**2. Mathematical formulation**

#### **2.1. Problem definition**

2 Effective and Sustainable Hydraulic Fracturing

depleted reservoir chosen for waste storage).

fracture cross-section as a pressurized Griffith (plane strain) crack.

impacts on the modeling of the classical PKN fracture.

different lengths are evaluated for a water injection project.

addressed elsewhere.

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

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

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

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

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 We consider a pre-existing, un-propped (zero opening) crack of length 2ℓ and height *h* within a linearly elastic, permeable rock characterized by the plane strain modulus *E* ′ and toughness *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 properties.

#### **2.2. Governing equations**

#### *2.2.1. Elasticity equation*

The elasticity equation

$$w(\mathbf{x}, z) = \frac{4\left(p\_f\left(\mathbf{x}\right) - \sigma\_{\text{min}}\right)}{E} \sqrt{\frac{\hbar^2}{4} - z^2} \tag{1}$$

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

$$w(\mathbf{x}) = \frac{2\hbar}{E} \left( p\_f \left( \mathbf{x} \right) - \sigma\_{\rm min} \right). \tag{2}$$

For the particular case of uniformly pressurized fracture, the fracture volume can be evaluated using the elasticity equation (1) as

$$V\_{crack} = \frac{\pi \hbar^2 \ell}{E'} \left(p\_f - \sigma\_{\rm min}\right). \tag{3}$$

633

<sup>√</sup>*πh*/4 [2]. The criterion

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

. (6)

*<sup>S</sup>* , (7)

. (8)

*2.2.3. Propagation condition*

expressed as

The stress intensity factor *KI* <sup>=</sup> <sup>√</sup>*GE*′ associated with the energy release rate *<sup>G</sup>* at the

for the propagation of a PKN fracture in mobile equilibrium (*KI* = *KIc*) can therefore be

*pf* (ℓ) <sup>−</sup> *<sup>σ</sup>min* <sup>=</sup> <sup>2</sup>*KIc*

The Green's function method can be used to solve an inhomogenous differential equation subjected to boundary conditions. For the fluid flow through the porous media, the

*<sup>∂</sup><sup>t</sup>* <sup>−</sup> *<sup>α</sup>*∇<sup>2</sup> *<sup>p</sup>* <sup>=</sup> *<sup>γ</sup>*˙

where *γ*˙ is the fluid source density (the rate of unit volume of injected fluid in a unit volume of material), *S* = *φct* and *α* = *k*/*µφct* are fluid storage and diffusivity coefficients, respectively, expressed in terms of the formation permeability *k*, formation bulk compressibility *ct*, and porosity *φ*. Due to the presence of the impermeable cap rock boundaries at *z* = ±*h*/2 and pressure equilibrium in a vertical cross section , the diffusion problem is two dimensional (2-D). The general 2-D boundary integral for the pressure perturbation due to a distribution of instantaneous sources *g*(*x*, *t*) [*L*/*T*] along a crack *y* = 0,

*∂p*

√*πh*

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

propagating PKN fracture edge is given by *KI* = (*pf* (ℓ) − *<sup>σ</sup>min*)

*2.2.4. Diffusivity equation and boundary integral representation*

diffusivity equation is given by [18]:


*pf*(*x*, *<sup>t</sup>*) − *<sup>p</sup>*<sup>0</sup> =

 *t*  ℓ

*g*(*x* ′ , *t* ′ )

4*πSα*(*t* − *t*

′ )

In this section, we study transient pressurization due to the injection of a fluid at a constant rate of flow into a pre-existing and stationary fracture. In order to facilitate the solution to this problem, we first revisit the fundamental solution to an auxiliary problem of a step pressure increase in crack [1] and introduce a new result for the large time asymptote of this problem. This fundamental solution is then used to formulate and solve a convolution

exp

<sup>−</sup> (*<sup>x</sup>* <sup>−</sup> *<sup>x</sup>*

4*α*(*t* − *t* ′ ) *dx*′ *dt*′

′ )2

−ℓ

integral equation governing the solution for the transient pressurization.

Consider a fracture subjected to a step pressure increase of magnitude *p*∗,

0

**3. Transient pressurization due to fluid injection**

**3.1. Auxiliary problem: step pressure increase**

**Figure 1.** A pre-existing PKN fracture with length 2ℓ and height *h*

#### *2.2.2. Fluid continuity*

#### *Local fluid continuity*

Following [15, 16], lubrication equation can be used to describe the flow of an incompressible fluid in a crack (Figure 1) as follows

$$\frac{\partial w}{\partial t} + \overline{\mathbf{g}}\left(\mathbf{x}, t\right) = \frac{1}{12\mu} \frac{\partial}{\partial \mathbf{x}} \left(w^3 \frac{\partial p\_f}{\partial \mathbf{x}}\right), \text{ } \overline{\mathbf{g}}\left(\mathbf{x}, t\right) = 2\mathbf{g}\left(\mathbf{x}, t\right) \quad \left(t > 0, |\mathbf{x}| < \ell\right), \tag{4}$$

where *g*¯ (*x*, *t*) is the fluid leak-off rate at the crack walls and *µ* is the viscosity of the injected fluid [17].

#### *Global fluid continuity*

The global volume balance of the fluid injected into the fracture is given by:

$$V\_{\text{inject}} = V\_{\text{crack}} + V\_{\text{leak\prime}} \tag{5}$$

in which *Vinject* indicates the cumulative volume of the fluid injected into the fracture and *Vleak* is the cumulative leak-off volume.

#### *2.2.3. Propagation condition*

4 Effective and Sustainable Hydraulic Fracturing

**Figure 1.** A pre-existing PKN fracture with length 2ℓ and height *h*

fluid in a crack (Figure 1) as follows

*<sup>∂</sup><sup>t</sup>* <sup>+</sup> *<sup>g</sup>*¯ (*x*, *<sup>t</sup>*) <sup>=</sup> <sup>1</sup>

*Vleak* is the cumulative leak-off volume.

12*µ*

*∂ ∂x <sup>w</sup>*<sup>3</sup> *<sup>∂</sup>pf ∂x* 

The global volume balance of the fluid injected into the fracture is given by:

Following [15, 16], lubrication equation can be used to describe the flow of an incompressible

where *g*¯ (*x*, *t*) is the fluid leak-off rate at the crack walls and *µ* is the viscosity of the injected

in which *Vinject* indicates the cumulative volume of the fluid injected into the fracture and

, *g*¯ (*x*, *t*) = 2*g* (*x*, *t*) (*t* > 0, |*x*| < ℓ), (4)

*Vinject* = *Vcrack* + *Vleak*, (5)

*2.2.2. Fluid continuity Local fluid continuity*

*∂w*

*Global fluid continuity*

fluid [17].

The stress intensity factor *KI* <sup>=</sup> <sup>√</sup>*GE*′ associated with the energy release rate *<sup>G</sup>* at the propagating PKN fracture edge is given by *KI* = (*pf* (ℓ) − *<sup>σ</sup>min*) <sup>√</sup>*πh*/4 [2]. The criterion for the propagation of a PKN fracture in mobile equilibrium (*KI* = *KIc*) can therefore be expressed as

$$p\_f\left(\ell\right) - \sigma\_{\rm min} = \frac{2K\_{\rm fc}}{\sqrt{\pi \hbar}}.\tag{6}$$

#### *2.2.4. Diffusivity equation and boundary integral representation*

The Green's function method can be used to solve an inhomogenous differential equation subjected to boundary conditions. For the fluid flow through the porous media, the diffusivity equation is given by [18]:

$$\frac{\partial p}{\partial t} - \mathfrak{a}\nabla^2 p = \frac{\dot{\gamma}}{\mathcal{S}}\,'\,\tag{7}$$

where *γ*˙ is the fluid source density (the rate of unit volume of injected fluid in a unit volume of material), *S* = *φct* and *α* = *k*/*µφct* are fluid storage and diffusivity coefficients, respectively, expressed in terms of the formation permeability *k*, formation bulk compressibility *ct*, and porosity *φ*. Due to the presence of the impermeable cap rock boundaries at *z* = ±*h*/2 and pressure equilibrium in a vertical cross section , the diffusion problem is two dimensional (2-D). The general 2-D boundary integral for the pressure perturbation due to a distribution of instantaneous sources *g*(*x*, *t*) [*L*/*T*] along a crack *y* = 0, |*x*| ≤ ℓ is given by [19]

$$p\_f(\mathbf{x}, t) - p\_0 = \int\_0^t \int\_{-\ell}^{\ell} \frac{\mathbf{g}(\mathbf{x}', t')}{4\pi \mathbf{S}a(t - t')} \exp\left(-\frac{(\mathbf{x} - \mathbf{x}')^2}{4a(t - t')}\right) d\mathbf{x}' dt'. \tag{8}$$

#### **3. Transient pressurization due to fluid injection**

In this section, we study transient pressurization due to the injection of a fluid at a constant rate of flow into a pre-existing and stationary fracture. In order to facilitate the solution to this problem, we first revisit the fundamental solution to an auxiliary problem of a step pressure increase in crack [1] and introduce a new result for the large time asymptote of this problem. This fundamental solution is then used to formulate and solve a convolution integral equation governing the solution for the transient pressurization.

#### **3.1. Auxiliary problem: step pressure increase**

Consider a fracture subjected to a step pressure increase of magnitude *p*∗,

$$p(\mathbf{x}, t) - p\_0 = p\_\* H\left(t\right), \quad \left(|\mathbf{x}| < \ell\right) \tag{9}$$

635

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

0.002

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

0.005

Τ = 0.01

Τ = 100, 300, 600, 1000

−<sup>1</sup>

(*τ* ≫ 1), (15)

�1.0 �0.5 0.0 0.5 1.0

�1.0 �0.5 0.0 0.5 1.0

Ξ � x

**Figure 2.** Comparison of the numerical solution for the normalized leak-off rate in the auxiliary problem (step pressure increase

Following [1] we solve (12) numerically for the Laplace image *ψ* (*ξ*,*s*) (−1 ≤ *ξ* ≤ 1 and <sup>10</sup>−<sup>9</sup> ≤ *<sup>s</sup>* ≤ <sup>10</sup>9) using *<sup>N</sup>* = 110 discretization nodes along the fracture, and then apply inverse numerical Laplace transform (Stehfest algorithm [20] with six terms) to tabulate the solution for the normalized leak-off rate *ψ*(*ξ*, *τ*). This solution is contrasted to the small and

The normalized cumulative leak-off from a fracture subjected to a step pressure increase can be obtained by integrating from fluid leak-off rate with respect to time and space and is given

asympt. num.

asympt. num.

 *π* 2 1 − *ξ*<sup>2</sup> ln *ω*2*τ* − 2*γ*

(a)

0

0.0

along the crack) with the small time (a) and large time (b) asymptotes.

*ψ*(*ξ*, *τ*) ≈

large time asymptotes in Figure 2.

where *ω* = 2.67.

by [1] :

0.1

Ψ � gx,t 2Π �p S

0.2

0.3

0.4

Α

0.5

(b)

2

Ψ � gx,t 2Π �p S

4

6

8

Α

10

where *H* (*t*) is a Heaviside function. To facilitate solution of (8) with (9), we rewrite it in the normalized form

$$1 = \frac{1}{2} \int\_{0}^{\tau} \int\_{-1}^{1} \psi(\underline{\xi}', \underline{\tau}') \exp\left(-\frac{(\underline{\xi} - \underline{\xi}')^2}{\underline{\tau} - \underline{\tau}'}\right) \frac{d\underline{\xi}' d\tau'}{\underline{\tau} - \underline{\tau}'},\tag{10}$$

where the nondimensional time (*τ*), coordinate (*ξ*), leak-off rate (*ψ*), and cumulative leak-off volume (Φ) are defined as

$$\tau = \frac{t}{t\_\*}, \quad \tilde{\xi} = \frac{\mathbf{x}}{\ell}, \text{ } \psi(\tilde{\xi}, \tau) = \frac{\ell \mathcal{g}\left(\mathbf{x}, t\right)}{2\pi a S p\_\*}, \text{ } \Phi(\tau) = \frac{2V\_{leak}(t)}{\pi \ell^2 h S p\_\*},\tag{11}$$

and *t*<sup>∗</sup> = ℓ2/4*α* is diffusion timescale. After applying Laplace transform, (10) becomes:

$$1/s = \int\_{-1}^{1} \psi(\underline{\zeta}', s) K\_0 \left(2\sqrt{s}|\underline{\zeta} - \underline{\zeta}'|\right) d\underline{\zeta}',\tag{12}$$

where *K*<sup>0</sup> is the modified Bessel function of the second kind, *s* is the Laplace transform parameter and *ψ*(*ξ*,*s*) is the Laplace image of *ψ*(*ξ*, *τ*).

Before integral convolution equation (12) is treated numerically, it is useful to consider its asymptotics for short and long injection times.

During the injection process when the characteristic lengthscale for fluid diffusion <sup>√</sup>*α<sup>t</sup>* is small compared to the crack size ℓ, or in terms of the normalize time, *τ* ≪ 1, the fluid diffusion pattern is approximately 1-D and the normalized leak-off rate is given by [1]:

$$
\psi(\tau) = \frac{2}{\pi^{3/2}\sqrt{\tau}} \quad (\tau \ll 1). \tag{13}
$$

As the injection time increases, the 1-D fluid diffusion pattern is no longer valid and a 2-D fluid diffusion pattern must be considered. We can show that for long enough injection times the Laplace image of the fluid leak-off rate is given by:

$$\psi(\mathfrak{f},s) = \left(-\frac{\pi}{2}\sqrt{1-\mathfrak{f}^2}s\left[\ln\left(s/4\right)+2\gamma\right]\right)^{-1} \quad (\tau \gg 1),\tag{14}$$

where *γ* = 0.5772 is the Euler's constant. The approximate image of (14) in actual time domain is:

**Figure 2.** Comparison of the numerical solution for the normalized leak-off rate in the auxiliary problem (step pressure increase along the crack) with the small time (a) and large time (b) asymptotes.

$$\Psi(\xi,\tau) \approx \left(\frac{\pi}{2}\sqrt{1-\xi^2}\left(\ln\left(\omega^2\tau\right)-2\gamma\right)\right)^{-1} \quad (\tau \gg 1),\tag{15}$$

where *ω* = 2.67.

6 Effective and Sustainable Hydraulic Fracturing

<sup>1</sup> <sup>=</sup> <sup>1</sup> 2 *τ*

*<sup>τ</sup>* <sup>=</sup> *<sup>t</sup> t*∗ 0

, *<sup>ξ</sup>* <sup>=</sup> *<sup>x</sup>* ℓ

> 1/*s* = 1

parameter and *ψ*(*ξ*,*s*) is the Laplace image of *ψ*(*ξ*, *τ*).

the Laplace image of the fluid leak-off rate is given by:

 −*π* 2 

*ψ*(*ξ*,*s*) =

domain is:

asymptotics for short and long injection times.

−1

*ψ*(*ξ* ′ ,*s*)*K*<sup>0</sup> 2 √

*<sup>ψ</sup>*(*τ*) = <sup>2</sup>

 1

*ψ*(*ξ* ′ , *τ* ′ ) exp 

−1

normalized form

volume (Φ) are defined as

*<sup>p</sup>*(*x*, *<sup>t</sup>*) − *<sup>p</sup>*<sup>0</sup> = *<sup>p</sup>*<sup>∗</sup> *<sup>H</sup>* (*t*), (|*x*| < ℓ) (9)

 *dξ* ′ *dτ* ′

, <sup>Φ</sup>(*τ*) = <sup>2</sup>*Vleak*(*t*)

*π*ℓ2*hSp*<sup>∗</sup>

*<sup>π</sup>*3/2√*<sup>τ</sup>* (*<sup>τ</sup>* <sup>≪</sup> <sup>1</sup>). (13)

−<sup>1</sup>

*<sup>τ</sup>* <sup>−</sup> *<sup>τ</sup>*′ , (10)

, (11)

, (12)

(*τ* ≫ 1), (14)

where *H* (*t*) is a Heaviside function. To facilitate solution of (8) with (9), we rewrite it in the

where the nondimensional time (*τ*), coordinate (*ξ*), leak-off rate (*ψ*), and cumulative leak-off

, *<sup>ψ</sup>*(*ξ*, *<sup>τ</sup>*) = <sup>ℓ</sup>*<sup>g</sup>* (*x*, *<sup>t</sup>*)

and *t*<sup>∗</sup> = ℓ2/4*α* is diffusion timescale. After applying Laplace transform, (10) becomes:

where *K*<sup>0</sup> is the modified Bessel function of the second kind, *s* is the Laplace transform

Before integral convolution equation (12) is treated numerically, it is useful to consider its

During the injection process when the characteristic lengthscale for fluid diffusion <sup>√</sup>*α<sup>t</sup>* is small compared to the crack size ℓ, or in terms of the normalize time, *τ* ≪ 1, the fluid diffusion pattern is approximately 1-D and the normalized leak-off rate is given by [1]:

As the injection time increases, the 1-D fluid diffusion pattern is no longer valid and a 2-D fluid diffusion pattern must be considered. We can show that for long enough injection times

1 − *ξ*2*s*[ln (*s*/4) + 2*γ*]

where *γ* = 0.5772 is the Euler's constant. The approximate image of (14) in actual time

2*παSp*<sup>∗</sup>

*s*|*ξ* − *ξ* ′ | *dξ* ′

<sup>−</sup>(*<sup>ξ</sup>* <sup>−</sup> *<sup>ξ</sup>* ′ )2 *τ* − *τ*′

> Following [1] we solve (12) numerically for the Laplace image *ψ* (*ξ*,*s*) (−1 ≤ *ξ* ≤ 1 and <sup>10</sup>−<sup>9</sup> ≤ *<sup>s</sup>* ≤ <sup>10</sup>9) using *<sup>N</sup>* = 110 discretization nodes along the fracture, and then apply inverse numerical Laplace transform (Stehfest algorithm [20] with six terms) to tabulate the solution for the normalized leak-off rate *ψ*(*ξ*, *τ*). This solution is contrasted to the small and large time asymptotes in Figure 2.

> The normalized cumulative leak-off from a fracture subjected to a step pressure increase can be obtained by integrating from fluid leak-off rate with respect to time and space and is given by [1] :

$$\Phi(\tau) = \int\_0^{\tau} \int\_{-1}^1 \Psi\left(\xi^{'}, \tau^{'}\right) d\xi^{'} d\tau^{'},\tag{16}$$

637

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

2-D 1-D

<sup>2</sup>*<sup>η</sup>* <sup>+</sup> *<sup>s</sup>*Φ(*s*) , (21)

.

, (22)

. (The 1-D

� 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1

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

B A

<sup>10</sup>�<sup>4</sup> 0.01 <sup>1</sup> <sup>100</sup> <sup>104</sup> <sup>10</sup>�<sup>4</sup>

**Figure 3.** Evolution of the fluid net-pressure during a constant-rate injection into a stationary crack within an

in which Φ(*s*) is the Laplace image of Φ(*τ*). This solution is then numerically inverted to the

Evolution of the normalized pressure Π during the transient pressurization of a crack is shown in Figure 3 for the case of an abnormally pressurized reservoir <sup>Σ</sup><sup>0</sup> ≈ 0 (*p*<sup>0</sup> ≈ *<sup>σ</sup>min*)

With the solution for the normalized pressure in hand, the onset of the fracture propagation

Consider an example of the fracture breakdown calculations for a water injection project in a sandstone formation [21] characterized by porosity *φ* = 0.1, permeability *k* = 10.132 md, pre-existing fracture height *h* = 30.48 m (assumed to span the height of the sandstone layer), minimum in-situ stress *<sup>σ</sup>min* <sup>=</sup> 28.8 MPa, bulk rock compressibility *ct* <sup>=</sup> 5.35 <sup>×</sup> <sup>10</sup>−<sup>10</sup> Pa-1, fluid viscosity *µ* = 1 cp, rock toughness *KIc* = 1 MPa m1/2, plane strain modulus

= 9.3 GPa. The reported injection rate was *Q*<sup>0</sup> = 0.00052 m3/s. The calculated values are *<sup>S</sup>* = 5.35 × <sup>10</sup>−<sup>11</sup> Pa-1 (storage parameter),*<sup>α</sup>* = 0.19 m2/s (diffusivity coefficient), *<sup>p</sup>*<sup>∗</sup> = *Q*0/(*πhSα*) = 0.537 MPa (characteristic pressure perturbation). The normalized breakdown

<sup>√</sup>*πhSαKIc Q*<sup>0</sup>

Τ � 4 Α t <sup>2</sup>

1 − 2Σ0*s*2Φ(*s*)

hSE'

abnormally-pressurized reservoir *<sup>p</sup>*<sup>0</sup> <sup>≈</sup> *<sup>σ</sup>min* for various values of the crack height-to-length ratio *<sup>h</sup>*/ℓ*SE*′

<sup>Π</sup>(*s*) = <sup>1</sup>

2*s*<sup>2</sup>

and for various values of the scaled crack height-to-length ratio *<sup>η</sup>* <sup>=</sup> *<sup>h</sup>*/ℓ*SE*′

diffusion solution to the same problem is shown by dashed lines for comparison).

<sup>Π</sup> <sup>=</sup> <sup>Π</sup>*<sup>B</sup>* with <sup>Π</sup>*<sup>B</sup>* <sup>=</sup> <sup>2</sup>

0.001

time domain using the Stehfest algorithm [20].

can be determined from the normalized form of (6):

where Π*<sup>B</sup>* is the normalized breakdown pressure.

*Example. Water injection project*

*E* ′ ��Π h S Α

*pf*�Σmin

*Q*0

0.01

0.1

1

#### **3.2. Transient pressurization problem**

Assuming a uniform pressure along the crack channel (the viscous pressure drop in the crack is negligible), and "instantaneous" pressure build-up at the beginning of injection from the initial pore pressure value *p*<sup>0</sup> to the value *pf* = *σmin* corresponding to the incipient crack opening, the cumulative leak-off volume *Vleak* can be obtained by the applying the Duhamel's theorem [1, 19]

$$V\_{leak} = \nu(t)(\sigma\_{min} - p\_0) + \int\_{0^+}^{t} \nu(t - t') \frac{dp\_f}{dt'} dt',\tag{17}$$

where *<sup>ν</sup>*(*t*) = *<sup>π</sup>* <sup>2</sup> <sup>ℓ</sup>2*hS*Φ(*t*/*t*∗) is the cumulative leak-off volume of the fracture subjected to a unit step pressure increase, as discussed in the previous section.

Equation (5) can be expressed in the case of fluid injection at a constant rate *Q*<sup>0</sup> as

$$Q\_0 t = \frac{\pi \hbar^2 \ell}{E} (p\_f(t) - \sigma\_{\rm min}) + \nu(t)(\sigma\_{\rm min} - p\_0) + \int\_{0^+}^t \nu(t - t') \frac{dp\_f}{dt'} dt',\tag{18}$$

where expression (3) for *Vcrack* was used. Let us now define a characteristic pressure perturbation *p*<sup>∗</sup> = *Q*0/(*πhSα*), which is then used to scale the net pressure and the initial effective stress

$$
\Pi = \frac{p\_f - \sigma\_{\min}}{p\_\*}, \qquad \Sigma\_0 = \frac{\sigma\_{\min} - p\_0}{p\_\*} \tag{19}
$$

respectively. Using normalized parameters (11) and (19), we convert (18) to the nondimensional form:

$$\frac{1}{4} = \eta \Pi(\tau) + \frac{1}{2} \Sigma\_0 \Phi(\tau) + \frac{1}{2} \int\_{0^+}^{\tau} \Phi(\tau - \tau') \frac{d\Pi}{d\tau'} d\tau', \qquad \left(\eta = \frac{h}{\ell SE}\right) \tag{20}$$

where Φ(*τ*) is the normalized cumulative leak-off rate in the auxiliary problem, (16), and *η* is a scaled crack height-to-length ratio. Applying the Laplace transform to (20) yields the solution for the Laplace image of the normalized pressure in the crack:

**Figure 3.** Evolution of the fluid net-pressure during a constant-rate injection into a stationary crack within an abnormally-pressurized reservoir *<sup>p</sup>*<sup>0</sup> <sup>≈</sup> *<sup>σ</sup>min* for various values of the crack height-to-length ratio *<sup>h</sup>*/ℓ*SE*′ .

$$\Pi(s) = \frac{1}{2s^2} \frac{1 - 2\Sigma\_0 s^2 \Phi(s)}{2\eta + s\Phi(s)},\tag{21}$$

in which Φ(*s*) is the Laplace image of Φ(*τ*). This solution is then numerically inverted to the time domain using the Stehfest algorithm [20].

Evolution of the normalized pressure Π during the transient pressurization of a crack is shown in Figure 3 for the case of an abnormally pressurized reservoir <sup>Σ</sup><sup>0</sup> ≈ 0 (*p*<sup>0</sup> ≈ *<sup>σ</sup>min*) and for various values of the scaled crack height-to-length ratio *<sup>η</sup>* <sup>=</sup> *<sup>h</sup>*/ℓ*SE*′ . (The 1-D diffusion solution to the same problem is shown by dashed lines for comparison).

With the solution for the normalized pressure in hand, the onset of the fracture propagation can be determined from the normalized form of (6):

$$
\Pi = \Pi\_B \quad \text{with} \quad \Pi\_B = \frac{2\sqrt{\pi h} S a K\_{Ic}}{Q\_0} \,\text{s}\tag{22}
$$

where Π*<sup>B</sup>* is the normalized breakdown pressure.

#### *Example. Water injection project*

8 Effective and Sustainable Hydraulic Fracturing

**3.2. Transient pressurization problem**

*<sup>Q</sup>*0*<sup>t</sup>* <sup>=</sup> *<sup>π</sup>h*2<sup>ℓ</sup>

theorem [1, 19]

where *<sup>ν</sup>*(*t*) = *<sup>π</sup>*

effective stress

nondimensional form:

*τ*

<sup>4</sup> <sup>=</sup> *<sup>η</sup>*<sup>Π</sup> (*τ*) <sup>+</sup>

1 2 Φ(*τ*) =

*Vleak* = *<sup>ν</sup>*(*t*)(*σmin* − *<sup>p</sup>*0) +

Equation (5) can be expressed in the case of fluid injection at a constant rate *Q*<sup>0</sup> as

*<sup>E</sup>*′ (*pf*(*t*) <sup>−</sup> *<sup>σ</sup>min*) + *<sup>ν</sup>*(*t*)(*σmin* <sup>−</sup> *<sup>p</sup>*0) +

<sup>Π</sup> <sup>=</sup> *pf* <sup>−</sup> *<sup>σ</sup>min p*∗

<sup>Σ</sup>0Φ(*τ*) + <sup>1</sup>

solution for the Laplace image of the normalized pressure in the crack:

2 *τ*

0+

where expression (3) for *Vcrack* was used. Let us now define a characteristic pressure perturbation *p*<sup>∗</sup> = *Q*0/(*πhSα*), which is then used to scale the net pressure and the initial

respectively. Using normalized parameters (11) and (19), we convert (18) to the

Φ(*τ* − *τ* ′ ) *d*Π *<sup>d</sup>τ*′ *<sup>d</sup><sup>τ</sup>* ′ ,

where Φ(*τ*) is the normalized cumulative leak-off rate in the auxiliary problem, (16), and *η* is a scaled crack height-to-length ratio. Applying the Laplace transform to (20) yields the

unit step pressure increase, as discussed in the previous section.

 *τ* 0

 <sup>1</sup> −1 *ψ ξ* ′ , *τ* ′ *dξ* ′ *dτ* ′

Assuming a uniform pressure along the crack channel (the viscous pressure drop in the crack is negligible), and "instantaneous" pressure build-up at the beginning of injection from the initial pore pressure value *p*<sup>0</sup> to the value *pf* = *σmin* corresponding to the incipient crack opening, the cumulative leak-off volume *Vleak* can be obtained by the applying the Duhamel's

> *t*

*ν*(*t* − *t* ′ ) *dpf dt*′ *dt*′

> *t*

*ν*(*t* − *t* ′ ) *dpf dt*′ *dt*′

> *<sup>η</sup>* <sup>=</sup> *<sup>h</sup>* ℓ*SE*′

0+

*p*∗

<sup>2</sup> <sup>ℓ</sup>2*hS*Φ(*t*/*t*∗) is the cumulative leak-off volume of the fracture subjected to a

, <sup>Σ</sup><sup>0</sup> <sup>=</sup> *<sup>σ</sup>min* <sup>−</sup> *<sup>p</sup>*<sup>0</sup>

0+

, (16)

, (17)

, (18)

(19)

(20)

Consider an example of the fracture breakdown calculations for a water injection project in a sandstone formation [21] characterized by porosity *φ* = 0.1, permeability *k* = 10.132 md, pre-existing fracture height *h* = 30.48 m (assumed to span the height of the sandstone layer), minimum in-situ stress *<sup>σ</sup>min* <sup>=</sup> 28.8 MPa, bulk rock compressibility *ct* <sup>=</sup> 5.35 <sup>×</sup> <sup>10</sup>−<sup>10</sup> Pa-1, fluid viscosity *µ* = 1 cp, rock toughness *KIc* = 1 MPa m1/2, plane strain modulus *E* ′ = 9.3 GPa. The reported injection rate was *Q*<sup>0</sup> = 0.00052 m3/s. The calculated values are *<sup>S</sup>* = 5.35 × <sup>10</sup>−<sup>11</sup> Pa-1 (storage parameter),*<sup>α</sup>* = 0.19 m2/s (diffusivity coefficient), *<sup>p</sup>*<sup>∗</sup> = *Q*0/(*πhSα*) = 0.537 MPa (characteristic pressure perturbation). The normalized breakdown 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 MPa). We chose two arbitrary fracture half-lengths <sup>ℓ</sup> <sup>=</sup> 100 m (*h*/ℓ*SE*′ = 0.61) and ℓ = 1000 m (*h*/ℓ*SE*′ = 0.061) to estimate the onset of fracture propagation from Figure 3 to be at *τ* = 2.5 (point A) and *τ* = 1.22 (point B), respectively. The corresponding dimensional breakdown times are 9 hrs (ℓ = 100 m) and 19 days (ℓ = 1000 m).

10.5772/56474

639

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

**References**

*Mech.*, 2013.

2003.

*Gas Control*, 1(62-68), 2007.

26(6):1235–1248, 1990.

222:937–949, 1961.

112:224–230, 1990.

1972. (SPE 3009).

[1] E. Detournay and A.H-D. Cheng. Plane strain analysis of a stationary hydraulic fracture

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

[2] D.I. Garagash. Fluid-driven crack tunneling in a layer. *to be submitted to ASME J. Appl.*

[3] J.B. Osborne and D.I. Garagash. Using CO2 as a fracture fluid in tight reservoirs.

[4] S. Bachu. Screening and ranking of sedimentary basins for sequestration of CO2 in geological media in response to climate change. *Environmental Geology*, 44(277–289),

[5] J. Bradshaw, S. Bachu, D. Bonijoly, R. Burruss, S. Holloway, N.P. Christensen, and O.M. Mathiassen. CO2 storage capacity estimation: issues and development. *Int. J. Greenhouse*

[6] A. Settari and G.M. Warren. Stimulation and field analysis of waterflood induced

[7] K. Pruess, J. S. Y. Wang, and Y. W. Tsang. On thermohydrologic conditions near high-level nuclear wastes emplaced in partially saturated fractured tuff: 1. simultion studies with explicit consideration of fracture effects. *Water Resources Res.*,

[8] T.K. Perkins and L.R. Kern. Widths of hydraulic fractures. *J. Pet. Tech., Trans. AIME*,

[9] E. Detournay, A.H-D. Cheng, and J.D. McLennan. A poroelastic PKN hydraulic fracture model based on an explicit moving mesh algorithm. *ASME J. Energy Res. Tech.*,

[10] Y. Kovalyshen and E. Detournay. A reexamination of the classical PKN model of

[11] Y. N. Gordayev and A. F. Zazovsky. Self-similar solution for deep-penetrating hydraulic

[12] Y. N. Gordayev and V. M. Entov. The pressure distribution around a growing crack. *J.*

[14] R.P. Nordgren. Propagation of vertical hydraulic fractures. *J. Pet. Tech.*, 253:306–314,

[15] D.A. Spence, P.W. Sharp, and D.L. Turcotte. Buoyancy-driven crack propagation: a

hydraulic fracture. *Transport in Porous Media*, 81:317–339, January 2009.

[13] Y. Kovalyshen. *Fluid-Driven Fracture in Poroelastic Medium*. PhD thesis, 2009.

mechanism for magma migration. *J. Fluid Mech.*, 174:135–153, 1987.

fracture propagation. *Transport in Porous Media*, 7:283–304, 1992.

*Appl. Maths Mechs.*, 61(6):1025–1029, 1997.

in a poroelastic medium. *Int. J. Solids Structures*, 27(13):1645–1662, 1991.

Technical report, Petroleum Technology Research Centre, 2012.

fracturing. Society of Petroleum Engineers, 1994.
