**3. Elastoplastic effect in fracturing**

regarding cohesive interface constitutive equations to model mechanical behaviour of preexisting cemented natural fractures. At the end, numerical results of implementing this model for several examples will be presented to examine the significance of induced thermal stress

Fracture is a mechanical discontinuity in the rock mass formed due to the presence of stress fields in earth's crust (Figure 2). There are wide scale ranges for fractures from micrometre (microfractures) to kilometres (lineaments). Presence of fractures in earth's crust can influence underground fluid flow and physical properties of rock like rock strength. Fractures can influence the velocity of elastic waves and rock elastic moduli [41]. Natural fractures are categorized into four groups [40] based on their genesis : (1) tensile fractures due to compres‐ sive stresses, (2) shear fractures due to compressive stresses, (3) tensile fractures due to unloading of compressive stresses, (3) natural hydraulic fractures. Despite indeterministic nature of the aforementioned mechanisms, a large number of outcrop studies have revealed pattern and identifiable organization in fracture orientation and spacing. Due to the limited access to the subsurface to map fractures and limited precision of seismic techniques, outcrops are the main source to speculate fracture's geometry in the subsurface. There are different distribution models used to describe fracture size like fracture length, aperture and tangential or perpendicular displacement due to fracture. Scale-limited laws (lognormal, exponential, gamma and power law) are methods in literature to characterize fracture systems [9], but it should be mentioned that scaling exponents alone cannot act as good criterion to define the whole pattern of fracture networks. Moreover, Bonnet et al. [9] showed that there is a linear relationship between rupture area and frequency scale of tensile fractures in seismometers acting. Field studies have confirmed the existence of a critical threshold that cracks with aperture less than this threshold are fully filled with digenetic materials [34]. Although microfractures are filled with calcite or quartz cements, laboratory measurements have proved that these filled natural fractures may still act as weak surfaces, or in other words, potential paths for rock failure. For instance, lab measurements for Barnett shale samples have shown tensile strength of cemented cracks to be about 10 times less than the tensile strength of intact rocks [27]. There exist some integrated models in the literature that can be utilized for this purpose [33]. By combining the knowledge of natural fracture patterns, cement properties and current in-situ stresses, it is possible to build a model to make a realistic prediction about the

distribution of natural fractures in the case of limited core and outcrop data.

quantities despite their small aperture and depth.

Proppants cannot move into microfractures opened during hydraulic fracturing due to their small aperture, which is less than a couple of microns. However, hydraulic pressure can open the microfractures if it goes beyond the local closure stress; therefore, activation of microfrac‐ tures is function of confining pressure and pore pressure. As it mentioned earlier, contact area between rock-fluid can be considerably affected by the presence of microfractures in large

in different situations.

778 Effective and Sustainable Hydraulic Fracturing

**2. Natural facture distribution**

The mechanical behaviour of quartz or calcite is essentially identified as elastic and brittle, however, clay/organic dominated regions can undergo significant plastic strains. Hence, it is not surprising that excessive fluid pressure present during hydraulic fracturing treatments may induce plastic deformations. This issue has been the subject of several studies in the literature [37], [38], [44]. For instance, it has been shown that plasticity causes shorter and wider fractures. However, most of these plastic deformations are due to high stress near the tip of the hydraulic fracture. The excess pressure in the main fracture may be only 1 or 2 MPa higher than the minimum in-situ stress, and this amount of additional stress may not cause a considerable plasticity unless in very weak formations

These papers were mainly focused on plastic deformations induced at the tip of fractures due to stress concentration at the tip of fractures, while plastic deformation of the surrounding rocks and its possible effects was out of the scope of these papers. Irreversible strain charac‐ terizes the plasticity when stress reaches a certain point. After this yielding point, the material shows elastoplasticity, which means its behaviour is somewhat plastic and also elastic. Equations (1) to (3) show general elastoplastic behaviour in three-dimensional problems for a strain increment *dεij* , where *Dijkl <sup>e</sup>* , *Q*, *dλ* and *σij* are elastic moduli tensor, plastic potential, plastic multiplier and components of stress tensor, respectively.

$$d\varepsilon\_{ij} = d\varepsilon\_{ij}^{\varepsilon} + d\varepsilon\_{ij}^{p} \tag{1}$$

hensive review of these models is given by Adachi et al. [1]. Griffith's criterion is the common method to model fracture propagation in all of these techniques. Fracture propagation in Griffith's criterion is a function of stress intensity factor and rock toughness. Griffith's criterion presumes the presence of an initial fracture and predicts its propagation, hence it is an appropriate method to model major hydraulic fracture propagation, but it cannot predict fracture nucleation. Since, we are interested in predicting fracture initiation on the surface of intact rock or along the cemented natural fractures; we cannot limit our analysis to Griffith's criterion. We use the cohesive interface technique to model reopening of cemented fractures. The cohesive interface model is a constitutive equation to model deformation of discontinui‐ ties, which can be easily applied to multiple cracks or incorporates their coalescence. Cohesive interfacial model could also be used to simulate fracture propagation with the advantage of removing stress singularity at the fracture tips [2]. Later laboratory experiments showed that nonlinear region added to cohesive crack models provides better prediction for fracture growth in granular cementious materials like rock and concrete [6]. The cohesive interfacial model considers a cohesive crack of zero width with traction transferring capacity, thus eliminates the stress singularity problem at the crack-tip. Additionally by nature, cohesive interface concept is the best fit for the problems with predefined fracture propagation paths like this problem; however, some sophisticated algorithms has been invented to adaptively add or remove cohesive elements in the computational model upon necessity [47]. In addition, cohesive interfacial models, despite their nonlinear nature, are easy to implement and we will see in the next section how we used this capability to model the initiation of microcracks during

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Cohesive interface model is mainly a nonlinear constitutive equation between the traction and jump in displacement between two bodies. Cohesive interface starts to fail when the applied traction exceeds a critical value and followed by softening, and both are described by traction separation law [43]. Traction-separation law has the flexibility to tune the parameters to find potential function solutions for fracture propagation in different materials [24], [29]. Bilinear law is the simplest form of traction separation law composed of two piecewise linear sections for pre- and post-failure situations (Figure 4). Despite its simplicity as constitutive equation, it has proven capabilities to model fracture behaviour in cementious materials [6]. Quadratic stress law is good candidate for mixed mode condition. In this model, damage initiates the quadratic interaction function involving nominal stress ratios reaching the value of one (Equation 7), while tn and ts represents the real values of normal and tangential tractions along

> ì üì ü ï ïï ï í ýí ý + = ï ïï ï î þî þ

ìï <sup>³</sup> <sup>=</sup> <sup>í</sup> ï £ î

*n*

*t*

*n n*

*n*

2 2 1 *n s no so t t t t*

, 0( ) 0 0( )

*t compression* (8)

*t t tension*

(7)

hydraulic fracturing.

the interface, respectively.

$$d\varepsilon\_{ij}^{\epsilon} = D\_{ijkl}^{\epsilon[-1]} d\sigma\_{kl} \tag{2}$$

$$d\boldsymbol{\omega}\_{ij}^p = d\boldsymbol{\lambda} \frac{\partial \mathbf{Q}}{\partial \sigma\_{ij}}\tag{3}$$

For flow rule and yield criterion, we used Drucker-Prager criterion, which is a well-known model to describe plastic behaviour of rocks. The Drucker-Prager criterion is an adjusted version of the Von Mises criterion for granular materials like soils and soft rocks. The yield surface for the Drucker-Prager criterion is a circular cone with the form shown in equation (4) where α and k are constants related to internal friction and cohesion of material. The stress at any point can be represented by the vector (σ1, σ2, σ3). This vector can be shown by a corresponding stress point on the π-plane which is constituted of vector *s* (σ1- σm, σ2- σm, σ3 σm) and ρ (σm, σm, σm), where σm equals to (σ1+σ2+σ3)/3 (Figure 3a). The constants can be obtained from the plot of failure in *J*<sup>2</sup> 1 2 and *J*1 space. Circumscribed Drucker-Prager and Inscribed Drucker-Prager are two criterions for description of Drucker-Prager criterion based on comparison with Mohr-Coulomb criterion (Figure 3b).

$$\|J\|\_{2}^{2} = k + aJ\_{1} \tag{4}$$

$$J\_1 = \frac{1}{3} (\sigma\_1 + \sigma\_2 + \sigma\_3) \tag{5}$$

$$J^{\frac{1}{2}\prime} = \sqrt{\frac{1}{6} \left( \left( \sigma\_1 - \sigma\_2 \right)^2 + \left( \sigma\_1 - \sigma\_3 \right)^2 + \left( \sigma\_2 - \sigma\_3 \right)^2 \right)}\tag{6}$$

#### **4. Initiation and propagation of cracks**

There is a considerable number of publications for modelling hydraulic fracturing treatments published since 1955, these solutions are varying from analytical and asymptotic solutions [21], [22], [32] to finite element or boundary element numerical schemes [11], [18], [31]. A compre‐ hensive review of these models is given by Adachi et al. [1]. Griffith's criterion is the common method to model fracture propagation in all of these techniques. Fracture propagation in Griffith's criterion is a function of stress intensity factor and rock toughness. Griffith's criterion presumes the presence of an initial fracture and predicts its propagation, hence it is an appropriate method to model major hydraulic fracture propagation, but it cannot predict fracture nucleation. Since, we are interested in predicting fracture initiation on the surface of intact rock or along the cemented natural fractures; we cannot limit our analysis to Griffith's criterion. We use the cohesive interface technique to model reopening of cemented fractures. The cohesive interface model is a constitutive equation to model deformation of discontinui‐ ties, which can be easily applied to multiple cracks or incorporates their coalescence. Cohesive interfacial model could also be used to simulate fracture propagation with the advantage of removing stress singularity at the fracture tips [2]. Later laboratory experiments showed that nonlinear region added to cohesive crack models provides better prediction for fracture growth in granular cementious materials like rock and concrete [6]. The cohesive interfacial model considers a cohesive crack of zero width with traction transferring capacity, thus eliminates the stress singularity problem at the crack-tip. Additionally by nature, cohesive interface concept is the best fit for the problems with predefined fracture propagation paths like this problem; however, some sophisticated algorithms has been invented to adaptively add or remove cohesive elements in the computational model upon necessity [47]. In addition, cohesive interfacial models, despite their nonlinear nature, are easy to implement and we will see in the next section how we used this capability to model the initiation of microcracks during hydraulic fracturing.

shows elastoplasticity, which means its behaviour is somewhat plastic and also elastic. Equations (1) to (3) show general elastoplastic behaviour in three-dimensional problems for a

> s

s

For flow rule and yield criterion, we used Drucker-Prager criterion, which is a well-known model to describe plastic behaviour of rocks. The Drucker-Prager criterion is an adjusted version of the Von Mises criterion for granular materials like soils and soft rocks. The yield surface for the Drucker-Prager criterion is a circular cone with the form shown in equation (4) where α and k are constants related to internal friction and cohesion of material. The stress at any point can be represented by the vector (σ1, σ2, σ3). This vector can be shown by a corresponding stress point on the π-plane which is constituted of vector *s* (σ1- σm, σ2- σm, σ3 σm) and ρ (σm, σm, σm), where σm equals to (σ1+σ2+σ3)/3 (Figure 3a). The constants can be

Inscribed Drucker-Prager are two criterions for description of Drucker-Prager criterion based

= +a

= ++ (sss) 1 123

(( ) ( ) ( ) ) = - +- +-

12 13 23

There is a considerable number of publications for modelling hydraulic fracturing treatments published since 1955, these solutions are varying from analytical and asymptotic solutions [21], [22], [32] to finite element or boundary element numerical schemes [11], [18], [31]. A compre‐

 ss

<sup>1</sup> <sup>222</sup> <sup>2</sup>

*ij*

are elastic moduli tensor, plastic potential,

*ij ij ij ddd* (1)

*ij ijkl kl d Dd* (2)

*<sup>Q</sup> d d* (3)

and *J*1 space. Circumscribed Drucker-Prager and

2 1 *J kJ* (4)

<sup>3</sup> *<sup>J</sup>* (5)

 ss

<sup>6</sup> *<sup>J</sup>* (6)

*<sup>e</sup>* , *Q*, *dλ* and *σij*

e

e

1 2

1 2

1

ss

eee= +*<sup>e</sup> <sup>p</sup>*


 l

¶ <sup>=</sup> ¶ *p ij*

strain increment *dεij*

780 Effective and Sustainable Hydraulic Fracturing

, where *Dijkl*

obtained from the plot of failure in *J*<sup>2</sup>

on comparison with Mohr-Coulomb criterion (Figure 3b).

1

**4. Initiation and propagation of cracks**

plastic multiplier and components of stress tensor, respectively.

Cohesive interface model is mainly a nonlinear constitutive equation between the traction and jump in displacement between two bodies. Cohesive interface starts to fail when the applied traction exceeds a critical value and followed by softening, and both are described by traction separation law [43]. Traction-separation law has the flexibility to tune the parameters to find potential function solutions for fracture propagation in different materials [24], [29]. Bilinear law is the simplest form of traction separation law composed of two piecewise linear sections for pre- and post-failure situations (Figure 4). Despite its simplicity as constitutive equation, it has proven capabilities to model fracture behaviour in cementious materials [6]. Quadratic stress law is good candidate for mixed mode condition. In this model, damage initiates the quadratic interaction function involving nominal stress ratios reaching the value of one (Equation 7), while tn and ts represents the real values of normal and tangential tractions along the interface, respectively.

$$\left\{ \frac{\left\{ t\_n \right\}}{t\_{no}} \right\}^2 + \left\{ \frac{\left\{ t\_s \right\}}{t\_{so}} \right\}^2 = \mathbf{1} \tag{7}$$

$$\begin{Bmatrix} t\_n \end{Bmatrix} = \begin{cases} t\_{n,} & t\_n \ge 0 \quad \text{(tension)}\\ 0 & t\_n \le 0 \quad \text{(compression)} \end{cases} \tag{8}$$

**Figure 3.** a) The Drucker-Prager criterion, (b) Yield envelopes for Circumscribed Drucker-Prager and Inscribed Drucker-Prager criterion (after Colmenares and Zoback, 2002 [15])

Fracture energy release, cohesive strength, initial cohesive stiffness, critical separation gap at complete failure and critical separation at damage initiation are key parameters to specify irreversible fracturing based on bilinear cohesive law. To model the mixed mode fracture propagation, Benzeggagh-Kenane (BK) fracture criterion is used [7].

For an incompressible and Newtonian fluid, equation (9) represents the continuity equation where the first term shows fracture capacity based on its width change and the second term represents the cross-sectional flow rate of the fracture (no leak-off from fracture into the formation). The tangential flow along the gap between two cohesive walls base on momentum equation for Poiseuille's flow pattern is represented in equation 10 where q, w, p and μ are local flow rate, local crack width, fluid pressure inside the fracture and fracturing fluid viscosity, respectively [4].

$$
\frac{\partial w}{\partial t} + \frac{\partial q}{\partial x} = 0 \tag{9}
$$

**Figure 4.** Linear softening law along the cohesive interface.

Commercial finite element software, ABAQUS (Dassault Systèmes Simulia Corp.) [20], is chosen for implementing cohesive crack methods to model secondary fractures initiation and propagation. We begin with the simplest possible geometry for the hydraulic fracture, i.e. a planar fracture. Extension of utilized techniques to non-planar hydraulic fractures does not require introducing any new concept and should be straightforward. Due to the symmetry of the problem with respect to the fracture plane, we only need to simulate half of the geometry. Figure 5 shows the numerical grid with the blue zone showing fracture surface, and the surrounding red zone showing the intact rock. To model pre-existing cemented natural fractures, cohesive elements have been embedded as parallel planes perpendicular to the fracture surface (Z-direction) with 5 cm spacing for this example. Hence, fluid pressure during pumping stage will be introduced only to the fracture surface (blue zone), and the rest of the model will be under the effect of in-situ stress only. In case that any part of the natural fractures (cohesive elements) reaches failure threshold, following the opening of the crack, fracturing fluid is supposed to reach the opened part of the fracture and pressurized it, which is consid‐ ered by removing failed cohesive elements and adding gap flow to our model to include fracturing fluid pressure and their cooling effect in natural fractures. Our preliminary models showed the significance of this effect on clustering of secondary fractures and their depth of

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The model assumed homogeneous and isotropic properties for mechanical and hydraulic properties of the rock. Additionally, fracturing of the rock is assumed as an irreversible process.

**5. Results and discussion**

penetration.

$$q = -\frac{w^3}{12\,\mu} \frac{\partial p}{\partial \mathbf{x}} \tag{10}$$

Analytical solutions for coupled thermo- poroelasticity and only restricted to simple geome‐ tries. Therefore, they are not pertinent for a system with numerous factures. Therefore, we used a commercial finite element package, ABAQUS, for modelling purposes.

**Figure 4.** Linear softening law along the cohesive interface.

### **5. Results and discussion**

(a) (b)

**Figure 3.** a) The Drucker-Prager criterion, (b) Yield envelopes for Circumscribed Drucker-Prager and Inscribed Drucker-

Fracture energy release, cohesive strength, initial cohesive stiffness, critical separation gap at complete failure and critical separation at damage initiation are key parameters to specify irreversible fracturing based on bilinear cohesive law. To model the mixed mode fracture

For an incompressible and Newtonian fluid, equation (9) represents the continuity equation where the first term shows fracture capacity based on its width change and the second term represents the cross-sectional flow rate of the fracture (no leak-off from fracture into the formation). The tangential flow along the gap between two cohesive walls base on momentum equation for Poiseuille's flow pattern is represented in equation 10 where q, w, p and μ are local flow rate, local crack width, fluid pressure inside the fracture and fracturing fluid

*t x* (9)

(10)

¶ ¶ + = ¶ ¶ <sup>0</sup> *<sup>w</sup> <sup>q</sup>*

m

Analytical solutions for coupled thermo- poroelasticity and only restricted to simple geome‐ tries. Therefore, they are not pertinent for a system with numerous factures. Therefore, we

*x*

¶ = - ¶ 3 12 *<sup>w</sup> <sup>p</sup> <sup>q</sup>*

used a commercial finite element package, ABAQUS, for modelling purposes.

propagation, Benzeggagh-Kenane (BK) fracture criterion is used [7].

Prager criterion (after Colmenares and Zoback, 2002 [15])

782 Effective and Sustainable Hydraulic Fracturing

viscosity, respectively [4].

Commercial finite element software, ABAQUS (Dassault Systèmes Simulia Corp.) [20], is chosen for implementing cohesive crack methods to model secondary fractures initiation and propagation. We begin with the simplest possible geometry for the hydraulic fracture, i.e. a planar fracture. Extension of utilized techniques to non-planar hydraulic fractures does not require introducing any new concept and should be straightforward. Due to the symmetry of the problem with respect to the fracture plane, we only need to simulate half of the geometry. Figure 5 shows the numerical grid with the blue zone showing fracture surface, and the surrounding red zone showing the intact rock. To model pre-existing cemented natural fractures, cohesive elements have been embedded as parallel planes perpendicular to the fracture surface (Z-direction) with 5 cm spacing for this example. Hence, fluid pressure during pumping stage will be introduced only to the fracture surface (blue zone), and the rest of the model will be under the effect of in-situ stress only. In case that any part of the natural fractures (cohesive elements) reaches failure threshold, following the opening of the crack, fracturing fluid is supposed to reach the opened part of the fracture and pressurized it, which is consid‐ ered by removing failed cohesive elements and adding gap flow to our model to include fracturing fluid pressure and their cooling effect in natural fractures. Our preliminary models showed the significance of this effect on clustering of secondary fractures and their depth of penetration.

The model assumed homogeneous and isotropic properties for mechanical and hydraulic properties of the rock. Additionally, fracturing of the rock is assumed as an irreversible process. Furthermore, we considered development of major fracture parallel to maximum principal stresses, and natural fractures are assumed to be fully cemented with digenetic cements like quartz or calcite. We used Drucker-Prager model to describe plastic behaviour of rock. A summary of mechanical properties considered for the rock is provided in Table1.

To estimate the parameters of the bilinear cohesive law, no new experimental step is needed. The classic lab tests to measure tensile strength and critical energy release rate should be enough to derive bilinear cohesive law parameters. Among various fracture testing techniques available for homogeneous materials, those which facilitate the stable advance of a fracture are more preferred for interface toughness (Gc) measurements. Examples include the double cantilever crack specimen [26] and Brazilian disk. The tensile strength (σmax) may also be measured using common Brazilian beam tests. Based on the definition of crack energy release rate and bilinear softening, the maximum separation at failure can be determined:

$$G\_c = \frac{1}{2} \sigma\_{\text{max}} \delta\_{\text{max}} \tag{11}$$

show that flaws initiation has noticeable dependency on cohesive stiffness. Small decrease in the cohesive stiffness leads to considerable increase in the cohesive damage, indicating increase of contact area between rock-fluid. As it showed in Figures 6 and 7, loading-unloading residual stress may be considerably effective in reactivating pre-existing natural fractures; moreover, it may have a significant influence on fracture reactivation when assisted by induced thermal stresses due to temperature difference between hydraulic fracturing fluid and formation rock

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**Figure 5.** The three-dimensional finite element model used here, which is a one-half of the fracture, is shown. Fracture

In Figures 8 and 9, initial temperature of 200°C for the rock is selected, and the expansion

to hydraulic fracturing fluid with temperature of 150°C. The temperature difference between hydraulic fracturing fluid and rock induced tensile stresses on the rock matrix. This tensile stress can be intensified by the induced tensile stresses due to elastic unloading of hydraulic fracture plastic deformation as shown in Figure 8. Figure 9 presents the pattern of reactivated natural fractures of Table 3 at the depth on-third of cohesive layer depth far away from the

10-5. The surface of hydraulic fracture is exposed

(Figure 9).

surface is meshed with a fine grids.

surface of hydraulic fracture.

coefficient of rock matrix is assumed to be 1.5x

Some parametric studies have been done on both cohesive parameters to observe their effects on the pattern of reactivated fractures; however, both parameters are basically a function of the composition of digenetic materials and environmental conditions at the time of their precipitation. Obviously, the values of tensile strength and energy release rate for fracture cements are much lower than the values for rock matrix. Strength of a cemented natural fracture is a function of cement type (composition) and its continuity [27]. Gale et al. [28] evaluated the rupture strength in Barnett shale at different depths for different lithofacies. Based on their published laboratory tests, we chose cohesive properties and other mechanical properties of rock. The values of 12 MPa and 3.2 Pa.m are considered for rupture strength and fracture toughness of cemented fractures, respectively. We assumed that net fracture fluid pressure, applied on blue zone in figure 5, is gradually increased to reach 2 MPa and then slowly bleed off to become equal to reservoir pressure. The induced tensile stress due to loading and unloading process during hydraulic fracturing can reactivate pre-existing fractures. Figures 6 and 7 show that the induced tensile stresses due to fluid pressure decline inside fracture can be as important as induced stresses in the loading process. Figure 6 shows reactivated fractures at different depth from the fracture surface at the peak of fracturing fluid pressure and before pressure decline due to leakoff; main mechanism for failure is shear associated with compressive stresses. However, the main mechanism for failure in Figure 7 is associated with residual tensile stresses induced in the unloading due to plasticity. Fractures in Figure 8 are reactivated due to not only plastic effect but also considering a temperature difference between the rock matrix and the hydraulic fracturing fluid.

Table 3 presents the effect of fracture cement strength, fracture toughness and cement resilience on the pattern of opened fractures i.e. failed cohesive elements. By increasing cement strength, the number of initiated cracks on the cohesive layer decreases (Figure 9 (a) and (b)); in addition, the decrease in cement toughness by nearly one-fifth makes considerable increase in the initiated cracks (Figure 9 (a) and (h)). The values for the failed cohesive elements in Table 3 show that flaws initiation has noticeable dependency on cohesive stiffness. Small decrease in the cohesive stiffness leads to considerable increase in the cohesive damage, indicating increase of contact area between rock-fluid. As it showed in Figures 6 and 7, loading-unloading residual stress may be considerably effective in reactivating pre-existing natural fractures; moreover, it may have a significant influence on fracture reactivation when assisted by induced thermal stresses due to temperature difference between hydraulic fracturing fluid and formation rock (Figure 9).

Furthermore, we considered development of major fracture parallel to maximum principal stresses, and natural fractures are assumed to be fully cemented with digenetic cements like quartz or calcite. We used Drucker-Prager model to describe plastic behaviour of rock. A

To estimate the parameters of the bilinear cohesive law, no new experimental step is needed. The classic lab tests to measure tensile strength and critical energy release rate should be enough to derive bilinear cohesive law parameters. Among various fracture testing techniques available for homogeneous materials, those which facilitate the stable advance of a fracture are more preferred for interface toughness (Gc) measurements. Examples include the double cantilever crack specimen [26] and Brazilian disk. The tensile strength (σmax) may also be measured using common Brazilian beam tests. Based on the definition of crack energy release

summary of mechanical properties considered for the rock is provided in Table1.

784 Effective and Sustainable Hydraulic Fracturing

rate and bilinear softening, the maximum separation at failure can be determined:

1 2

Some parametric studies have been done on both cohesive parameters to observe their effects on the pattern of reactivated fractures; however, both parameters are basically a function of the composition of digenetic materials and environmental conditions at the time of their precipitation. Obviously, the values of tensile strength and energy release rate for fracture cements are much lower than the values for rock matrix. Strength of a cemented natural fracture is a function of cement type (composition) and its continuity [27]. Gale et al. [28] evaluated the rupture strength in Barnett shale at different depths for different lithofacies. Based on their published laboratory tests, we chose cohesive properties and other mechanical properties of rock. The values of 12 MPa and 3.2 Pa.m are considered for rupture strength and fracture toughness of cemented fractures, respectively. We assumed that net fracture fluid pressure, applied on blue zone in figure 5, is gradually increased to reach 2 MPa and then slowly bleed off to become equal to reservoir pressure. The induced tensile stress due to loading and unloading process during hydraulic fracturing can reactivate pre-existing fractures. Figures 6 and 7 show that the induced tensile stresses due to fluid pressure decline inside fracture can be as important as induced stresses in the loading process. Figure 6 shows reactivated fractures at different depth from the fracture surface at the peak of fracturing fluid pressure and before pressure decline due to leakoff; main mechanism for failure is shear associated with compressive stresses. However, the main mechanism for failure in Figure 7 is associated with residual tensile stresses induced in the unloading due to plasticity. Fractures in Figure 8 are reactivated due to not only plastic effect but also considering a temperature

*Gc* (11)

= s dmax max

difference between the rock matrix and the hydraulic fracturing fluid.

Table 3 presents the effect of fracture cement strength, fracture toughness and cement resilience on the pattern of opened fractures i.e. failed cohesive elements. By increasing cement strength, the number of initiated cracks on the cohesive layer decreases (Figure 9 (a) and (b)); in addition, the decrease in cement toughness by nearly one-fifth makes considerable increase in the initiated cracks (Figure 9 (a) and (h)). The values for the failed cohesive elements in Table 3

**Figure 5.** The three-dimensional finite element model used here, which is a one-half of the fracture, is shown. Fracture surface is meshed with a fine grids.

In Figures 8 and 9, initial temperature of 200°C for the rock is selected, and the expansion coefficient of rock matrix is assumed to be 1.5x 10-5. The surface of hydraulic fracture is exposed to hydraulic fracturing fluid with temperature of 150°C. The temperature difference between hydraulic fracturing fluid and rock induced tensile stresses on the rock matrix. This tensile stress can be intensified by the induced tensile stresses due to elastic unloading of hydraulic fracture plastic deformation as shown in Figure 8. Figure 9 presents the pattern of reactivated natural fractures of Table 3 at the depth on-third of cohesive layer depth far away from the surface of hydraulic fracture.


1

2

1 2

> 3 4 5

3

4

1

5

6

7

8

9

2

depths

3

4

5

6

16 Effective and Sustainable Hydraulic Fracturing

**reactivated fractures at 10cm depth from surface reactivated fractures at 20cm depth from surface reactivated fractures at surface of hydraulic fracture** 

(a) (b) (c)

(a) (b) (c)

(a) (b) (c)

**Figure 8.** Reactivated fractures by thermo-plasticity for case 5 of Table 2 at different depths

**Fig. 8.** Reactivated fractures by thermo-plasticity for case 5 of Table 2 at different

**Figure 7.** Reactivated fractures after unloading for case 5 of Table 2 at different depths

**Fig. 7.** Reactivated fractures after unloading for case 5 of Table 2 at different depths

18 Effective and Sustainable Hydraulic Fracturing

**reactivated fractures at surface of hydraulic fracture reactivated fractures at 10cm depth from surface reactivated fractures at 20cm depth from surface** 

**Figure 6.** Reactivated fractures after loading for case 5 of Table 2 at different depths

**reactivated fractures at surface of hydraulic fracture** 

**Fig. 6.** Reactivated fractures after loading for case 5 of Table 2 at different depths

Secondary Fractures and Their Potential Impacts 17

**reactivated fractures at 10cm depth from surface reactivated fractures at 20cm depth from surface** 

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**Table 1.** Rock properties in simulation


#### **Table 2.** Cohesive parameters


**Table 3.** Failed cohesive elements

**Fig. 6.** Reactivated fractures after loading for case 5 of Table 2 at different depths

16 Effective and Sustainable Hydraulic Fracturing

**Figure 6.** Reactivated fractures after loading for case 5 of Table 2 at different depths

Young's modulus 26 GPa Rock density 2100 Kg/m3 Rock friction angle 30° Rock dilation angle 20° Poisson's ratio 0.27 Rock yield stress 30MPa

**Hydraulic pressure**

**(MPa) Cohesive stiffness (GPa)**

**Failed cohesive elements by Thermoplasticity**

**Studied parameter** 1

2

1 2

> 3 4 5

3

4

1

5

6

7

8

9

2

3

4

5

6

Reference case

Tensile strength

Cohesive Stiffness

Cement toughness

Tensile strength

Cohesive Stiffness

Cohesive Stiffness

Cement toughness

**Cement toughness (Pa.m)**

**Failed cohesive elements after loading**

1 14700 3406 4692 5511

2 14700 0 0 7923

3 14700 0 0 0

4 14700 0 0 1

5 14700 415 1247 1683

6 14700 0 0 0

7 14700 18 74 154

8 14700 298 484 2440

 12 3.2 2 6.4 1.2 3.2 2 6.4 12 3.2 2 0.64 12 32 2 6.4 20 3.2 2 6.4 12 3.2 2 1 12 3.2 2 3.4 12 15 6.4 2

> **Failed cohesive elements after unloading**

**Table 1.** Rock properties in simulation

786 Effective and Sustainable Hydraulic Fracturing

**Case Tensile strength**

**Table 2.** Cohesive parameters

**Table 3.** Failed cohesive elements

**Total cohesive elements**

**Case**

**(MPa)**

**Fig. 7.** Reactivated fractures after unloading for case 5 of Table 2 at different depths

18 Effective and Sustainable Hydraulic Fracturing

**Figure 7.** Reactivated fractures after unloading for case 5 of Table 2 at different depths

**Fig. 8.** Reactivated fractures by thermo-plasticity for case 5 of Table 2 at different

depths **Figure 8.** Reactivated fractures by thermo-plasticity for case 5 of Table 2 at different depths

1

2

3

4

5

6

in the rock. The effect of energy release rate and cohesive tensile strength investigated and it was shown that the decrease of these parameters can activate more natural fractures but not in a uniform pattern. Moreover, the simulation showed that the effect of plasticity can be more considerable when it would be helped by thermal stress induced by temperature difference

Secondary Fractures and Their Potential Impacts on Hydraulic Fractures Efficiency

http://dx.doi.org/10.5772/56360

789

and J.E. Olson2

[1] Adachi, J, Siebrits, E, Peirce, A, & Desroches, J. Computer simulation of hydraulic fractures, International Journal of Rock Mechanics & Mining Sciences 44, (2007). ,

[2] Barenblatt, G. K. the mathematical theory of equilibrium cracks in brittle fractures.

[3] Barr, D. T. Thermal cracking in nonporous geothermal reservoirs. Master Thesis, De‐

[4] Batchelor, G. K. An introduction to fluid dynamics. Cambridge University Press,

[5] Bazant, Z. P, Wahab, M, & Instability, A. B. and spacing of cooling or shrinkage

[6] Bazant, Z. P, & Planas, J. Fracture and size effect in concrete and other quasibrittle

[7] Benzeggagh, M. L, & Kenane, M. Measurement of mixed-mode delamination fracture toughness of unidirectional glass\_epoxy Composites with Mixed-mode Bending Ap‐

[8] Biot, M. A, Masse, L, & Medlin, W. L. Temperature analysis in hydraulic fracturing.

[9] Bonnet, E, Bour, O, Odling, N. E, Davy, P, Main, I, Cowie, P, & Berkowitz, B. Scaling

of fracture systems in geological media. Reviews of Geophysics, (2001).

between rock matrix and fracturing fluid.

, Milad Ahmadi1

**Author details**

**References**

739-757.

Arash Dahi Taleghani1

1 Louisiana State University, USA

2 The University of Texas at Austin, USA

Journal of Applied Mechanics, (1962).

Journal of Petroleum Technology, (1987).

Cambridge, UK, (1967).

partment of Mechanical Engineering, MIT, (1980).

cracks. Journal of the Engineering Mechanical division, (1979).

materials. Published by CRC Press; first edition, (1998). 084938284

paratus. Journal of Composites Science and Technology, (1996).

**Fig. 9.** Reactivated fractures by thermoplasticity for different cases of Table 2 at onethird depth from the surface of hydraulic fracture **Figure 9.** Reactivated fractures by thermoplasticity for different cases of Table 2 at one-third depth from the surface of hydraulic fracture

#### **6. Conclusion**

Induced tensile stresses are expected to occur in rocks with significant plastic behaviour during loading and unloading of hydraulic fractures. These induced stresses can open pre-existing natural fractures in the formation and even open the cemented natural fractures. These activated fractures can provide more rock-fluid contact area. The size of these natural fractures is much less than the main hydraulic fracture but presence of these fractures in considerable numbers can significantly increase the contact area between the wellbore and formation. The path of initiation and propagation of these fractures can be induced by natural fractures. To study their effect, a three-dimensional finite element model with cohesive interfaces embedded in the rock. The effect of energy release rate and cohesive tensile strength investigated and it was shown that the decrease of these parameters can activate more natural fractures but not in a uniform pattern. Moreover, the simulation showed that the effect of plasticity can be more considerable when it would be helped by thermal stress induced by temperature difference between rock matrix and fracturing fluid.
