**Abstract**

We investigated the problem of a hydraulic fracture propagation through a weakly cohesive frictional discontinuity for different conditions of fracture toughness, in situ stresses, fracture intersection angle, injection parameters and permeability of the pre-existing fracture. The parametric sensitivity of the fracture interaction process, in terms of crossing versus arresting of the hydraulic fracture at the discontinuity, was performed using numerical simulations through an extensive parameter space representative of hydraulic fracturing field conditions. The effect of the pre-existing fracture permeability on the crossing behavior was analyzed using a simple analytical model. We showed that the injection rate and viscosity of fracturing fluid are the key parameters controlling the crossing/non-crossing interaction behavior, in addition to already known fracture interaction angle and in-situ stress parameters. We have also found that the pre-existing fracture hydraulic aperture, when as large as that of the hydraulic fracture aperture, has significant influence on the interaction and may more likely cause the hydraulic fracture to arrest.

### **1. Introduction**

The main function of a hydraulic fracture (HF) treatment is to effectively increase reservoir permeability and drainage by creating one or more conductive fractures that connect to the wellbore [1-2]. The stimulation treatments are especially necessary in low-permeability unconventional source rocks such as shales, which are not economical without fracturing [3] and sometimes even subsequent refracturing [4]. The modeling of HF propagation is important

© 2013 Chuprakov et al.; 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.

to the design of the treatment and the ability to evaluate post-treatment production. Most fracture propagation models assume an oversimplified single planar geometry of fracture propagation [1, 5]. However, in highly heterogeneous and naturally fractured formations, the geometry of HFs can be complex because of their interaction with preexisting discontinuities in the rock, such as natural fractures, faults, and bedding interfaces [3, 6-7], which will be referred as NFs. For example, it is well known that an HF can be arrested by an NF or can reinitiate after the contact [8-16]. The result of the interaction depends on the in-situ stresses [10, 15, 17-18]; friction [12], cohesion [19], and permeable properties of the NF [20]; the rheological properties of the injecting fluid; and injection flow rate [21-22].

fractures (perpendicular to the plane of Fig. 1) are far more comparable to or significantly exceed the horizontal size of the fractures affected by the interaction (in a plane of Fig. 1), the consideration of fractures can be reduced to plain-strain geometry. In this work we use a plain-strain model for the interacting fractures in the cross-sectional plane defined by the

**L**

**Q, µ** x

**l22**

**Figure 1.** Schematic representation of the problem statement where a plain-strain (Khristianovich-Zheltov-DeClerk)

Suppose that the HF propagates from the injection point (point *l*11= (0;0) in Fig. 1) toward the NF perpendicular to the minimum horizontal far-field stress σ2, coinciding with axis *Oy*. When the HF reaches the NF (point *l*12=(*L*;0) in Fig. 1) it forms a T-shape contact with an angle *β*. Maximum and minimum far-field stresses, σ1 and σ2, acting parallel to x- and y-axes respec‐ tively, are constant and uniformly distributed. The rock is assumed impermeable, isotropic,

Following Zhang and Jeffrey [12], the elasticity equations for the system of interacting fractures can be written as the following sum of the contributions from each fracture with coordinates

where integration is performed along the fracture path, *N* is the number of fractures (initially *N* = 2), *Gij* is the hypersingular Green's functions for this problem [30-32], *E* is Young's modulus, *ν* is Poisson's ratio. The equations describe the integral relationship between the net normal and shear stress applied at the fracture, σn and *τ* respectively*,* and fracture opening and sliding, *w* and *v* respectively. Frictional slippage at the NF obeys the following Mohr-Coulomb friction

**<sup>1</sup> l12**

**NF**

G11(x, y, ξ)w(ξ) + G12(x, y, ξ)v(ξ) dξ

(1)

G21(x, y, ξ)w(ξ) + G22(x, y, ξ)v(ξ) dξ

**2**

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

b

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159

y

model is considered with a one-wing HF is propagating toward a discontinuity (NF)

of their tips *li*1, *li*2*,* where *i* = 1,2, (1 refers to the HF, 2 refers to the NF):

l i2

l i2 **HF**

1D flow direction (Fig. 1).

and elastic.

{

σn(x, y) - σ<sup>n</sup>

*τ*(x, y) - τ*<sup>∞</sup>*(x, y)= ∑

*<sup>∞</sup>*(x, y)= ∑ i=1 N E <sup>2</sup>π(1 - <sup>ν</sup>2) *∫* l i1

criterion with cohesion *C* and friction coefficient *λ*:

i=1 N E <sup>2</sup>π(1 - <sup>ν</sup>2) *∫* l i1

To predict the ultimate geometry of HFs, one needs to predict the result of every HF-NF interaction [23]. Several theoretical, numerical, and experimental studies have focused on this task [8, 10, 24-26]. One of the simplest analytical criterion for predicting the outcome of the HF-NF orthogonal interaction was developed by Renshaw and Pollard [18] and extended to arbitrary angle of interaction by Gu et al. [15]. This criterion predicts initiation of the secondary fracture (SF) on the opposite side of the interface influenced only by the friction, cohesion, and in-situ stresses. However, the HF activation and stress field near the intersection point are strongly dependent on the opening of the HF at contact and hence on pumping rate and fluid viscosity — a key point that has been ignored up to now [27]. Recent theoretical developments and laboratory experiments [21-22] have shown it is possible to derive an analytical crossing model taking into account the effect of flow rate and fluid viscosity on the HF-NF crossing behavior [27]. The results of this new HF-NF crossing model in a fracturing simulator envi‐ ronment are presented in a companion paper [28].

This paper offers dimensionless formulation and interpretation of the problem of fracture interaction. The numerical investigations are made by means of the code MineHF2D developed by Zhang et al. [20, 29]. This work focuses primarily on the parametric sensitivity analysis of the problem in terms of HF reinitiation at the NF and takes into account a more rigorous initiation criterion based on stress and energy premises.

The content of the paper is organized as follows. First, we define the problem of the HF-NF interaction mathematically and select all independent dimensionless parameters that influence the result. Next, we discuss the results of the parametrical study in terms of crossing or arresting behavior in the selected parametric diagrams. Finally, we concentrate on the particular effect of the hydraulic permeability of the NF by means of analytical and numerical models.

### **2. Problem statement**

Consider the interaction between an HF and a preexisting discontinuity that represents a mechanically closed fracture with finite permeability, which is higher than that of the sur‐ rounding rock. In what follows the discontinuity will be referred to as an NF.

In nature, both the HF and the NF have certain 3D extents in the vertical and horizontal directions and their interaction should be described in 3D. If the vertical heights of the fractures (perpendicular to the plane of Fig. 1) are far more comparable to or significantly exceed the horizontal size of the fractures affected by the interaction (in a plane of Fig. 1), the consideration of fractures can be reduced to plain-strain geometry. In this work we use a plain-strain model for the interacting fractures in the cross-sectional plane defined by the 1D flow direction (Fig. 1).

to the design of the treatment and the ability to evaluate post-treatment production. Most fracture propagation models assume an oversimplified single planar geometry of fracture propagation [1, 5]. However, in highly heterogeneous and naturally fractured formations, the geometry of HFs can be complex because of their interaction with preexisting discontinuities in the rock, such as natural fractures, faults, and bedding interfaces [3, 6-7], which will be referred as NFs. For example, it is well known that an HF can be arrested by an NF or can reinitiate after the contact [8-16]. The result of the interaction depends on the in-situ stresses [10, 15, 17-18]; friction [12], cohesion [19], and permeable properties of the NF [20]; the

To predict the ultimate geometry of HFs, one needs to predict the result of every HF-NF interaction [23]. Several theoretical, numerical, and experimental studies have focused on this task [8, 10, 24-26]. One of the simplest analytical criterion for predicting the outcome of the HF-NF orthogonal interaction was developed by Renshaw and Pollard [18] and extended to arbitrary angle of interaction by Gu et al. [15]. This criterion predicts initiation of the secondary fracture (SF) on the opposite side of the interface influenced only by the friction, cohesion, and in-situ stresses. However, the HF activation and stress field near the intersection point are strongly dependent on the opening of the HF at contact and hence on pumping rate and fluid viscosity — a key point that has been ignored up to now [27]. Recent theoretical developments and laboratory experiments [21-22] have shown it is possible to derive an analytical crossing model taking into account the effect of flow rate and fluid viscosity on the HF-NF crossing behavior [27]. The results of this new HF-NF crossing model in a fracturing simulator envi‐

This paper offers dimensionless formulation and interpretation of the problem of fracture interaction. The numerical investigations are made by means of the code MineHF2D developed by Zhang et al. [20, 29]. This work focuses primarily on the parametric sensitivity analysis of the problem in terms of HF reinitiation at the NF and takes into account a more rigorous

The content of the paper is organized as follows. First, we define the problem of the HF-NF interaction mathematically and select all independent dimensionless parameters that influence the result. Next, we discuss the results of the parametrical study in terms of crossing or arresting behavior in the selected parametric diagrams. Finally, we concentrate on the particular effect of the hydraulic permeability of the NF by means of analytical and numerical

Consider the interaction between an HF and a preexisting discontinuity that represents a mechanically closed fracture with finite permeability, which is higher than that of the sur‐

In nature, both the HF and the NF have certain 3D extents in the vertical and horizontal directions and their interaction should be described in 3D. If the vertical heights of the

rounding rock. In what follows the discontinuity will be referred to as an NF.

rheological properties of the injecting fluid; and injection flow rate [21-22].

ronment are presented in a companion paper [28].

158 Effective and Sustainable Hydraulic Fracturing

initiation criterion based on stress and energy premises.

models.

**2. Problem statement**

**Figure 1.** Schematic representation of the problem statement where a plain-strain (Khristianovich-Zheltov-DeClerk) model is considered with a one-wing HF is propagating toward a discontinuity (NF)

Suppose that the HF propagates from the injection point (point *l*11= (0;0) in Fig. 1) toward the NF perpendicular to the minimum horizontal far-field stress σ2, coinciding with axis *Oy*. When the HF reaches the NF (point *l*12=(*L*;0) in Fig. 1) it forms a T-shape contact with an angle *β*. Maximum and minimum far-field stresses, σ1 and σ2, acting parallel to x- and y-axes respec‐ tively, are constant and uniformly distributed. The rock is assumed impermeable, isotropic, and elastic.

Following Zhang and Jeffrey [12], the elasticity equations for the system of interacting fractures can be written as the following sum of the contributions from each fracture with coordinates of their tips *li*1, *li*2*,* where *i* = 1,2, (1 refers to the HF, 2 refers to the NF):

$$\begin{cases} \sigma\_{\text{n}} \langle \mathbf{x}, \mathbf{y} \rangle \cdot \sigma\_{\text{n}}''' \langle \mathbf{x}, \mathbf{y} \rangle = \sum\_{i=1}^{N} \frac{\mathcal{E}}{2\pi \langle 1 \cdot \mathbf{v}^{\dagger} \rangle} \Big[ \mathcal{G}\_{11} \langle \mathbf{x}, \mathbf{y}, \xi \rangle \text{w} \langle \xi \rangle + \mathcal{G}\_{12} \langle \mathbf{x}, \mathbf{y}, \xi \rangle \text{v} \langle \xi \rangle \Big] \text{d}\xi \\\\ \tau \langle \mathbf{x}, \mathbf{y} \rangle - \tau''' \langle \mathbf{x}, \mathbf{y} \rangle = \sum\_{i=1}^{N} \frac{\mathcal{E}}{2\pi \langle 1 \cdot \mathbf{v}^{\dagger} \rangle} \Big[ \mathcal{G}\_{21} \langle \mathbf{x}, \mathbf{y}, \xi \rangle \text{w} \langle \xi \rangle + \mathcal{G}\_{22} \langle \mathbf{x}, \mathbf{y}, \xi \rangle \text{v} \langle \xi \rangle \Big] \text{d}\xi \end{cases} \tag{1}$$

where integration is performed along the fracture path, *N* is the number of fractures (initially *N* = 2), *Gij* is the hypersingular Green's functions for this problem [30-32], *E* is Young's modulus, *ν* is Poisson's ratio. The equations describe the integral relationship between the net normal and shear stress applied at the fracture, σn and *τ* respectively*,* and fracture opening and sliding, *w* and *v* respectively. Frictional slippage at the NF obeys the following Mohr-Coulomb friction criterion with cohesion *C* and friction coefficient *λ*:

$$\left| \pi \right| = \lambda (\sigma\_n - p\_f) + \mathbb{C} \tag{2}$$

where *q* is the 1D flow rate inside the fracture, and *μ* is the dynamic fluid viscosity. For fluid flow inside the NF, the same equations can be written with slight modification of the total

+ = (8)

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161

<sup>+</sup> = - (9)

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

<sup>0</sup> *qx t Q* ( 0, ) = = (10)

(11)

( ) <sup>0</sup> *<sup>h</sup> dw w dq dt dx*

> <sup>3</sup> ( ) 12 *<sup>f</sup> <sup>h</sup> w w dp*

m*dx*

When the HF and the NF are in contact at the junction point, the fluid flux is required to satisfy the local continuity equations, meaning that the income flux from the HF is equal to the flux outgoing to the NF. Additionally, the fluid pressure profile along the HF and the NF must

have a common value at the junction point. The coupling condition at the junction is

(HF) (NF) ( ) ( ' 0)

*f f p xL p x qx L qx qx*

( ) ( ' 0) ( ' 0)

The imposed boundary conditions at the inlet and junction point are sketched in Fig. 2.

== = = = =+ + =-

+

*q*

At the inlet of the HF the fluid flow is prescribed as constant:

hydraulic opening, so that

**Figure 2.** Boundary conditions of the problem

where *pf* is the fluid pressure in the NF. The mechanically closed NF can possess a finite hydraulic permeability, which is significantly higher than that of the surrounding rock. To describe the NF permeability the concept of hydraulic aperture *wh* is introduced. Hydraulic aperture is an imaginary opening of the closed crack simulating the residual conductivity of the NF and does not contribute to stress change. The permeability of the NF, *k,* and the hydraulic opening, *wh,* are related by

$$k = \frac{w\_h^2}{12} \tag{3}$$

If the fluid penetrates the closed NF, the hydraulic aperture *wh* can be changed depending on the infiltrated fluid pressure in the NF. When the fluid pressure exceeds the normal stress applied to the NF, the NF will open mechanically, leading to the associated increase of its hydraulic aperture, which is the sum of mechanical and hydraulic opening, and

$$k = \frac{\left(w + w\_h\right)^2}{12} \tag{4}$$

To describe the dependency of the hydraulic aperture *wh* on the fluid pressure *pf* , a nonlinear spring model is used [20]:

$$\frac{dw\_h}{dp\_f} = \mathcal{X}w\_h\tag{5}$$

where *χ* is the empirical constant of order of 10-8 Pa-1 - 10-6 Pa-1 that characterizes the compliance of a NF with respect to the increase of net pressure inside the fracture.

Restricting our study to incompressible and Newtonian fluids, the fluid flow in the growing HF is described by the following continuity equation and Poiseulle's law:

$$\frac{d\mathbf{w}}{dt} + \frac{d\boldsymbol{q}}{d\mathbf{x}} = \mathbf{0} \tag{6}$$

$$q = -\frac{w^3}{12\,\mu} \frac{dp\_f}{dx} \tag{7}$$

where *q* is the 1D flow rate inside the fracture, and *μ* is the dynamic fluid viscosity. For fluid flow inside the NF, the same equations can be written with slight modification of the total hydraulic opening, so that

$$\frac{d(w+w\_h)}{dt} + \frac{dq}{dx} = 0\tag{8}$$

$$q = -\frac{(w + w\_h)^3}{12\,\mu} \frac{dp\_f}{d\mathbf{x}}\tag{9}$$

At the inlet of the HF the fluid flow is prescribed as constant:

( ) *n f*

hydraulic permeability, which is significantly higher than that of the surrounding rock. To describe the NF permeability the concept of hydraulic aperture *wh* is introduced. Hydraulic aperture is an imaginary opening of the closed crack simulating the residual conductivity of the NF and does not contribute to stress change. The permeability of the NF, *k,* and the

> 2 12

If the fluid penetrates the closed NF, the hydraulic aperture *wh* can be changed depending on the infiltrated fluid pressure in the NF. When the fluid pressure exceeds the normal stress applied to the NF, the NF will open mechanically, leading to the associated increase of its

> ( ) <sup>2</sup> 12 *w wh <sup>k</sup>* +

hydraulic aperture, which is the sum of mechanical and hydraulic opening, and

To describe the dependency of the hydraulic aperture *wh* on the fluid pressure *pf*

*h*

*f dw*

of a NF with respect to the increase of net pressure inside the fracture.

HF is described by the following continuity equation and Poiseulle's law:

*q*

*dp* <sup>=</sup> c

*h*

Pa-1 - 10-6

*w*

Restricting our study to incompressible and Newtonian fluids, the fluid flow in the growing

<sup>0</sup> *dw dq dt dx*

> 3 12 *<sup>f</sup> w dp*

m

is the fluid pressure in the NF. The mechanically closed NF can possess a finite

= -+ *p C* (2)

*wh <sup>k</sup>* <sup>=</sup> (3)

= (4)

(5)

+ = (6)

*dx* = - (7)

Pa-1 that characterizes the compliance

, a nonlinear

t ls

where *pf*

hydraulic opening, *wh,* are related by

160 Effective and Sustainable Hydraulic Fracturing

spring model is used [20]:

where *χ* is the empirical constant of order of 10-8

$$q(\mathbf{x} = \mathbf{0}, t) = \mathbf{Q}\_0 \tag{10}$$

When the HF and the NF are in contact at the junction point, the fluid flux is required to satisfy the local continuity equations, meaning that the income flux from the HF is equal to the flux outgoing to the NF. Additionally, the fluid pressure profile along the HF and the NF must have a common value at the junction point. The coupling condition at the junction is

$$\begin{aligned} p\_f^{(\text{HF})} (\mathbf{x} = L) &= p\_f^{(\text{NF})} (\mathbf{x}' = \mathbf{0})\\ q(\mathbf{x} = L) &= q(\mathbf{x}' = +\mathbf{0}) + q(\mathbf{x}' = -\mathbf{0}) \end{aligned} \tag{11}$$

The imposed boundary conditions at the inlet and junction point are sketched in Fig. 2.

**Figure 2.** Boundary conditions of the problem

The condition for the HF tip propagation implies the quasi-static growth of the Mode I fracture such that Mode I stress intensity factor at the tip of the HF, *KI* , equals the fracture toughness of rock *KIC*:

$$\mathbf{K}\_{\mathbf{l}} = \frac{\mathbf{w}^{\prime}\mathbf{x}^{\prime}\mathbf{E}}{4(\mathbf{1} - \mathbf{v}^{2})} \sqrt{\frac{\pi}{2}} \frac{1}{\sqrt{\mathbf{h}\_{\text{(op)}} - \mathbf{x}}} \Big|\_{\mathbf{h}\_{\text{(op)}} \times \mathbf{x} \ll \mathbf{l}\_{\text{(op)}}} = \mathbf{K}\_{\text{IC}} \tag{12}$$

where *θ* is the deflection angle from the original fracture tip growth orientation.

NF interaction are schematically drawn in Fig. 3.

**Figure 3.** Schematic diagram of possible HF-NF interaction scenarios

tion with the NF.

20, 24, 29].

In what follows we are interested in the ultimate result of the HF-NF interaction. After the contact with the NF, the HF can either stay arrested by the NF or reinitiate at the NF and continue propagation to the remainder of the rock behind the NF. Possible outcomes of HF-

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

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163

In the case of arresting, the NF is extensively activated in opening and sliding such that the tensile stresses near the HF tip become insufficient for continuous fracture propagation or initiation of a new crack behind the NF. After the contact, the fracturing fluid is injected into the NF and the dilated NF becomes a part of a complex hydraulic fracture network. In contrast, if the NF appears to be frictionally or cohesively strong, the HF will reinitiate at the NF and continue its propagation into the remainder of the rock. The fracture often reiniti‐ ates at the offset positions, forming a more complex kinked fracture path after the interac‐

In this work, the problem of the HF-NF interaction is solved by means of the computational code developed by Australia's Commonwealth Scientific and Industrial Research Organiza‐ tion, CSIRO. The details of numerical scheme implemented in this code can be found in [12-13,

where l(op) is the half-length of the open fracture zone. A condition similar to zero toughness can be written for the tip of the open zone created at the NF when the sliding zone propagates farther than the open zone. At the tip of the NF sliding zone we have then

$$\mathbf{K}\_{\mathrm{II}}\,^{\mathrm{(NF)}} = \frac{\mathbf{v}(\mathbf{x})\mathbf{E}}{\mathbf{4}\{\mathbf{1}\cdot\mathbf{v}^{2}\}} \sqrt{\frac{\pi}{2}} \frac{1}{\sqrt{\mathbf{l}\_{(\mathrm{d0})}\cdot\mathbf{x}}} \begin{vmatrix} \\ \\ \mathbf{l}\_{(\mathrm{d0})}\times\mathbf{e}\mathbf{l}\_{\mathrm{l}\_{(\mathrm{d0})}} \end{vmatrix} \leq \mathbf{K}\_{\mathrm{IIC}}\,^{\mathrm{(NF)}}\tag{13}$$

where l(sl) is the half-length of the sliding fracture zone. If the sliding zone and open zone coincide at cohesive NF, the following mixed mode criterion is used instead of (13):

$$\begin{cases} \mathbf{K}\_{\rm I}^{\rm (NF)} = \mathbf{K}\_{\rm IC}^{\rm (NF)} \\\\ \mathbf{K}\_{\rm II}^{\rm (NF)} = \mathbf{K}\_{\rm IIC}^{\rm (NF)} \end{cases} \tag{14}$$

Modeling of the new fracture initiation has been traditionally based on stress criterion only [26]. Following Leguillon [33], this work presents an extension in which the initiation of a new fracture must meet the joint stress and energy criteria, which is dependent on the initial length of the fracture initiated, *δ l*. To create a new tensile crack, the stress acting normally to the crack must exceed tensile strength of the rock, *T*0. Energy criterion for crack creation is simplified here to the requirement that the stress intensity factor at the initiated crack tip, *KI* , exceeds fracture toughness of the rock, *KIC*. Using the sign convention for the tensile stress to be negative, the joint initiation criterion used reads

$$\begin{cases} \sigma\_{\tau\tau}(\mathcal{S}l) \le -T\_0 \\ K\_l(\mathcal{S}l) > K\_{l\mathcal{C}} \end{cases} \tag{15}$$

where *σττ* is the tangential normal stress parallel to the NF and evaluated along the initiated crack path. Once the new crack is initiated, its further growth is determined by the following criterion of mixed-mode fracture propagation [34]

$$\left[K\_I \cos^2\frac{\theta}{2} - \frac{3}{2}K\_{II}\sin\theta\right] \cos\frac{\theta}{2} = K\_{IC} \tag{16}$$

where *θ* is the deflection angle from the original fracture tip growth orientation.

The condition for the HF tip propagation implies the quasi-static growth of the Mode I fracture such that Mode I stress intensity factor at the tip of the HF, *KI* , equals the fracture toughness

where l(op) is the half-length of the open fracture zone. A condition similar to zero toughness can be written for the tip of the open zone created at the NF when the sliding zone propagates

where l(sl) is the half-length of the sliding fracture zone. If the sliding zone and open zone

Modeling of the new fracture initiation has been traditionally based on stress criterion only [26]. Following Leguillon [33], this work presents an extension in which the initiation of a new fracture must meet the joint stress and energy criteria, which is dependent on the initial length of the fracture initiated, *δ l*. To create a new tensile crack, the stress acting normally to the crack must exceed tensile strength of the rock, *T*0. Energy criterion for crack creation is simplified here to the requirement that the stress intensity factor at the initiated crack tip, *KI* , exceeds fracture toughness of the rock, *KIC*. Using the sign convention for the tensile stress to be

> <sup>0</sup> ( ) ( ) *I IC l T*

where *σττ* is the tangential normal stress parallel to the NF and evaluated along the initiated crack path. Once the new crack is initiated, its further growth is determined by the following

q

 q

ë û (16)

*KlK* tt s d

ìï £ - <sup>í</sup> ï > î

d

<sup>2</sup> <sup>3</sup> cos sin cos 22 2 *ΚΚ Κ <sup>I</sup> II IC*

é ù - = ê ú

q

l(op)-x≪l(op)

l(sl)-x≪l(sl)

≤KIIC

=KIC (12)

(NF) (13)

(15)

(NF) (14)

KI= w(x)E 4(1 - ν2)

> <sup>=</sup> v(x)E 4(1 - ν2)

KII (NF)

negative, the joint initiation criterion used reads

criterion of mixed-mode fracture propagation [34]

π 2

farther than the open zone. At the tip of the NF sliding zone we have then

{ KI (NF) =KIC (NF)

KII (NF) =KIIC

π 2

1 l(sl) - <sup>x</sup> <sup>|</sup>

coincide at cohesive NF, the following mixed mode criterion is used instead of (13):

1 l(op) - <sup>x</sup> <sup>|</sup>

of rock *KIC*:

162 Effective and Sustainable Hydraulic Fracturing

In what follows we are interested in the ultimate result of the HF-NF interaction. After the contact with the NF, the HF can either stay arrested by the NF or reinitiate at the NF and continue propagation to the remainder of the rock behind the NF. Possible outcomes of HF-NF interaction are schematically drawn in Fig. 3.

In the case of arresting, the NF is extensively activated in opening and sliding such that the tensile stresses near the HF tip become insufficient for continuous fracture propagation or initiation of a new crack behind the NF. After the contact, the fracturing fluid is injected into the NF and the dilated NF becomes a part of a complex hydraulic fracture network. In contrast, if the NF appears to be frictionally or cohesively strong, the HF will reinitiate at the NF and continue its propagation into the remainder of the rock. The fracture often reiniti‐ ates at the offset positions, forming a more complex kinked fracture path after the interac‐ tion with the NF.

In this work, the problem of the HF-NF interaction is solved by means of the computational code developed by Australia's Commonwealth Scientific and Industrial Research Organiza‐ tion, CSIRO. The details of numerical scheme implemented in this code can be found in [12-13, 20, 24, 29].

**Figure 3.** Schematic diagram of possible HF-NF interaction scenarios

### **3. Parameterization of HF-NF problem**

The complete system of coupled equations (1, 5-7) subject to the boundary and initial condi‐ tions (2, 10-14) and criteria for crack initiation and propagation (15-16) is a complex multipar‐ ameterized problem. The aim of the work is to study parametric sensitivity of the result of HF-NF interaction, and the challenge of this undertaking substantially grows with the number of the independent parameters in the study. Our first initiative is to come up with proper parameterization of the problem that will be used in the numerical computations.

Let us normalize the equations by the convenient choice of the scaling. First we introduce the displacement scale *W* for the opening, shear displacement of the fractures, and the residual

where *Ω*, Ψ and *Ω<sup>h</sup>* are the dimensionless opening, sliding, and residual NF opening, respec‐

where *q*˜ is the dimensionless flow rate. Similarly, we introduce the dimensionless fluid

t

ts

g

, ,, *f nn i i pP P P P*

(,) (,), *ij ij xy xyL l L* = =

s

=P =S =S =S

The coordinates, including coordinates of fracture tips, are scaled by the length of the contacted

After substitution of (17-21) into equations (1-13) we obtain these seven dimensionless groups

', , , 1 , '

 (NF)' (NF)' ( ) ( )

*KL K L K L WE WE WE*

*TQ W P E W Q WL Q L PL Q*

 a

= = = ==

, , '' ' *IC NF IC NF IIC*

Choosing the viscosity scaling by setting *α<sup>i</sup>* = 1 in (22), we define the following expressions for

2 36 4 4 4

'

*E L E Q*

, , '

*LQ E Q <sup>L</sup> WP T*

' '' '

3

m

m

12 3 4

*IC IC IIC*

== =

tively. The fluid flow rate is scaled by the injection rate at the inlet *Q = Q0* as

, , *w Wv Ww W h h* =W =Y =W (17)

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

, with the stress and pressure scale *P*

<sup>0</sup> *q qQ* = % (18)

(19)

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165

(20)

(22)

*t tT* = (21)

0

 a

 k

2 3

mm== = (23)

aperture of the NF as

fracture *L* as

the chosen scales:

pressure, *Π*, and stress components, *Σ<sup>i</sup>*

and the time is scaled by a certain time scale *T* as

'

kk

aa

There are 14 dimensional parameters that impact the solution: injection rate into fracture, *Q*0; fracturing fluid viscosity, *μ*; plain-strain Young's modulus, *E'= E/*(1 – *ν*<sup>2</sup> ); Mode I fracture toughness of the rock, *KIC* ; tensile strength of the rock, *T*0; maximum in-situ stress, σ1; minimum in-situ stress, σ2; length of the HF at contact with the NF, *L*; angle between the fractures, *β*; friction coefficient, λ; Mode I NF toughness, *KIC* (NF) Mode II NF toughness, *KIIC* (NF) ; hydraulic permeability of the NF, *k*, which is coupled to the residual NF opening as *k* =*wh* <sup>2</sup> / 12; and NF compliance, *χ*. For the sake of concise expressions, we introduce the following notations for viscosity *μ' =* 12*μ*, for the fracture toughness in rock *KIC* ' = 32 / *πKIC*, and at the NF *KIC* (NF)' = 32 / *πKIC* (NF) , *KIIC* (NF)' = 32 / *πKIIC* (NF) .


Table 1 summarizes the expected range of dimensional parameters for the majority of HF jobs in unconventional reservoirs from various sources [35-36]. The data were obtained from laboratory experiments and field measurements.

**Table 1.** Range of the problem's dimensional parameters specific to gas shale fracturing jobs

Let us normalize the equations by the convenient choice of the scaling. First we introduce the displacement scale *W* for the opening, shear displacement of the fractures, and the residual aperture of the NF as

$$
\varpi = \Omega \mathcal{W} \,, \quad \upsilon = \Psi \mathcal{W} \,, \quad \varpi\_h = \Omega\_h \mathcal{W} \,\tag{17}
$$

where *Ω*, Ψ and *Ω<sup>h</sup>* are the dimensionless opening, sliding, and residual NF opening, respec‐ tively. The fluid flow rate is scaled by the injection rate at the inlet *Q = Q0* as

$$
\eta = \tilde{q}Q\_0 \tag{18}
$$

where *q*˜ is the dimensionless flow rate. Similarly, we introduce the dimensionless fluid pressure, *Π*, and stress components, *Σ<sup>i</sup>* , with the stress and pressure scale *P*

$$\sigma\_f = \Pi P, \quad \sigma\_n = \Sigma\_n P, \quad \pi = \Sigma\_\tau P, \quad \sigma\_i = \Sigma\_i P \tag{19}$$

The coordinates, including coordinates of fracture tips, are scaled by the length of the contacted fracture *L* as

$$\mathbf{r}(\mathbf{x}, \mathbf{y}) = (\overline{\mathbf{x}}, \overline{\mathbf{y}}) L, \quad \mathbf{l}\_{\mathrm{ij}} = \mathbf{y}\_{\mathrm{ij}} L \tag{20}$$

and the time is scaled by a certain time scale *T* as

**3. Parameterization of HF-NF problem**

164 Effective and Sustainable Hydraulic Fracturing

The complete system of coupled equations (1, 5-7) subject to the boundary and initial condi‐ tions (2, 10-14) and criteria for crack initiation and propagation (15-16) is a complex multipar‐ ameterized problem. The aim of the work is to study parametric sensitivity of the result of HF-NF interaction, and the challenge of this undertaking substantially grows with the number of the independent parameters in the study. Our first initiative is to come up with proper

There are 14 dimensional parameters that impact the solution: injection rate into fracture, *Q*0;

toughness of the rock, *KIC* ; tensile strength of the rock, *T*0; maximum in-situ stress, σ1; minimum in-situ stress, σ2; length of the HF at contact with the NF, *L*; angle between the

hydraulic permeability of the NF, *k*, which is coupled to the residual NF opening as

<sup>2</sup> / 12; and NF compliance, *χ*. For the sake of concise expressions, we introduce the

(NF) .

*Parameter Range*

= 32 / *πKIIC*

Table 1 summarizes the expected range of dimensional parameters for the majority of HF jobs in unconventional reservoirs from various sources [35-36]. The data were obtained from

> *L Distance between HF and NF 1–10 m µ Fluid viscosity 1–1000 cP E Young's modulus 9–110 GPa* ν *Poisson's coefficient 0.11 –0.252 KIC Mode I fracture toughness 0.1–2.7 MPa(m1/2)*

*KIC (NF)*/Κ*IC (HF) Toughness ratio for NF vs. rock matrix 0–0.5 KIIC (NF)/KIC (NF) Toughness ratio for NF ~1*

**Table 1.** Range of the problem's dimensional parameters specific to gas shale fracturing jobs

σ*<sup>1</sup> Maximum in-situ stress 13–105 MPa* σ*<sup>2</sup> Minimum in-situ stress 11–100 MPa* λ *Friction coefficient at the NF 0.2–1 kh Permeability of NF 1 md–1 darcy* β *Fracture interaction angles 30°*

(NF)

); Mode I fracture

= 32 / *πKIC*,

(NF) ;

Mode II NF toughness, *KIIC*

'

*0.01–0.25 m3/s mmoposdvrgenmk m3/sec*

*–90°*

parameterization of the problem that will be used in the numerical computations.

following notations for viscosity *μ' =* 12*μ*, for the fracture toughness in rock *KIC*

fracturing fluid viscosity, *μ*; plain-strain Young's modulus, *E'= E/*(1 – *ν*<sup>2</sup>

fractures, *β*; friction coefficient, λ; Mode I NF toughness, *KIC*

(NF) , *KIIC* (NF)'

*Q Volumetric injection rate*

= 32 / *πKIC*

laboratory experiments and field measurements.

*k* =*wh*

and at the NF *KIC*

(NF)'

$$t = tT \tag{21}$$

After substitution of (17-21) into equations (1-13) we obtain these seven dimensionless groups

$$\begin{aligned} \alpha\_1 &= \frac{T\mathbb{Q}}{\mathbb{W}\mathbb{L}}, \qquad \alpha\_2 = \frac{W^3 P}{\mathbb{Q}\mu^\* \mathbb{L}}, \qquad \alpha\_3 = \frac{E^\* W}{\mathbb{P}\mathbb{L}}, \qquad \alpha\_4 = \frac{\mathbb{Q}\_0}{\mathbb{Q}} = 1, \\\ \kappa\_{\text{IC}} &= \frac{K\_{\text{IC}}^\circ \sqrt{\mathbb{L}}}{\text{WE}^\circ}, \qquad \kappa\_{\text{IC}}^{\text{(NF)}} = \frac{K\_{\text{IC}}^{\text{(NF)}} \sqrt{\mathbb{L}}}{\text{WE}^\circ}, \qquad \kappa\_{\text{IC}}^{\text{(NF)}} = \frac{K\_{\text{IC}}^{\text{(NF)}} \sqrt{\mathbb{L}}}{\text{WE}^\circ} \end{aligned} \tag{22}$$

Choosing the viscosity scaling by setting *α<sup>i</sup>* = 1 in (22), we define the following expressions for the chosen scales:

$$W = \sqrt[4]{\frac{L^2 \mathbb{Q} \mu^{\prime}}{E^{\prime}}}, \quad P = \sqrt[4]{\frac{E^{\prime 3} \mathbb{Q} \mu^{\prime}}{L^2}}, \quad T = \sqrt[4]{\frac{L^6 \mu^{\prime}}{E^{\prime} \mathbb{Q}^3}} \tag{23}$$

After the scaling we obtain nine dimensionless parameters (*κIC*, *κIC* (NF) , *κIIC* (NF) , *Σ*1, *Σ*2, *Ω<sup>h</sup>* , *β*, *λ*, *χ*) that characterize our dimensionless problem. We also note the parameters have nonequal sensitivity to the result of fracture interaction. For example, all previous theoretical and experimental studies have shown that the significant effect of the relative difference of the applied stresses can be written as *ΔΣ* =(*Σ*<sup>1</sup> −*Σ*2)/ *Σ*2, rather than as the mean stress. The effect of NF compliance and the difference between Mode I and Mode II NF toughness can be neglected in the first attempt of parametric study, so we assume here *κIC* (*NF* ) =*κIIC* (*NF* ) . Consequently, we restrict our parametric analysis with the following six parameters { *κIC*, *ΔΣ*, *β*, *λ*, *κIC* (*NF* ) , *Ω<sup>h</sup>* }.

**4.1. Dimensionless toughness** *κIC* **vs. angle** *β* **diagram**

outcome as shown by arrows at the bottom of Fig. 4.

to the orientation of the HF is far from 90°.

**4.3.** *λ* **vs.** *κIC***(NF) /** *κIC* **diagram**

becomes more favorable.

**4.2.** *κIC* **vs.***ΔΣ* **diagram**

Fig. 4 shows the results of a large number of numerical computations performed with various independent parameters of the problem. They are projected on the diagram of dimensionless toughness versus interaction angle. These results represent the final outcome of the HF-NF interaction in terms of either fracture crossing or arresting at the NF. We observe that the HF crosses the NF predominantly at large angles of intersection (i.e., close to 90°) and smaller values of dimensionless toughness as expected. Note that the dimensionless toughness *κIC* we used for plotting depends not only on the Mode I toughness but also on the injection rate and fluid viscosity, as seen from Eqs. (22-23). If the fluid viscosity *μ* or the flow rate at the fracture inlet *Q* increases, the value of the dimensionless toughness decreases and the crossing behavior

Hydraulic Fracture Propagation Across a Weak Discontinuity Controlled by Fluid Injection

http://dx.doi.org/10.5772/55941

167

The computations have been performed for several values of in-situ rock stress difference *ΔΣ*. To denote this important parameter in the plotted results, we have used different colors for different values of *ΔΣ*. To separate the crossing or arrest tendency at small stress contrast from the similar tendency at large stress contrast, we have presented the results (Fig. 4) in two diagrams for the small and large values of *ΔΣ* respectively. After using such a representation of the results, one can still observe that some points remained nonseparated and overlapped. For example, in the bottom diagram, the results from the computations with different stress contrast coincide at the same point in the angle-toughness diagram. They can have opposite

The next series of parametric diagrams represent the results of numerical experiments in the dimensional toughness vs. stress contrast cross section. This representation better emphasizes the role of the dimensionless relative stress and with the flow rate and viscosity that are inversely proportional to the dimensionless toughness. Plots in Fig. 5 clearly show that the higher relative stress difference *ΔΣ*, the larger the region of the dimensionless toughness values where fracture crossing occurs. The highest threshold value of the dimensionless toughness for crossing behavior increases with the stress contrast; i.e., in the formations with high stress contrast the crossing behavior will most probably happen even with low viscosity and/or low pumping rate during fracturing jobs. At the same time, it is unlikely that the rock with relatively small stress difference *ΔΣ* <1 will support fracture crossing unless the dimensionless tough‐ ness value is sufficiently decreased by stronger injection power (larger pumping rates or higher fluid viscosity). This is especially the case if the preferential inclination of the NFs with respect

Next we analyzed the influence of the friction coefficient *λ* and the relative NF Mode I toughness *κIC*(NF) /*κIC* on the result of HF-NF interaction (Fig. 6). In this series of numerical tests we observe a sharp boundary for the friction coefficient values where the crossing behavior

This study is mainly intended for the oil and gas industry, so the comprehensive numerical study in infinite limits of these parameters in the present study is not necessary. In what follows we limit the range of the dimensionless parameter values to the practical range by use of the compilation of dimensional parameters of the problem shown in Table 1. Using the scaling introduced previously it is possible to calculate the corresponding range of the dimensionless parameters (Table 2).


**Table 2.** Range of dimensionless parameters calculated from Table 1 using introduced scaling

The following parametric study is restricted to the range of values of dimensionless parameters shown in Table 2, as practically required.
