**Author details**

Erfan Sarvaramini and Dmitry I. Garagash

Dalhousie University, Department of Civil and Resource Engineering, Halifax, Nova Scotia, Canada

#### **References**

10 Effective and Sustainable Hydraulic Fracturing

m (*h*/ℓ*SE*′

**4. Conclusions**

are reported elsewhere.

**Acknowledgments**

**Author details**

Canada

net-pressure, (22), is <sup>Π</sup>*<sup>B</sup>* = 0.38 (or, in dimensional terms, (*pf* − *<sup>σ</sup>min*)*<sup>B</sup>* = *<sup>p</sup>*∗Π*<sup>B</sup>* = 0.204

*τ* = 2.5 (point A) and *τ* = 1.22 (point B), respectively. The corresponding dimensional

Important applications of injection of a low viscosity fluid into a pre-existing fracture, such as waterflooding and supercritical CO2 injection in geological sequestration, necessitate comprehensive studies of mechanical and hydraulically properties of fractures from the beginning of injection until the onset of fracture propagation. In this study, we considered a low viscosity fluid injection into a pre-existing, un-propped crack of a PKN geometry. We focus on the case of a critically-overpressured reservoir and initially closed (un-propped) crack. The extension of this work to propped cracks and more general reservoir conditions

The analysis assumes negligible viscous dissipation during injection of a low viscosity fluid at a sufficiently slow injection rate [22], and, as a result, approximately uniform pressure distribution in the crack. Furthermore, the poroelastic effects are also neglected in this study. To outline the validity of the latter assumption, we can show that the later stages of transient pressurization (the so-called leak-off dominated regime when the injection time ≫ diffusion timescale ℓ2/4*α*) with and without poroelasticity effects are identical. However, the generated poroelastic backstress which tends to close the fracture may cause a delay in the initiation of crack propagation when compared to the case where poroelastic effects are neglected. In addition, for certain ranges of fracture and fluid properties and field operating condition the backstress may become large enough to prevent the fracture from propagation indefinitely. We evaluated the evolution of the fluid pressure inside the fracture during the transient pressurization by considering 2-D fluid diffusion from the fracture into the surrounding porous rock. As the fracture is pressurized, the condition for the onset of its propagation (breakdown condition) is eventually reached. We quantified how the fracture breakdown condition depends upon the rock and fluid properties, the in-situ stress and the fluid injection rate. The history of the transient pressurization prior to breakdown can be used to provide

We acknowledge support for this work from the Natural Science and Engineering Research

Dalhousie University, Department of Civil and Resource Engineering, Halifax, Nova Scotia,

= 0.061) to estimate the onset of fracture propagation from Figure 3 to be at

= 0.61) and ℓ = 1000

MPa). We chose two arbitrary fracture half-lengths <sup>ℓ</sup> <sup>=</sup> 100 m (*h*/ℓ*SE*′

breakdown times are 9 hrs (ℓ = 100 m) and 19 days (ℓ = 1000 m).

the initial conditions for the fracture propagation problem.

Council of Canada under Discovery Grant No. 371606.

Erfan Sarvaramini and Dmitry I. Garagash


[16] J. R. Lister. Buoyancy-driven fluid fracture: The effects of material toughness and of low-viscosity precursors. *J. Fluid Mech.*, 210:263–280, 1990.

**Chapter 31**

. It

**Modified Formulation, ε-Regularization and the**

**Efficient Solution of Hydraulic Fracture Problems**

Analytical models and numerical simulation are important means to increase understanding and to enhance efficiency of hydraulic fracturing. The reasons for developing and using them are clearly explained, for instance, by Mack and Warpinski [1]. Thus, there is no need to dwell on them. Rather we focus on improving analytical and numeric methods used to the date. Our objective is to suggest new approaches for developing accurate, robust and

The approaches discussed in the paper stem from the fact [2,3] that the conventional formu‐ lation of the hydraulic fracture problem (for example, see [7]), when neglecting the lag and fixing the position of the fracture front at a time step, is ill-posed. This feature has not been reported for more than three decades of studying hydraulic fractures because of two rea‐ sons. Numerical simulators, based on the conventional formulation (for example, see [7,8]), employ quite rough meshes, which themselves serve as specific 'regularizators'. On the oth‐ er hand, rare solutions of model problems either also employed rough meshes [9], or they were obtained by solving the initial value (Cauchy) problem [10-12] rather than the boun‐ dary value problem (a discussion of the difference may be found in references [3,4]). The disclosure of the mentioned fact has led to (i) explicit formulation of the speed equation (SE)

hydraulic fractures and (iii) distinguishing the particle velocity as a preferable variable2

1 To the authors'knowledge, Kemp [11] was the first who clearly distinguished the speed equation when revisiting the Nordgren's problem. When introducing the SE, numbered (5) in his paper, Kemp wrote (p. 289): "Nowhere is (5) mentioned. (5) is called the Stefan condition and is always present in moving boundary problems". Kemp used the SE to solve the Nordgren's problem as an initial value rather than boundary value problem. This excluded solving ill-posed

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

, (ii) comprehension of its significance for proper numerical simulation of

© 2013 Linkov and Mishuris; 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,

stable simulators on the basis of recent analytical and computational findings [2-6].

Alexander M. Linkov and Gennady Mishuris

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/56218

**1. Introduction**

in its general form1

boundary value problem.

