**Apendix: Analytical solutions for plane strain Kristianovic-Geertsma-de Klerk (KGD) hydraulic fractures**

The solution of a plane strain KGD hydraulic fracture in an infinite elastic body depends on the injection rate *Q*0 and on the three material parameters *E*' , *K*' , and *μ*' , which are defined as [21]

*E*' =*E* / 1 - *ν* <sup>2</sup> , *K*' =(32 / *π*)1/2 *K*I*c*, *μ*' =12*μ*

For the plane strain KGD hydraulic fracture, the crack opening *w*(*x*, *t*), crack length (half length) *l*(*t*), and net fluid pressure *p*(*x*, *t*) can be expressed as [21]

$$\begin{aligned} \varphi\_t(\mathbf{x}, t) &= \epsilon(t) \mathsf{L}(t) \mathsf{L}[\xi, \mathsf{F}\_t, P(t)] = \epsilon(t) \mathsf{L}(t) \gamma[P(t)] \overline{\mathsf{L}}(\xi) \\ p(\mathbf{x}, t) &= \epsilon(t) \mathsf{E}^\dagger \Pi[\xi, P(t)] \mathbf{I} \end{aligned} \tag{34}$$
 
$$l(t) = \gamma[P(t)] \mathsf{L}(t)$$

where *ξ* = *x* / *l*(*t*) is the scaled coordinate (0≤*ξ* ≤1), ϵ(*t*) is a small dimensionless parameter, *L*(*t*) denotes a length scale of the same order as the fracture length *l*(*t*), *P*(*t*) is the dimensionless evolution parameter, and *γ P*(*t*) is dimensionless fracture length.

The evolution parameter *P*(*t*) can be interpreted as a dimensionless toughness *κ* in the viscosity scaling [21]

$$\kappa = K \left( E^{\top 3} \mu^{\prime} \mathbf{Q}\_{0} \right)^{-1/4} \tag{35}$$

or as a dimensionless viscosity ℳ in the toughness scaling [21]

$$\mathcal{M} = \mu^{\stackrel{\cdot}{\cdot}} \mathrm{E}^{\stackrel{\cdot}{\cdot}} \mathbb{Q}\_0 / \mathrm{K}^{\stackrel{\cdot}{\cdot}} \tag{36}$$

For the toughness scaling, denoted by a subscript *k*, the small parameter ϵ(*t*) and the length scale *L*(*t*) take the explicit forms [21]

$$L\_{\vec{k}}(t) = \left(\mathbf{K}^{\prime 4} / \boldsymbol{E}^{\prime 4} \mathbf{Q}\_{0} t\right)^{1/3}, \ L\_{\vec{k}}(t) = \left(\boldsymbol{E}^{\prime} \mathbf{Q}\_{0} t / \boldsymbol{K}^{\prime}\right)^{2/3} \tag{37}$$

The solution for the zero viscosity case is given by [21]

assigned to the appropriate nodes of the proposed element to describe the fluid flow within the crack and its contribution to the crack deformation. Verification of the user-defined element has been made by comparing the FEM predictions with the analytical solutions available in the literature. The preliminary result presented here is a first attempt to the promising

**Apendix: Analytical solutions for plane strain Kristianovic-Geertsma-de**

The solution of a plane strain KGD hydraulic fracture in an infinite elastic body depends on

For the plane strain KGD hydraulic fracture, the crack opening *w*(*x*, *t*), crack length (half

ϵ

Π *ξ*, *P*(*t*)

ϵ

denotes a length scale of the same order as the fracture length *l*(*t*), *P*(*t*) is the dimensionless

The evolution parameter *P*(*t*) can be interpreted as a dimensionless toughness *κ* in the viscosity

, *K*'

(*t*)*L*(*t*)*<sup>γ</sup> <sup>P</sup>*(*t*) <sup>Ω</sup>¯(*ξ*)

, and *μ*'

(*t*) is a small dimensionless parameter, *L*(*t*)

*<sup>Q</sup>*0)-1/4 (35)

*Q*<sup>0</sup> / *K*' (36)

ϵ

)2/3 (37)

(*t*) and the length

, which are defined as

(34)

application of the XFEM to the hydraulic fracture simulation.

the injection rate *Q*0 and on the three material parameters *E*'

*K*I*c*, *μ*'

ϵ

=12*μ*

(*t*)*L*(*t*)Ω *ξ*, *P*(*t*) =

ϵ(*t*)*E*'

*l*(*t*)=*γ P*(*t*) *L*(*t*)

*p*(*x*, *t*)=

evolution parameter, and *γ P*(*t*) is dimensionless fracture length.

or as a dimensionless viscosity ℳ in the toughness scaling [21]

*<sup>k</sup>* (*t*)=(*K*'4 / *E*'4

*κ* =*K*' (*E*'3 *μ*'

ℳ=*μ*' *E*'3

For the toughness scaling, denoted by a subscript *k*, the small parameter

*<sup>Q</sup>*0*t*)1/3

, *Lk* (*t*)=(*E*'

*Q*0*t* / *K*'

length) *l*(*t*), and net fluid pressure *p*(*x*, *t*) can be expressed as [21]

**Klerk (KGD) hydraulic fractures**

736 Effective and Sustainable Hydraulic Fracturing

=(32 / *π*)1/2

*w*(*x*, *t*)=

where *ξ* = *x* / *l*(*t*) is the scaled coordinate (0≤*ξ* ≤1),

scale *L*(*t*) take the explicit forms [21]

ϵ

[21] *E*'

=*E* / 1 - *ν* <sup>2</sup>

scaling [21]

, *K*'

$$
\gamma\_{k0} = 2 \left/ \pi^{2/3} \right. \tag{38}
$$

$$
\overline{\Omega}\_{k0}(\xi) = \sqrt{1 - \xi^2} \Big/ \pi^{1/3} \tag{39}
$$

$$
\Pi\_{k0} = \pi^{1/3} / 8 \tag{40}
$$

For the viscosity scaling, denoted by a subscript *m*, the small parameter ϵ(*t*) and the length scale *L*(*t*) take the explicit forms [21]

$$\mathbf{e}\_m(t) = \left(\mu^\top/\to^\cdot t\right)^{1/3}, \quad \mathbf{L}\_m(t) = \left(\mathbf{E}^\cdot Q\_0^3 t^4/\mu^\prime\right)^{1/6} \tag{41}$$

The first order approximation of the zero toughness solution is [21]

$$
\gamma\_{m0} \cong 0.616\tag{42}
$$

$$\overline{\Omega}\_{m0}(\boldsymbol{\xi}) = A\_0 \{ \mathbf{1} \cdot \boldsymbol{\xi}^{-2} \}^{2/3} + A\_1^{(1)} \{ \mathbf{1} \cdot \boldsymbol{\xi}^{-2} \}^{5/3} + B^{(1)} \left[ \mathbf{4} \sqrt{\mathbf{1} \cdot \boldsymbol{\xi}^2} + 2 \boldsymbol{\xi}^2 \ln \left| \frac{\mathbf{1} \cdot \boldsymbol{\sqrt{1 \cdot \boldsymbol{\xi}^2}}}{\mathbf{1} \cdot \boldsymbol{\sqrt{1 \cdot \boldsymbol{\xi}^2}}} \right| \right] \tag{43}$$

$$\mathbf{T}\_{m0}^{(1)} = \frac{1}{3\pi} B\left(\frac{1}{2}, \,\,\frac{2}{3}\right) \mathbf{I}\_0 F\_1\left(\mathbf{-}\frac{1}{6}, \,\, 1; \frac{1}{2}; \, \xi^2\right) + \frac{10}{7} A\_1^{(1)} F\_1\left(\mathbf{-}\frac{7}{6}, \, 1; \frac{1}{2}; \, \xi^2\right) + B^{(1)}(2 - \pi \, \, \vert \, \xi \, \vert) \tag{44}$$

where *A*<sup>0</sup> =31/2 , *A*<sup>1</sup> (1) ≅ - 0.156, and *B*(1) ≅0.0663; *B* is Euler beta function, and *F*1 is hypergeo‐ metric function. Thus, Ω¯ *m*0 (0)≅1.84.

#### **Acknowledgements**

The author would like to thank Dr. Rob Jeffrey for the support of this work. Furthermore, the author thanks CSIRO CESRE for support and for granting permission to publish.

#### **Author details**

Zuorong Chen\*

CSIRO Earth Science & Resource Engineering, Melbourne, Australia

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An ABAQUS Implementation of the XFEM for Hydraulic Fracture Problems

http://dx.doi.org/10.5772/56287

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**Chapter 37**

**Stress Intensity Factor Determination for**

**Discontinuity Method with Applications to**

**Propagation Paths**

http://dx.doi.org/10.5772/56308

**Abstract**

Farrokh Sheibani and Jon Olson

Additional information is available at the end of the chapter

SIF distribution along the upper front of the fracture.

**Three-Dimensional Crack Using the Displacement**

**Hydraulic Fracture Height Growth and Non-Planar**

Stress intensity factor determination plays a central role in linearly elastic fracture mechanics (LEFM) problems. Fracture propagation is controlled by the stress field near the crack tip. Because this stress field is asymptotic dominant or singular, it is characterized by the stress intensity factor (SIF). Since many rock types show brittle elastic behaviour under hydrocarbon reservoir conditions, LEFM can be satisfactorily used for studying hydraulic fracture devel‐ opment. The purpose of this paper is to describe a numerical method to evaluate the stress intensity factor in Mode I, II and III at the tip of an arbitrarily-shaped, embedded cracks. The stress intensity factor is evaluated directly based on displacement discontinuities (DD) using a three-dimensional displacement discontinuity, boundary element method based on the equations of proposed in [1]. The boundary element formulation incorporates the fundamental closed-form analytical solution to a rectangular discontinuity in a homogenous, isotropic and linearly elastic half space. The accuracy of the stress intensity factor calculation is satisfactorily examined for rectangular, penny-shaped and elliptical planar cracks. Accurate and fast evaluation of the stress intensity factor for planar cracks shows the proposed procedure is robust for SIF calculation and crack propagation purposes. The empirical constant proposed by [2] relating crack tip element displacement discontinuity and SIF values provides surpris‐ ingly accurate results for planar cracks with limited numbers of constant DD elements. Using the described numerical model, we study how fracturing from misaligned horizontal well‐ bores might results in non-uniform height growth of the hydraulic fracture by evaluating of

> © 2013 Sheibani and Olson; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

> © 2013 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

and reproduction in any medium, provided the original work is properly cited.

**Stress Intensity Factor Determination for Three-Dimensional Crack Using the Displacement Discontinuity Method with Applications to Hydraulic Fracture Height Growth and Non-Planar Propagation Paths**

Farrokh Sheibani and Jon Olson

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56308
