tip of crack a reaches crack b)

$$J_{\text{H}} ({\mathbf{x}}) = \left\{ \begin{aligned} &H^{a} \left( {\mathbf{x}} \right)\quad {\text{if}}\,H^{b} ({\mathbf{x}} )= H^{b} ({\mathbf{x}}_{\text{pc}} )\hfill \\ &0\quad \, \quad \quad {\text{if }}\,H^{b} ({\mathbf{x}} )= - H^{b} ({\mathbf{x}}_{\text{pc}} ) ,\hfill \\ \end{aligned} \right.$$

where Ha(x) is the Heaviside enrichment function at point x for crack a, Hb(x) is the Heaviside enrichment function at point x for crack b, and xpc is an arbitrary point on crack a.

Crack Tip Enrichment Functions

$$\left\{ {B_{\alpha } ({\mathbf{x}})} \right\} = \left\{ {\sqrt r { \sin }\frac{{\theta_{\text{r}} }}{ 2},\sqrt r { \cos }\frac{{\theta_{\text{r}} }}{ 2},\sqrt r { \sin }\frac{{\theta_{\text{r}} }}{ 2}{ \sin }\theta_{\text{r}} ,\sqrt r { \cos }\frac{{\theta_{\text{r}} }}{ 2}{ \sin }\theta_{\text{r}} } \right\},$$

where r and θr are the polar coordinates of point x in the coordinate system centred on the tip of the crack with the x axis aligned with the crack direction.

Appendix 2

$$\begin{aligned} & {\mathbf{K}} = \int\limits_{\varOmega } {({\mathbf{B}}^{\alpha } )^{\text{T}} {\mathbf{DB}}^{\beta } {\text{d}}\varOmega } \\ & {\mathbf{f}}^{\text{ext}} = \int\limits_{\varOmega } {({\mathbf{N}}^{\alpha } )^{\text{T}} \rho {\mathbf{b}}{\text{d}}\varOmega } - \int\limits_{{\varGamma_{t} }} {({\mathbf{N}}^{\alpha } )^{\text{T}} {\bar{\mathbf{t}}}{\text{d}}\varGamma } \\ & {\mathbf{f}}_{p}^{\text{int}} = \int\limits_{{\varGamma_{d} }} {\left[\kern-0.15em\left[ {{\mathbf{N}}^{\alpha } } \right]\kern-0.15em\right]^{\text{T}} p_{\text{f}} {\mathbf{n}}_{{\varGamma {\text{d}}}} {\text{d}}\varGamma } \\ & {\mathbf{f}}_{\text{cont}}^{\text{int}} = \int\limits_{{\varGamma_{d} }} {\left[\kern-0.15em\left[ {{\mathbf{N}}^{\alpha } } \right]\kern-0.15em\right]^{T} {\mathbf{f}}_{\text{cont}} {\text{d}}\varGamma } . \\ \end{aligned}$$

The strain vector ε can be written as \({\varvec{\upvarepsilon}} = [\begin{array}{*{20}c} {{\mathbf{B}}^{\text{std}} } & {{\mathbf{B}}^{\text{enr}} } \\ \end{array} ]{\bar{\mathbf{U}}}\), where Bstd and Benr ((α, β) ϵ (std, enr)) are standard discretized gradient operators and enriched discretized gradient operators, respectively. D is the elasticity material property matrix of the continuum body.

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Zhou, Y., Yang, D., Zhang, X. et al. A Numerical Method for Fracture Crossing Based on Average Stress Levels. Rock Mech Rock Eng 53, 4471–4485 (2020). https://doi.org/10.1007/s00603-020-02054-x

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  • Hydraulic fracture
  • Natural fracture
  • Crossing criterion
  • Fracture interaction
  • Average stress
  • XFEM