**4. Simulation results**

(15<sup>o</sup> )

)

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

**Parameter Symbol Value**

Elastic modulus *Ec* 30 GPa

Poisson's ratio μ 0.25

Internal friction angle ϕ 37

**Table 1.** Rock material mechanical parameter

496 Effective and Sustainable Hydraulic Fracturing

**Table 2.** Bedding material mechanical parameter

**Elastic modulus**

**(***Ec***)**

Uniaxial compressive strength *fc* 200 MPa

**Parameter Symbol Value**

Elastic modulus *Ec* 3.0 GPa

Uniaxial compressive strength *fc* 20 MPa

Coefficient of permeability *K* 0.00864 m/d

30GPa 200MPa 3.0GPa(1/10) 200MPa(1) 30GPa 200MPa 1.5GPa(1/20) 200MPa(1) 30GPa 200MPa 0.5GPa(1/60) 200MPa(1) 30GPa 200MPa 30GPa(1) 20MPa(1/10) 30GPa 200MPa 30GPa(1) 10MPa(1/20) 30GPa 200MPa 30GPa(1) 3.33MPa(1/60) 30GPa 200MPa 3.0GPa(1/10) 20MPa(1/10)

**Table 3.** Change of elastic modulus and uniaxial compressive strength values of bedding material

**Elastic modulus**

**Uniaxial compressive**

**strength (***fc***)**

**(***Ec***)**

Poisson's ratio μ 0.25

Internal friction angle ϕ 37

**Rock material Bedding material**

**Uniaxial compressive**

**strength (***fc***)**

Homogeneity index *m* 2

Coefficient of permeability *K* 0.000864 m/d

Homogeneity index *m* 2

#### **4.1. The effect of perforation angles**

The initiation pressure, the breakdown pressure and the fracture evolution of seven rock specimens with different perforation angles under constant confining pressure and increasing hydraulic pressure are simulated. The results reflect the damage evolution process of rock specimen, which causes the macroscopic damage by microscopic under hydraulic fracturing and is consistent with the experimental result in [4]. Pore pressure and the minimum principal stress distribution of specimens with different perforation angles which achieved by numerical simulation are shown from figure 4 to figure 10. The comparison of the numerical simulation result and the experimental result which has the same perforation angle (60o ) and under the same ground stress difference (5MPa) is shown in figure 11, and the values of the initiation and the breakdown pressure are shown in figure 12.

)

)

Figure 4 Pore pressure and the minimum principal stress distribution in fracture evolution process (0<sup>o</sup> **Figure 4.** Pore pressure and the minimum principal stress distribution in fracture evolution process (0o) Minimum principal stress

Figure 4 Pore pressure and the minimum principal stress distribution in fracture evolution process (0<sup>o</sup>

Figure 5 Pore pressure and the minimum principal stress distribution in fracture evolution process (15<sup>o</sup> **Figure 5.** Pore pressure and the minimum principal stress distribution in fracture evolution process (15o)

Pore pressure

Minimum principal stress

Pore pressure

(15<sup>o</sup> )

> (45<sup>o</sup> )

Figure 5 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 4 Pore pressure and the minimum principal stress distribution in fracture evolution process (0<sup>o</sup>

)

Figure 7 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 8 Pore pressure and the minimum principal stress distribution in fracture evolution process

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Figure 9 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 9.** Pore pressure and the minimum principal stress distribution in fracture evolution process (75o) Figure 9 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 10 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 11.** Comparison of numerical simulation and experimental results in [4] which has the same perforation angle

Figure 10 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 10.** Pore pressure and the minimum principal stress distribution in fracture evolution process (90o)

**Turning fracture** 

**Turning fracture** 

(45<sup>o</sup> ) Pore pressure

Pore pressure

Pore pressure

Pore pressure

Minimum principal stress

(60o) and under the same ground stress difference (5MPa)

Minimum principal stress

Minimum principal stress

Minimum principal stress

(60<sup>o</sup> )

(75<sup>o</sup> )

(90<sup>o</sup> ) (75<sup>o</sup> )

Pore pressure

(90<sup>o</sup> )

Minimum principal stress

**Figure 6.** Pore pressure and the minimum principal stress distribution in fracture evolution process (30o)

Figure 7 Pore pressure and the minimum principal stress distribution in fracture evolution process (45<sup>o</sup> ) **Figure 7.** Pore pressure and the minimum principal stress distribution in fracture evolution process (45o) Figure 7 Pore pressure and the minimum principal stress distribution in fracture evolution process

Minimum principal stress

(60<sup>o</sup> ) Figure 8 Pore pressure and the minimum principal stress distribution in fracture evolution process (60<sup>o</sup> ) **Figure 8.** Pore pressure and the minimum principal stress distribution in fracture evolution process (60o)

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

Figure 8 Pore pressure and the minimum principal stress distribution in fracture evolution process Minimum principal stress Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles… http://dx.doi.org/10.5772/56012 499

Figure 7 Pore pressure and the minimum principal stress distribution in fracture evolution process

(45<sup>o</sup> ) Pore pressure

Pore pressure

Minimum principal stress

(60<sup>o</sup> )

(75<sup>o</sup> )

(90<sup>o</sup> ) (75<sup>o</sup> )

**Figure 9.** Pore pressure and the minimum principal stress distribution in fracture evolution process (75o) Figure 9 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 4 Pore pressure and the minimum principal stress distribution in fracture evolution process (0<sup>o</sup>

Pore pressure

Minimum principal stress

Pore pressure

498 Effective and Sustainable Hydraulic Fracturing

Minimum principal stress

Pore pressure

Minimum principal stress

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Figure 5 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 6.** Pore pressure and the minimum principal stress distribution in fracture evolution process (30o)

Figure 7 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 7.** Pore pressure and the minimum principal stress distribution in fracture evolution process (45o) Figure 7 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 8 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 8 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 8.** Pore pressure and the minimum principal stress distribution in fracture evolution process (60o)

(15<sup>o</sup> )

> (45<sup>o</sup> )

> > (45<sup>o</sup> )

(60<sup>o</sup> )

> (60<sup>o</sup> )

)

Figure 10 Pore pressure and the minimum principal stress distribution in fracture evolution process (90<sup>o</sup> ) **Figure 10.** Pore pressure and the minimum principal stress distribution in fracture evolution process (90o) Figure 10 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 11.** Comparison of numerical simulation and experimental results in [4] which has the same perforation angle (60o) and under the same ground stress difference (5MPa)

ground stress difference (5MPa)

Figure 9. Pore pressure and the minimum principal stress distribution in fracture evolution process (75o)

Figure 10.Pore pressure and the minimum principal stress distribution in fracture evolution process (90o)

Figure 11.Comparison of numerical simulation and experimental results in [4] which has the same perforation angle (60o) and under the same

Figure 13 Pore pressure and the minimum principal stress distribution in fracture evolution process

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Figure 14 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 15 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 15.** Pore pressure and the minimum principal stress distribution in fracture evolution process (30o) Figure 15 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 16 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 16 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 16.** Pore pressure and the minimum principal stress distribution in fracture evolution process (45o)

Figure 17 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 14.** Pore pressure and the minimum principal stress distribution in fracture evolution process (15o) Figure 14 Pore pressure and the minimum principal stress distribution in fracture evolution process

(0o ) Pore pressure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Minimum principal stress

Pore pressure

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

(15<sup>o</sup> )

(15<sup>o</sup> )

(30<sup>o</sup> )

> (30<sup>o</sup> )

(45<sup>o</sup> )

(45<sup>o</sup> )

(60<sup>o</sup> )

Figure 12.Changes of initiation and breakdown pressure of different perforation angle specimens **Figure 12.** Changes of initiation and breakdown pressure of different perforation angle specimens

#### **4.2. The effect of bedding angles 4.2. The effect of bedding angles**

(0o )

Pore pressure

Pore pressure

Minimum principal stress

(15<sup>o</sup> )

The initiation pressure, the breakdown pressure and the fracture evolution of seven rock specimens with different bedding angles are simulated. Pore pressure and the minimum principal stress distribution achieved by numerical simulation are shown from figure 13 to figure 19, and the values of initiation and breakdown pressure shown in figure 20. The initiation pressure, the breakdown pressure and the fracture evolution of seven rock specimens with different bedding angles are simulated. Pore pressure and the minimum principal stress distribution achieved by numerical simulation are shown from figure 13 to figure 19, and the values of initiation and breakdown pressure shown in figure 20.

Figure 13 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 14 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 15.Pore pressure and the minimum principal stress distribution in fracture evolution process (30o)

Figure 16.Pore pressure and the minimum principal stress distribution in fracture evolution process (45o)

**Figure 13.** Pore pressure and the minimum principal stress distribution in fracture evolution process (0o)

Figure 13 Pore pressure and the minimum principal stress distribution in fracture evolution process Minimum principal stress Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles… http://dx.doi.org/10.5772/56012 501

Figure 14 Pore pressure and the minimum principal stress distribution in fracture evolution process (15<sup>o</sup> ) **Figure 14.** Pore pressure and the minimum principal stress distribution in fracture evolution process (15o) Figure 14 Pore pressure and the minimum principal stress distribution in fracture evolution process

Minimum principal stress

Minimum principal stress

Pore pressure

Minimum principal stress

(0o )

(15<sup>o</sup> )

)

(30<sup>o</sup> )

(60<sup>o</sup> ) Pore pressure

Figure 9. Pore pressure and the minimum principal stress distribution in fracture evolution process (75o)

Figure 10.Pore pressure and the minimum principal stress distribution in fracture evolution process (90o)

Figure 12.Changes of initiation and breakdown pressure of different perforation angle specimens

Breakdown pressure Initiation pressure

figure 13 to figure 19, and the values of initiation and breakdown pressure shown in figure 20.

Figure 13.Pore pressure and the minimum principal stress distribution in fracture evolution process (0o)

Figure 14.Pore pressure and the minimum principal stress distribution in fracture evolution process (15o)

Figure 15.Pore pressure and the minimum principal stress distribution in fracture evolution process (30o)

Figure 16.Pore pressure and the minimum principal stress distribution in fracture evolution process (45o)

ground stress difference (5MPa)

0

**4.2. The effect of bedding angles**

Pore pressure

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

(0o )

(15<sup>o</sup> ) 5

10

15

Hydraulic pressure(MPa)

20

25

500 Effective and Sustainable Hydraulic Fracturing

**4.2. The effect of bedding angles** 

**Figure 12.** Changes of initiation and breakdown pressure of different perforation angle specimens

figure 19, and the values of initiation and breakdown pressure shown in figure 20.

0 10 20 30 40 50 60 70 80 90 100 Perforation angle

The initiation pressure, the breakdown pressure and the fracture evolution of seven rock specimens with different bedding angles are simulated. Pore pressure and the minimum principal stress distribution achieved by numerical simulation are shown from figure 13 to

Figure 13 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 13.** Pore pressure and the minimum principal stress distribution in fracture evolution process (0o)

Figure 14 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 15 Pore pressure and the minimum principal stress distribution in fracture evolution process (30<sup>o</sup> **Figure 15.** Pore pressure and the minimum principal stress distribution in fracture evolution process (30o) Figure 15 Pore pressure and the minimum principal stress distribution in fracture evolution process

(45<sup>o</sup> ) Figure 16 Pore pressure and the minimum principal stress distribution in fracture evolution process (45<sup>o</sup> ) **Figure 16.** Pore pressure and the minimum principal stress distribution in fracture evolution process (45o)

Figure 16 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 17 Pore pressure and the minimum principal stress distribution in fracture evolution process

(30<sup>o</sup> )

(45<sup>o</sup> )

)

)

(75<sup>o</sup> )

Figure 16 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 15 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 18 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 19 Pore pressure and the minimum principal stress distribution in fracture evolution process

Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles…

0 10 20 30 40 50 60 70 80 90 100 Bedding angle

breakdown pressure and the fracture evolution of seven rock specimens with different bedding materials are simulated. Pore pressure and the minimum principal stress distribution achieved by numerical simulation are shown from figure 21 to figure 27 and the values of initiation and

Figure 21 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 21.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus

Figure 22 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 20.** Changes of initiation and breakdown pressure of different bedding angle specimens

Breakdown pressure Initiation pressure

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bedding angle for example, the initiation pressure, the

(75<sup>o</sup> )

(90<sup>o</sup> )

0

**4.3. The effect of bedding materials**

Taking the rock specimen of 60o

breakdown pressure shown in figure 28.

Pore pressure

Minimum principal stress

(elastic modulus value is 3.0GPa)

value is 3.0GPa)

Pore pressure

Minimum principal stress

(elastic modulus value is 1.5GPa)

Pore pressure

5

10

Hydraulic pressure(MPa)

15

20

25

Pore pressure

Minimum principal stress

Pore pressure

Minimum principal stress

Minimum principal stress

Pore pressure

Figure 17 Pore pressure and the minimum principal stress distribution in fracture evolution process (60<sup>o</sup> ) **Figure 17.** Pore pressure and the minimum principal stress distribution in fracture evolution process (60o)

Figure 18 Pore pressure and the minimum principal stress distribution in fracture evolution process (75<sup>o</sup> **Figure 18.** Pore pressure and the minimum principal stress distribution in fracture evolution process (75o) Figure 18 Pore pressure and the minimum principal stress distribution in fracture evolution process

(90<sup>o</sup> ) Figure 19 Pore pressure and the minimum principal stress distribution in fracture evolution process (90<sup>o</sup> **Figure 19.** Pore pressure and the minimum principal stress distribution in fracture evolution process (90o)

Hydraulic pressure(MPa)

Hydraulic pressure(MPa)

Figure 19 Pore pressure and the minimum principal stress distribution in fracture evolution process

0 10 20 30 40 50 60 70 80 90 100 Bedding angle

0 10 20 30 40 50 60 70 80 90 100 Bedding angle

Breakdown pressure Initiation pressure

Breakdown pressure Initiation pressure

Minimum principal stress Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles… http://dx.doi.org/10.5772/56012 503

Figure 19 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 18 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 20.** Changes of initiation and breakdown pressure of different bedding angle specimens

#### **4.3. The effect of bedding materials**

(elastic modulus value is 3.0GPa)

Pore pressure

Minimum principal stress

(elastic modulus value is 1.5GPa)

Pore pressure

(75<sup>o</sup> ) Pore pressure

Pore pressure

Minimum principal stress

(90<sup>o</sup> )

Figure 15 Pore pressure and the minimum principal stress distribution in fracture evolution process

Minimum principal stress

Pore pressure

Minimum principal stress

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Minimum principal stress

Figure 16 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 17 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 18 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 18.** Pore pressure and the minimum principal stress distribution in fracture evolution process (75o) Figure 18 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 19 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 19 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 19.** Pore pressure and the minimum principal stress distribution in fracture evolution process (90o)

0 10 20 30 40 50 60 70 80 90 100 Bedding angle

0 10 20 30 40 50 60 70 80 90 100 Bedding angle

Breakdown pressure Initiation pressure

Breakdown pressure Initiation pressure

**Figure 17.** Pore pressure and the minimum principal stress distribution in fracture evolution process (60o)

(30<sup>o</sup> )

(45<sup>o</sup> )

502 Effective and Sustainable Hydraulic Fracturing

(60<sup>o</sup> )

(75<sup>o</sup> )

(75<sup>o</sup> )

(90<sup>o</sup> )

(90<sup>o</sup> )

Hydraulic pressure(MPa)

Hydraulic pressure(MPa)

Taking the rock specimen of 60o bedding angle for example, the initiation pressure, the breakdown pressure and the fracture evolution of seven rock specimens with different bedding materials are simulated. Pore pressure and the minimum principal stress distribution achieved by numerical simulation are shown from figure 21 to figure 27 and the values of initiation and breakdown pressure shown in figure 28.

**Figure 21.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus value is 3.0GPa)

Figure 22 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 21 Pore pressure and the minimum principal stress distribution in fracture evolution process

(elastic modulus value is 3.0GPa)

(elastic modulus value is 1.5GPa)

(elastic modulus value is 0.5GPa)

Pore pressure

Figure 21 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 23 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 24 Pore pressure and the minimum principal stress distribution in fracture evolution process

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Figure 25 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 26 Pore pressure and the minimum principal stress distribution in fracture evolution process

pressive strength value is 3.33MPa) Figure 26 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 26.** Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial com‐

Figure 27 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 27 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 27.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus

(elastic modulus value is 3.0GPa and uniaxial compressive strength value is 20MPa)

(elastic modulus value is 3.0GPa and uniaxial compressive strength value is 20MPa)

**Figure 25.** Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial com‐

(elastic modulus value is 0.5GPa)

Pore pressure

Minimum principal stress

(uniaxial compressive strength value is 20MPa)

Pore pressure

Minimum principal stress

(uniaxial compressive strength value is 10MPa)

Pore pressure

Pore pressure

(uniaxial compressive strength value is 3.33MPa)

Minimum principal stress

(uniaxial compressive strength value is 3.33MPa)

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

value is 3.0GPa and uniaxial compressive strength value is 20MPa)

Minimum principal stress

pressive strength value is 10MPa)

Minimum principal stress

Figure 22 Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus value is 1.5GPa) **Figure 22.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus value is 1.5GPa) Figure 22 Pore pressure and the minimum principal stress distribution in fracture evolution process Minimum principal stress

Figure 23 Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus value is 0.5GPa) **Figure 23.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus value is 0.5GPa) Figure 23 Pore pressure and the minimum principal stress distribution in fracture evolution process Minimum principal stress

Figure 24 Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial compressive strength value is 20MPa) **Figure 24.** Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial com‐ pressive strength value is 20MPa)

Figure 25 Pore pressure and the minimum principal stress distribution in fracture evolution process

(uniaxial compressive strength value is 10MPa)

Minimum principal stress

Pore pressure

Figure 23 Pore pressure and the minimum principal stress distribution in fracture evolution process

(elastic modulus value is 0.5GPa)

Pore pressure

Minimum principal stress

(uniaxial compressive strength value is 20MPa)

(uniaxial compressive strength value is 3.33MPa)

Figure 21 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 22 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 22 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 23 Pore pressure and the minimum principal stress distribution in fracture evolution process

Figure 24 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 24.** Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial com‐

Figure 25 Pore pressure and the minimum principal stress distribution in fracture evolution process

value is 0.5GPa) Figure 23 Pore pressure and the minimum principal stress distribution in fracture evolution process

**Figure 23.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus

**Figure 22.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus

(elastic modulus value is 3.0GPa)

504 Effective and Sustainable Hydraulic Fracturing

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

Pore pressure

Minimum principal stress

(elastic modulus value is 1.5GPa)

(elastic modulus value is 1.5GPa)

(elastic modulus value is 0.5GPa)

(elastic modulus value is 0.5GPa)

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

(uniaxial compressive strength value is 20MPa)

pressive strength value is 20MPa)

Pore pressure

(uniaxial compressive strength value is 10MPa)

Minimum principal stress

value is 1.5GPa)

Pore pressure

Pore pressure

Minimum principal stress

Minimum principal stress

Figure 25 Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial compressive strength value is 10MPa) **Figure 25.** Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial com‐ pressive strength value is 10MPa)

Figure 26 Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial compressive strength value is 3.33MPa) **Figure 26.** Pore pressure and the minimum principal stress distribution in fracture evolution process (uniaxial com‐ pressive strength value is 3.33MPa) Figure 26 Pore pressure and the minimum principal stress distribution in fracture evolution process Minimum principal stress

Figure 27 Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus value is 3.0GPa and uniaxial compressive strength value is 20MPa) **Figure 27.** Pore pressure and the minimum principal stress distribution in fracture evolution process (elastic modulus value is 3.0GPa and uniaxial compressive strength value is 20MPa)

Minimum principal stress

Figure 28 Changes of initiation and breakdown pressure of different bedding material specimens

Figure 27 Pore pressure and the minimum principal stress distribution in fracture evolution process

released from parts of the high stress area and transferred to the fracture tip, which makes the fracture propagate continually under the constant hydraulic pressure. The speed of fracture propagation become faster and faster and the main fracture and the secondary fractures are

Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles…

From figure 4 to figure 10, it can be concluded that no matter how the perforation azimuth changes, the fracture is still initiating on the perforation tip, which is because of the casing. But after the fracture initiate, the fracture propagation will turn to the horizontal direction (the maximum principal stress direction) gradually under the increasing hydraulic pressure and form a turning fracture finally. The perforation angle is bigger, the fracture turning will be more obvious and the turning distance will be bigger. Fracture propagation is always de‐ viating from the perforation direction to the maximum principal stress direction (horizontal direction), which proves that the effect of perforation angle on the direction of fracture prop‐ agation is small and the maximum principal stress control the final fracture propagation di‐

The results also show that the perforation angle determines the initiation and the break‐ down pressure of rock specimens. With the perforation angle increases, the initiation pres‐ sure are 15.2MPa, 15.2MPa, 15.3MPa, 15.2MPa, 15.2MPa, 20.1MPa and 21.4MPa respectively, and the breakdown pressure 16.8MPa, 16.7MPa, 17.1MPa, 19.4MPa, 19.4MPa, 22.5MPa and 23.2MPa respectively. The initiation pressure and the breakdown pressure are divided into

increase of α, the values of initiation pressure will increase gradually; When α≤30<sup>o</sup>

breakdown pressure will increase obviously and with the increase of α, the values of break‐ down pressure will increase gradually, of which the increase rate is smaller than that of ini‐

initiation and breakdown pressure are small, which may help reduce fracturing cost and im‐ prove the fracturing efficiency. The comparison of numerical simulation results and the ex‐

(5MPa) is shown in figure 11 and we can find that the macroscopic fracture propagation of

The results of first group simulation indicates that the maximum principal stress determines the fracture propagation direction, and the effect of bedding angles of rock specimens under the same confining pressure on fracture propagation will be studied in the second group.

From the simulation results of the second group, we can conclude that the fracture initiation and the propagation pattern of rock specimens under constant confining pressure are chang‐ ing gradually as bedding angle increases. From figure 13 to figure 19, we can see that, when

with the tension failure bedding. Because of the stress accumulation, there exist a high ten‐ sile stress area on the fracture tip and because the bedding material is weaker than rock ma‐

ues of breakdown pressure are small and basically constant, while as α>30<sup>o</sup>

the numerical simulation is basically consistent with the experimental results.


perimental results with the same perforation angle (60o

~15o

, the values of initiation pressure are small and basically

is the best perforation azimuth area and the values of

), the initiation and propagation of fracture are only along

, the val‐

, the value of

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) and ground stress difference

, the values of initiation pressure increase obviously and with the

connected finally.

rection.

two stages (figure 12): When α≤60<sup>o</sup>

tiation pressure. Therefore, 0o

bedding angle α is small (0o

constant, while as α>60<sup>o</sup>

(elastic modulus value is 3.0GPa and uniaxial compressive strength value is 20MPa)

From the simulation results of the first group, we can see that the hydraulic fracturing process of the **Figure 28.** Changes of initiation and breakdown pressure of different bedding material specimens

#### three stages: **5. Discussions**

5 Discussions

1. Stress accumulation stage In this stage, there doesn't appear any fracture and broken element, but as the pore pressure increases step by step, the stress is accumulating on perforation tip gradually and forming a high minimum From the simulation results of the first group, we can see that the hydraulic fracturing process of the rock specimens with different perforation angle under constant confining pressure is divided into three stages:

rock specimens with different perforation angle under constant confining pressure is divided into

principal stress area (green zone). Because of the tensile strength of rock is far less than the **1.** Stress accumulation stage

compressive strength, it can be speculated that the fracture initiation will be happened on the perforation tip where tensile stress is the largest; 2. Steady propagation stage The fracture will initiate and propagate on perforation tip when the minimum principal stress In this stage, there doesn't appear any fracture and broken element, but as the pore pressure increases step by step, the stress is accumulating on perforation tip gradually and forming a high minimum principal stress area (green zone). Because of the tensile strength of rock is far less than the compressive strength, it can be speculated that the fracture initiation will be happened on the perforation tip where tensile stress is the largest;

accumulates to a certain point (tensile strength). In this stage, lots of micro fractures will appear on **2.** Steady propagation stage

the main fracture tip as the loading step increases, and distributing as an umbrella and disconnected to each other; 3. Unsteady propagation stage The fracture will initiate and propagate on perforation tip when the minimum principal stress accumulates to a certain point (tensile strength). In this stage, lots of micro fractures will appear on the main fracture tip as the loading step increases, and distributing as an umbrella and disconnected to each other;

**3.** Unsteady propagation stage

As the number of micro fractures increase, some micro fractures connect to each other and become secondary fractures. In this stage, in the process of fracture propagation, stress is released from parts of the high stress area and transferred to the fracture tip, which makes the fracture propagate continually under the constant hydraulic pressure. The speed of fracture propagation become faster and faster and the main fracture and the secondary fractures are connected finally.

Figure 27 Pore pressure and the minimum principal stress distribution in fracture evolution process

(elastic modulus value is 3.0GPa and uniaxial compressive strength value is 20MPa)

(1/10,1/10) (1/10,1) (1/20,1) (1/

happened on the perforation tip where tensile stress is the largest;

3

**Figure 28.** Changes of initiation and breakdown pressure of different bedding material specimens

4

The ratio of bedding material and rock material(elastic modulus,uniaxial compressive strength)

From the simulation results of the first group, we can see that the hydraulic fracturing process of the rock specimens with different perforation angle under constant confining pressure is divided into

In this stage, there doesn't appear any fracture and broken element, but as the pore pressure increases step by step, the stress is accumulating on perforation tip gradually and forming a high minimum principal stress area (green zone). Because of the tensile strength of rock is far less than the compressive strength, it can be speculated that the fracture initiation will be happened on the

From the simulation results of the first group, we can see that the hydraulic fracturing process of the rock specimens with different perforation angle under constant confining pressure is

In this stage, there doesn't appear any fracture and broken element, but as the pore pressure increases step by step, the stress is accumulating on perforation tip gradually and forming a high minimum principal stress area (green zone). Because of the tensile strength of rock is far less than the compressive strength, it can be speculated that the fracture initiation will be

The fracture will initiate and propagate on perforation tip when the minimum principal stress accumulates to a certain point (tensile strength). In this stage, lots of micro fractures will appear on the main fracture tip as the loading step increases, and distributing as an umbrella and disconnected to

The fracture will initiate and propagate on perforation tip when the minimum principal stress accumulates to a certain point (tensile strength). In this stage, lots of micro fractures will appear on the main fracture tip as the loading step increases, and distributing as an umbrella and

As the number of micro fractures increase, some micro fractures connect to each other and become secondary fractures. In this stage, in the process of fracture propagation, stress is

Figure 28 Changes of initiation and breakdown pressure of different bedding material specimens

60,1) (1,1/10) (1,1/20) (1,1/60)

6

7

8

Breakdown pressure Initiation pressure

5

2

0

1. Stress accumulation stage

divided into three stages:

**1.** Stress accumulation stage

**5. Discussions**

2. Steady propagation stage

3. Unsteady propagation stage

disconnected to each other;

**3.** Unsteady propagation stage

**2.** Steady propagation stage

0

1

Minimum principal stress

perforation tip where tensile stress is the largest;

5

10

Hydraulic pressure(MPa)

5 Discussions

three stages:

each other;

15

20

25

506 Effective and Sustainable Hydraulic Fracturing

From figure 4 to figure 10, it can be concluded that no matter how the perforation azimuth changes, the fracture is still initiating on the perforation tip, which is because of the casing. But after the fracture initiate, the fracture propagation will turn to the horizontal direction (the maximum principal stress direction) gradually under the increasing hydraulic pressure and form a turning fracture finally. The perforation angle is bigger, the fracture turning will be more obvious and the turning distance will be bigger. Fracture propagation is always de‐ viating from the perforation direction to the maximum principal stress direction (horizontal direction), which proves that the effect of perforation angle on the direction of fracture prop‐ agation is small and the maximum principal stress control the final fracture propagation di‐ rection.

The results also show that the perforation angle determines the initiation and the break‐ down pressure of rock specimens. With the perforation angle increases, the initiation pres‐ sure are 15.2MPa, 15.2MPa, 15.3MPa, 15.2MPa, 15.2MPa, 20.1MPa and 21.4MPa respectively, and the breakdown pressure 16.8MPa, 16.7MPa, 17.1MPa, 19.4MPa, 19.4MPa, 22.5MPa and 23.2MPa respectively. The initiation pressure and the breakdown pressure are divided into two stages (figure 12): When α≤60<sup>o</sup> , the values of initiation pressure are small and basically constant, while as α>60<sup>o</sup> , the values of initiation pressure increase obviously and with the increase of α, the values of initiation pressure will increase gradually; When α≤30<sup>o</sup> , the val‐ ues of breakdown pressure are small and basically constant, while as α>30<sup>o</sup> , the value of breakdown pressure will increase obviously and with the increase of α, the values of break‐ down pressure will increase gradually, of which the increase rate is smaller than that of ini‐ tiation pressure. Therefore, 0o -30o is the best perforation azimuth area and the values of initiation and breakdown pressure are small, which may help reduce fracturing cost and im‐ prove the fracturing efficiency. The comparison of numerical simulation results and the ex‐ perimental results with the same perforation angle (60o ) and ground stress difference (5MPa) is shown in figure 11 and we can find that the macroscopic fracture propagation of the numerical simulation is basically consistent with the experimental results.

The results of first group simulation indicates that the maximum principal stress determines the fracture propagation direction, and the effect of bedding angles of rock specimens under the same confining pressure on fracture propagation will be studied in the second group.

From the simulation results of the second group, we can conclude that the fracture initiation and the propagation pattern of rock specimens under constant confining pressure are chang‐ ing gradually as bedding angle increases. From figure 13 to figure 19, we can see that, when bedding angle α is small (0o ~15o ), the initiation and propagation of fracture are only along with the tension failure bedding. Because of the stress accumulation, there exist a high ten‐ sile stress area on the fracture tip and because the bedding material is weaker than rock ma‐ terial, the fracture propagation is along the cracked bedding and form a straight fracture eventually. In this case, the bedding plane determines the fracture evolution.

(bedding elastic modulus/rock elastic modulus is 1/10, 1/20, 1/60), the pattern of fracture propagation will be unchanged. Because of the reduction of elastic modulus, the initiation pressure reduced (15.5 MPa, 15.2 MPa, 14.7 MPa) and the breakdown pressure increased slightly (19.7 MPa, 19.9 MPa, 20.4 MPa), however, both of the reduction and the increase can be ignored because the values of the initiation and the breakdown pressure are almost con‐ stant (figure 28). As the stiffness of the bedding material (bedding elastic modulus/rock elas‐ tic modulus is 1) is constant but the strength decreased (bedding strength/rock strength is 1/10, 1/20, 1/60), the pattern of the fracture propagation will be unchanged, and the values of initiation (14.8 MPa, 14.7 MPa, 12.8 MPa) and breakdown pressure (20 MPa, 19.2 MPa, 18.4 MPa) decreased gradually with almost the same decrease rates (figure 28). As both of the stiffness and strength are decreased (bedding elastic modulus/rock elastic modulus is 1/10, bedding strength/rock strength is 1/10), the initiation pressure, the breakdown pressure and the pattern of fracture propagation are almost the same as the condition of (1, 1/10). As sug‐ gested above, the stiffness of bedding material has little influence on initiation pressure, breakdown pressure and fracture evolution of rock specimens, except that the strength de‐

Numerical Simulation of Hydraulic Fracturing in Heterogeneous Rock: The Effect of Perforation Angles…

In summary, the damage process of rock specimen are determined by the maximum princi‐ pal stress, the bedding angle and the strength of bedding material, while the effect of perfo‐

perforation angle turns bigger, the fracture turning will be more obvious and the turning distance bigger. The effect of perforation angle on fracture propagation direction is small,

, a turning fracture will be formed, and if the

http://dx.doi.org/10.5772/56012

509

is the best perforation angle area. The initiation and

Based on the simulation results of three groups, the following can be concluded:


and the maximum principal stress controls the fracture propagation direction.

the breakdown pressure can be predicted through the numerical simulation.

**2.** The initiation and the breakdown pressure of specimens with different perforation angles

**3.** The influence of bedding angle on initiation pressure, breakdown pressure, fracture shape and fracture propagation pattern is great. As the bedding angle increases, the bedding plane and the maximum principal stress will control the fracture evolution respectively and the initiation and the breakdown pressure are in a linear growth with the similar rates. The specimen will be in the most unstable situation as the bedding plane paralleling to

**4.** The stiffness of bedding material has little influence on damage process of rock specimens, except that the strength controls it. With the decrease of bedding material strength, the initiation and the breakdown pressure will decrease gradually with the similar decrease

ration angle and stiffness is small and can be ignored.

**1.** When perforation angle is larger than 0o

are divided into two stages and 0o

the maximum principal stress direction.

termines them.

**6. Conclusions**

rates.

When bedding angle α increases slightly (30o ~45o ), the initiation and propagation of fracture is still along the tension failure bedding. With the increase of loading step, there is a high tensile stress area on the main fracture tip which is along the cracked bedding and the bed‐ dings in the high tensile stress area are cracked and form secondary fractures paralleling to the main fracture, moreover, with the increase of bedding angle, the number of secondary fractures is increasing gradually. Because of the advantage of main fracture, the fracture propagation is still along the main fracture bedding.

When the bedding angle α continues to increase (60o ), the fracture will turn from along the tension failure bedding to the horizontal direction that is the main fracture and the secon‐ dary fractures paralleling to the main fracture is still initiating and propagating in bedding plane with the horizontal secondary fracture initiating at the same time and connecting the main fracture and the parallel secondary fractures gradually. In this case, the bedding plane and the maximum principal stress determine the fracture evolution together.

When bedding angle α is big (75<sup>o</sup> ~90o ), the initiation and propagation of fracture is no longer along the bedding plane. Because of the heterogeneous characteristics of rock and bedding materials, different strength elements are in random distribution causing an uneven stress distribution and the local stress concentration thus making the fracture become bend and rough, but the general trend is the maximum principal stress direction. In this case, the ef‐ fect of bedding plane on fracture evolution is almost disappeared, but the maximum princi‐ pal stress controls the fracture initiation and propagation. Comparing figure 4 and figure 19, we can find that the existence of bedding influences the fracture shape greatly in the same condition as the maximum principal stress controls the fracture evolution.

From the numerical simulation results, as bedding angle increase, the values of initiation pressure are 13.3MPa, 13.7MPa, 14.2MPa, 16.8MPa, 17.1MPa, 16.9MPa and 18.2MPa respec‐ tively, and the values of breakdown pressure 15.3MPa, 16.4MPa, 17.4MPa, 18MPa, 21.2MPa, 21.3MPa and 20.5MPa respectively. Both of the values of initiation and breakdown pressure are in a linear growth (figure 20) with the growth rate similar and as the bedding plane is parallel to the maximum principal stress direction (bedding angle is 0o ), the specimen is in the most unstable situation.

Because the fracture propagation is determined by the maximum principal stress and the bedding plane together when bedding angle is 60o seeing from the second group simulation, taking the bedding angle of 60o for example, in the third group, the effect of strength and stiffness of bedding material on fracture evolution will be studied under the combined ef‐ fects of the maximum principal stress and the bedding plane.

In the third group, the rock specimens with the same bedding angle but different materials are under the constant confining and increasing hydraulic pressure. As the strength of bed‐ ding material is constant (bedding strength/rock strength is 1), but the stiffness decreased

(bedding elastic modulus/rock elastic modulus is 1/10, 1/20, 1/60), the pattern of fracture propagation will be unchanged. Because of the reduction of elastic modulus, the initiation pressure reduced (15.5 MPa, 15.2 MPa, 14.7 MPa) and the breakdown pressure increased slightly (19.7 MPa, 19.9 MPa, 20.4 MPa), however, both of the reduction and the increase can be ignored because the values of the initiation and the breakdown pressure are almost con‐ stant (figure 28). As the stiffness of the bedding material (bedding elastic modulus/rock elas‐ tic modulus is 1) is constant but the strength decreased (bedding strength/rock strength is 1/10, 1/20, 1/60), the pattern of the fracture propagation will be unchanged, and the values of initiation (14.8 MPa, 14.7 MPa, 12.8 MPa) and breakdown pressure (20 MPa, 19.2 MPa, 18.4 MPa) decreased gradually with almost the same decrease rates (figure 28). As both of the stiffness and strength are decreased (bedding elastic modulus/rock elastic modulus is 1/10, bedding strength/rock strength is 1/10), the initiation pressure, the breakdown pressure and the pattern of fracture propagation are almost the same as the condition of (1, 1/10). As sug‐ gested above, the stiffness of bedding material has little influence on initiation pressure, breakdown pressure and fracture evolution of rock specimens, except that the strength de‐ termines them.

In summary, the damage process of rock specimen are determined by the maximum princi‐ pal stress, the bedding angle and the strength of bedding material, while the effect of perfo‐ ration angle and stiffness is small and can be ignored.
