A Numerical Method for Fracture Crossing Based on Average Stress Levels

Abstract

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.

Appendices

Appendix 1: Enrichment Functions for Displacement

1.1 Heaviside Enrichment Function

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

(14)

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)

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

(15)

where H^a(x) is the Heaviside enrichment function at point x for crack a, H^b(x) is the Heaviside enrichment function at point x for crack b, and x_pc is an arbitrary point on crack a.

1.2 Crack Tip Enrichment Functions

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

(16)

where r and θ_r 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

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

(17)

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 B^std and B^enr ((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.