**1. Introduction**

In this study, we are interested in the initiation of hydraulic fractures from an open-hole horizontal well. Horizontal wells are drilled preferably in the direction of the minimum

© 2013 Abbas and Lecampion; 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.

horizontal stress in order to create hydraulic fractures perpendicular to the wellbore and therefore maximize the drainage from the reservoir. During the pressurization of the wellbore, tangential tensile stress is generated which can result in the initiation of longitudinal fractures parallel to the wellbore axis [1,2]. In an isotropic elastic medium, depending upon the stress field, the initiation of fractures transverse to the wellbore axis is favored by creating an initial flaw, i.e. an axisymmetric notch, of sufficient depth [3] (see Figure. 1 for a sketch). We focus solely on transverse hydraulic fractures in this contribution.

(andinthewellbore), *p* = *p <sup>f</sup>* - *σ<sup>h</sup>* thenetpressureopeningthefracturefaceand*w* thecorrespond‐ ing fracture width. The axial stress variation azimuthally around the wellbore is given by [9]

Initiation and Breakdown of an Axisymmetric Hydraulic Fracture Transverse to a Horizontal Wellbore

where *r* and *θ* are the polar coordinates centered at the wellbore, *σ<sup>H</sup>* is the maximum horizontal stress, *σv* is the vertical stress, *ν* is the poisson's ratio and *a* is the wellbore radius. The azimuthal average of the axial stress (i.e. integrating the above equation from *θ* =0 to *θ* =2*π*) reduces to *σ<sup>h</sup>* . We will neglect the azimuthal variation of the axial stress close to the wellbore wall as a first approximation. The axial stress therefore reduces to the minimum horizontal stress. Under

**Figure 1.** Sketch of a transverse fracture propagating from a horizontal wellbore (top), axisymmetric model (bottom)

The relation between the fracture width and the net loading acting on the crack is ex‐ pressed by a hyper-singular boundary integral equation following the method of distribut‐

> ) <sup>∂</sup> *<sup>w</sup>*(*<sup>r</sup>* ' )

<sup>∂</sup> *<sup>r</sup>* ' *dr* ' (2)

<sup>2</sup>*<sup>π</sup> ∫ a a*+l *J*(*r*, *r* '

where the singular kernel *J*(*r*, *r '*) is given by [11]. It provides the normal stress due to an axisymmetric dislocation around a wellbore of radius *a*. Details of this kernel are recalled in

*<sup>p</sup>*(*r*) <sup>=</sup> *<sup>p</sup> <sup>f</sup>* (*r*) - *<sup>σ</sup><sup>h</sup>* <sup>=</sup> *<sup>E</sup>* '

**2.1. Elasticity**

ed dislocations [10]:

Appendix A for completeness.

*<sup>r</sup>* <sup>2</sup> cos <sup>2</sup>*<sup>θ</sup>* (1)

http://dx.doi.org/10.5772/56262

403

*<sup>σ</sup><sup>a</sup>* <sup>=</sup>*σ<sup>h</sup>* - <sup>2</sup>*ν*(*σ<sup>H</sup>* - *<sup>σ</sup>v*) *<sup>a</sup>* <sup>2</sup>

such an approximation, the model is truly axi-symmetric, i.e. independent of *θ*.

In the initial stage of propagation, these transverse fractures can be idealized as axisymmetric (radial) fractures around the wellbore until they hit a stress barrier or other type of heteroge‐ neities. We investigate the initiation and propagation of such a fracture under constant injection of a Newtonian fluid, focusing on the case of "tight" rocks where leak-off is typically negligible. In particular, we are interested in clarifying when the effects associated with the near-wellbore region dissipate and no longer affect the hydraulic fracture propagation and how this transition takes place.

The initiation and the early stage of the propagation of such a hydraulic fracture is affected by two "near wellbore" effects: i) the finiteness of the wellbore and the initial flaw length and ii) a transient phenomenon associated with the release of the fluid stored by compressibility in the wellbore during the pressurization phase prior to the initiation of the fracture. This second effect is ultimately linked to the compressibility of the injection system (i.e. mostly the fluid volume stored by compressibility within the wellbore). These effects have been investigated for the case of plane-strain fractures in [4,5] and for the case of axisymmetric hydraulic fractures driven by an inviscid fluid (i.e. zero-viscosity) in [6]. In this paper, we extend these contribu‐ tions to the case of viscous flow in axi-symmetric fractures.

A detailed dimensional analysis is performed indicating various time scales, length scales and dimensionless numbers controlling the problem. A numerical solver, along the lines of previous contributions [7,8], is presented. A series of numerical simulations are performed in order to study the transition from the near-wellbore propagation regime to the case of a radial hydraulic fracture in an infinite medium under constant injection rate.
