**Abstract**

22 Effective and Sustainable Hydraulic Fracturing

628 Effective and Sustainable Hydraulic Fracturing

*Min. Sci.*, 26(6):507–513, 1989.

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*Formation Damage, The Hague, The Netherlands*, May 2001. (SPE 68974).

The aim of the present work is to investigate injection of a low-viscosity fluid into a pre-existing fracture within a linear elastic, permeable rock, as may occur in waterflooding and supercritical CO2 injection. In conventional hydraulic fracturing, high viscosity and cake building properties of injected fluid limit diffusion to a 1-D boundary layer incasing the crack. In the case of injection of low viscosity fluid into a fracture, diffusion will take place over wider range of scales, from 1-D to 2-D, thus, necessitating a new approach. In addition, the dissipation of energy associated with fracturing of the rock dominates the energy expended to flow a low viscosity fluid into the crack channel. As a result, the rock fracture toughness is an important parameter in evaluating the propagation driven by a low-viscosity fluid. We consider a pre-existing, un-propped, stationary Perkins, Kern and Nordgren's (PKN) fracture into which a low viscosity fluid is injected under a constant flow rate. The fundamental solution to the auxiliary problem of a step pressure increase in a fracture [1] is used to formulate and solve the convolution integral equation governing the transient crack pressurization under the assumption of negligible viscous dissipation. The propagation criterion for a PKN crack [2] is then used to evaluate the onset of propagation. The obtained solution for transient pressurization of a stationary crack provides initial conditions to the fracture propagation problem.

### **1. Introduction**

The problem of injection of a low-viscosity fluid into a pre-existing fracture may arise in several rock engineering areas, such as, injection of liquid waste (e.g., supercritical CO2) into deep geological formations for storage [3,4,5], waterflooding process to increase recovery from an oil reservoir [6], and control of possible leaks from pre-existing fractures around

<sup>©2012</sup> Sarvaramini and Garagash, licensee InTech. This is an open access chapter 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 Sarvaramini and Garagash; 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.

2 Effective and Sustainable Hydraulic Fracturing

radioactive and nuclear wastes storage sites [7]. These fractures could be either of natural origin or man-made (e.g., hydraulic fractures used to stimulate production from a now depleted reservoir chosen for waste storage).

10.5772/56474

631

and toughness

′

http://dx.doi.org/10.5772/56474

<sup>4</sup> <sup>−</sup> *<sup>z</sup>*2, (1)

. (2)

. (3)

**2. Mathematical formulation**

We consider a pre-existing, un-propped (zero opening) crack of length 2ℓ and height *h* within

Pressurization of a PKN Fracture in a Permeable Rock During Injection of a Low Viscosity Fluid

*KIc* (Figure 1). The crack is aligned perpendicular to the minimum in-situ stress *σmin* and is loaded internally by fluid pressure *pf* , generated by the fluid injection at the crack center at a constant rate *Qo*. The following assumptions are used in this work. 1) The crack height is small compared to the length, such that the deformation field in any vertical cross-section that is not immediately close to the crack edges (*x* = ±ℓ) is approximately plane-strain, and the fluid pressure is equilibrated within a vertical crack cross-section (the PKN assumptions). 2) The minimum in-situ stress *σmin* and the initial reservoir pore pressure *p*<sup>0</sup> are uniform along the crack. 3) Initial reservoir pore pressure *p*<sup>0</sup> is approximately equal to the minimum in-situ stress *σmin*, allowing the crack to open immediately upon the start of the injection; or alternatively, time *t*<sup>0</sup> from the onset of injection that is required to pressurize the initially closed crack (*p*<sup>0</sup> < *<sup>σ</sup>min*) to the point of incipient opening (*pf*(*t*0) = *<sup>σ</sup>min*) is small compared to the timescale of interest (e.g., the time to the onset of the fracture propagation). 4) Injected fluid is of a low viscosity (and/or the rate of injection is slow), such that the viscous pressure drop in the crack is negligible, or, in other words, the fluid pressure is uniform in the crack. 5) The crack is confined between two impermeable layers, which, together with the assumption of pressure equilibrium within a vertical crack cross-section, suggests a 2-D fluid diffusion within the permeable rock layer. 6) The injected and reservoir fluid have similar rheological

a linearly elastic, permeable rock characterized by the plane strain modulus *E*

**2.1. Problem definition**

properties.

**2.2. Governing equations**

*w*(*x*, *z*) =

opening of PKN fracture at mid height (*z* = 0) is

evaluated using the elasticity equation (1) as

4 

*<sup>w</sup>*(*x*) = <sup>2</sup>*<sup>h</sup> E*′ 

*Vcrack* <sup>=</sup> *<sup>π</sup>h*2<sup>ℓ</sup>

*E*′ 

*pf* (*x*) <sup>−</sup> *<sup>σ</sup>min E*′

is used to relate the opening of a PKN fracture *<sup>w</sup>* to the net pressure *pf* − *<sup>σ</sup>min*, which is assumed to be equilibrated in a vertical cross-section of the crack, *∂pf* /*∂z* = 0, [14]. The

For the particular case of uniformly pressurized fracture, the fracture volume can be

*pf* (*x*) <sup>−</sup> *<sup>σ</sup>min*

*pf* <sup>−</sup> *<sup>σ</sup>min*

*h*<sup>2</sup>

*2.2.1. Elasticity equation* The elasticity equation

This paper attempts to study injection of a low viscosity fluid into a pre-existing un-propped fracture of the Perkins and Kern and Nordgren (PKN) geometry within a linearly elastic, permeable rock. In the classical PKN model, the fracture length is much larger than the fracture height [8] with the latter confined to a permeable (reservoir) layer sandwiched between two impermeable (cap) rock layers. This assumption allows to model a vertical fracture cross-section as a pressurized Griffith (plane strain) crack.

Until recently, in part due to the lack of a reliable fracture breakdown criterion for a PKN fracture, studies of the PKN fracture propagation have been bounded to the limiting regime corresponding to the dominance of the viscous dissipation in the fluid flow in the crack channel, i.e. when the rock toughness can be neglected [9, 10]. This particular dissipation regime is favored when a high viscosity fracturing fluid and/or high injection rates are used, or at late stages of fracture growth (long fractures). Moreover, for sufficiently large time, the history of injection prior to the onset of the viscosity dominated regime may have minor impacts on the modeling of the classical PKN fracture.

In unconventional hydraulic fracturing (injection of a low-viscosity fluid), on one hand, the dissipation of energy to extend the fracture in the rock may not be negligible compared to the viscous dissipation. On the other hand, the injection history prior to the onset of propagation may not be neglected. With this in mind, we investigate fluid injection into a stationary, pre-existing fracture up to the onset of the propagation, which is defined by the recently introduced propagation criteria for a PKN fracture [2]. The corresponding transient pressurization and leak-off history prior to the breakdown will provide initial conditions for the problem of a propagating PKN fracture in the toughness dominated regime, to be addressed elsewhere.

Contrary to conventional hydraulic fracture where high viscosity and cake-building properties of injected fluid limit the leak-off to a 1-D boundary layer incasing the crack, the low viscosity fluid allows for diffusion over a wider range of scales from 1-D to 2-D. Although, several investigations looked at the propagation of a fracture driven by a low viscosity fluid, when fluid diffusion is fully two-dimensional [11, 12, 13], the study of injection into a stationary, pre-existing fracture has not yet received due attention. One of the foci of this study is to identify solutions corresponding to the limiting cases of the small and large injection time (1-D and fully-developed 2-D diffusion, respectively), and the solution in the intermediate regime corresponding to the evolution between the two limiting cases.

This paper is organized as follows. In section 2 we define and formulate the problem. In Section 3 we first revisit the problem of a step pressure increase in a crack [1], which we then use to formulate and solve the problem of transient pressurization of a crack due to a constant rate of fluid injection. The criterion of PKN propagation [2] is used to evaluate the onset of the fracture propagation. We illustrate the results of this study by considering a case study in which the transient pressurization and the breakdown of pre-existing fractures of different lengths are evaluated for a water injection project.
