Geomech. Geophys. Geo-energ. Geo-resour. https://doi.org/10.1007/s40948-017-0077-z

ORIGINAL ARTICLE

The study on mechanics of hydraulic fracture propagation direction in shale and numerical simulation Bohu Zhang

. Binxiang Ji . Weifeng Liu

Received: 28 September 2017 / Accepted: 4 December 2017 Ó Springer International Publishing AG, part of Springer Nature 2017

Abstract Disciplinarians of hydraulic fractures propagation direction can make engineering decisions and suggestions for shale gas reservoir reformation. Based on the strain energy density factor theory, the relationships of fracture surface water pressure, fracture dip, and confining pressure ratio on hydraulic fracture propagation angle are studied. The extended finite element method is used to analysis the influences of bedding plane, natural fracture and horizontal stress difference on fracture propagation characteristics. When water pressure is less than maximum horizontal principal stress (rH), there is a critical fracture surface inclination angle, and fractures propagation direction is reversed along the original fracture line. When the pressure is equal to rH, fracture propagation angle increases with dip angle of the fracture surface, and is independent of the stress state. When water pressure is greater than rH, fracture propagation angle increases first and then decreases. When the angle between bedding plane and rH is small, hydraulic fractures propagation direction extends along the rH, conversely, along the bedding plane. The smaller horizontal stress difference is, the easier hydraulic fracture tends to be perpendicular to natural fracture, and the hydraulic fractures propagation direction easier to parallel to natural fracture in natural fracture. B. Zhang (&)  B. Ji  W. Liu School of Geoscience and Technology, Southwest Petroleum University, Chengdu 610500, Sichuan, China e-mail: [email protected]

Keywords Hydraulic fracturing  Fracture propagation  Mechanics  Natural fracture

1 Introduction Hydraulic fracturing technology is one of the core technology in the exploitation of shale gas. Shale gas reservoirs which are buried deep have some characteristic including low porosity, low permeability, significantly anisotropic, nature fracture, and bedding plane development electrode, which makes it difficult to control and estimate in shale gas reservoir fracturing hydraulic fracture propagation (Curtis 2002; Montgomery and Smith 2010; Clark et al. 2012). Scholars domestic and overseas mainly studied the hydraulic fractures propagation direction (HFPD) from fracture mechanics theory and numerical simulation. Fracture propagation direction can be obtained by maximum circumferential stress (Zhou et al. 2016; Erdogan and Sih 1963), strain energy density theory (Sih 1973), critical energy release rate (Hussain and Pu 1975), and hydraulic fracture factor was less considered. Guo et al. (2014), studied the effects of horizontal stress difference coefficient, flow rate, viscosity of fracturing fluid on HFPD. Tan et al. (2017), studied the multiple factors on fracture vertical propagation under horizontal bedding plane, and proposed five patterns for initiation and propagation. Verified that variable rates can reactive natural fracture and leading to complex natural fracture network by Cheng et al. (2015). The

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Chen 1997), derived from the Fig. 1 and hydraulic fracturing conditions, the normal stress on the fracture surface (p) and the shear stress on the fracture surface (q) are obtained as:  p ¼ ry sin2 b þ rx cos2 b ð1Þ q ¼ ry  rx sin b cos b where p is parallel to x0 , q is parallel to y0 , rx is the normal components of stress parallel to x, ry is the normal components of stress parallel to y, b is the angle between the fracture and the vertical compressive stress. Assuming that the fracture surface is straight, the stress function in the plane can be obtained as follow: UðzÞ ¼

z½p þ iðq  p tan /Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 z 2  a2

ð2Þ

where u is the friction coefficient of the fracture surface, a is half length of the fracture, z is the independent variable, /ðzÞ is the stress function. Stress intensity factor can be obtained as follow (Shi et al. 2013):

p 2a

influences of bedding plane on hydraulic fracture was studied (Dan et al. 2015; Sun et al. 2016), and it was shown that the HFPD were jointly controlled by in situ stress state and bedding plane. In numerical analysis, used the cohesive-zonefinite-element model, Guo et al. (2015), discussed the effects of in situ stress state on natural fracture and hydraulic fracture. Scholars used discrete element method (DEM) and particle flow code (PFC) software to study the hydraulic fracture propagation in natural fracture network (Behnia et al. 2015; Fu et al. 2013; Zou et al. 2016; Marina et al. 2015; Zhou et al. 2017). The displace discontinuity method (DDM) was used to study the effects of multi-fracture propagation and stress shadow on fracture propagation pattern (Olson 2008; Olson and Taleghani 2009; Zhang et al. 2017; Wu and Olson 2013). Based on DEM, scholars used the universal distinct element code (UDEC) software to study natural fracture properties and the factors affecting the shape of hydraulic fracture (Wasantha et al. 2017; Nagel et al. 2011,2013), but they only used isotropic and simple porous media flow model. Hydraulic fracture propagation was studied systematically using XFEM, ignoring the filtration of the fracturing fluid filtration (Arash 2009, 2011a, b; Keshavarzi and Mohammadi 2012). The in situ stress and the orthotropy of the material determined the fracture propagation by two-dimensional fluid–solid coupled numerical model (Wang et al. 2016). In the theoretical and numerical study, hydraulic conditions, natural fracture, shale inhomogeneity have great influence on hydraulic fractures propagation direction. Based on fracture mechanics, in this paper, hydraulic condition factor was conditioned to study the relationship between HFPD and fracture surface water pressure (pw). Based on XFEM the effects of different bedding plane directions and natural fracture on the hydraulic fractures propagation direction are studied.

q 2 Analysis of inclined HFPD under confining pressure 2.1 Calculation of stress intensity factor of inclined fracture Central inclined fracture under confining pressure is showed in Fig. 1. According to the pseudo-traction method and the superposition technique (Zhao and

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pw Fig. 1 Slant fracture under confining pressure and pw (Zhang et al. 2012)

Geomech. Geophys. Geo-energ. Geo-resour.

pffiffiffiffiffiffi K ¼ KI  iKII ¼ ½p þ iðq  p tan /Þ pa

ð3Þ

Considering the pw (Fig. 1), the effective normal stress of the fracture surface is:

2 S ¼ a11 KI2 þ 2a12 KI KII þ a22 KII2 þ a33 KIII

ð8Þ

The stress intensity of type I crack and type II crack can be obtained by the upper form:   pffiffiffiffiffiffi 2 2 KI ¼   ry sin b  rx cos bpþffiffiffiffiffiPffi w pa : ð6Þ KII ¼ ry  rx sin b cos b pa

where S is the strain density factor, which is a parameter describing the singularity of strain energy density field at the tip of the fracture. The expression of the strain energy density factor of inclined hydraulic fracture under confining pressure can be obtained by substituting the stress intensity factor of Eq. (6) into formula (9). Strain energy density criterion is mainly based on two conditions to determine the fracture propagation direction. Firstly, fracture begins to expand when the minimum strain energy factor is equal to the inherent critical strain energy factor. Secondly, the direction in which fracture expand forward, that is, the direction of the minimum strain energy density factor. According to the above conditions, the hydraulic fracture propagation angle h can be calculated from the following formula.

2.2 HFPD analysis

oS ¼0 oh

p ¼ ry sin2 b þ rx cos2 bpw

ð4Þ

where pw is fracture surface water pressure. Hydraulic fracturing is considered to be mainly a tensile process (Maxwell 2014). Without considering the friction coefficient of the fracture surface (i.e., u = 0), the stress function can be expressed as:    K ¼  ry sin2 b þ rx cos2 bPw ð5Þ   pffiffiffiffiffiffi þ i ry  rx sin b cos b pa

Stain energy density factor is a fracture mechanics theory based on local strain energy density field. The calculation procedure is simple and has a wide range of applications. It shows great advantages in solving the problem of complex fracture propagation (Sih and Madenci 1983; Sih and Hong 1989; Sih 2012). Strain energy density of type I-II composite fracture represented by stress intensity factor: W¼

 1 2 a11 KI2 þ 2a12 KI KII þ a22 KII2 þ a33 KIII r

ð7Þ

where 8 1 > > ½ð3  4l  cos hÞð1 þ cos hÞ > a11 ¼ > 16pG > > > > 1 > > ½2 sin hðcos h  1 þ 2lÞ < a12 ¼ 16pE 1 > > > ½4ð1  cos hÞð1  lÞ þ ð1 þ cos hÞð3 cos h  1Þ a22 ¼ > > 16pG > > > > 1 > : a33 ¼ 4pG E where G is modulus of elasticity in shear, G ¼ 2ð1þl Þ, E is elastic modulus, G is the shear modulus, l is the Poisson ratio. Strain energy density factor of the fracture type I-II composite fracture can be expressed as (6):

o2 S [0 oh2

ð9Þ

Define n as the ratio of the minimum horizonal principal stress (rh) to the maximum principal (rH), can be expressed as (11): n¼

rh rH

ð10Þ

when E = 10 GPa, l = 0.23, a = 0.1 m, rH = 2 MPa. pw was taken 1, 2 and 3 MPa respectively. The influences of the pw on h and the dip angle (b) between fracture surface and rH were analyzed when the pw is less, equal or greater than rH, and the result are shown in Figs. 2, 3 and 4 (n = 0, 0.2, 0.4, 0.6, corresponding to the red, green, blue and black curves of the four colors in the graph). As shown in Fig. 2, when the pw is less than rH, n less than 0.4, there is a dip angle (b) between fracture surface and rH. HFPD is reversed along the original fracture line (the change of the propagation angle, plus and minus sign) at the two sides of the critical value. When the n is greater than 0.6, the HFPD is always at one side of the original fracture. With the b increase, hydraulic fracture propagation angle varied first increase and then decrease. As shown in Fig. 3, when pw is equal to the rH, the hydraulic fracture propagation angle increase with b which is not affected by the n.

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3 Numerical simulation 3.1 Influence of bedding plane on HFPD in shale

Fig. 2 The relation between h and b when pw is 1 MPa

Fig. 3 The relation between h and b when pw is 2 MPa

Fig. 4 The relation between h and b when pw is 3 MPa

As shown in Fig. 4, when Pw is greater than rH, the hydraulic fracture propagation angle increased first and then decreased with b in the trend, and decrease with the increase of the n. When the b is 0° or 90° which means that fracture is parallel or perpendicular to the rH, the fracture propagation is 0°. Fracture presents a single failure pattern that always propagated along the original fracture line.

A two-dimensional numerical model of hydraulic fracturing in shale reservoir is established to analyze the crack initiation and propagation of hydraulic fracture after injecting fracturing fluid in shale reservoir. Material parameters are setting to transverse isotropy (TI). The material direction is expressed by two mutually perpendicular local coordinate axes 1–2, The plane perpendicular to the axis 2 is an isotropic plane, which is bedding plane. In the Fig. 5, the red line indicates axis 1 and the blue line indicates axis 2. The size of the model is 10 m * 10 m. Perforation length is 0.25 m, which start and end point is at the middle of left of the model. The direction is the axis 2. Rock and numerical parameters were showed in Table 1. Injection rate of fracturing fluid gradually increase from 0 to the specified injection rate between 1 and 10 s, and remained unchanged then. The maximum horizonal principal stress was X direction, and the minimum horizonal principal stress is Y direction. Bedding plane inclination was defined as 0° when the bedding plane direction (the direction of the 1 axis) was consistent with the rH direction (X direction). Considering counterclockwise for positive six dip angles of 0°, 15°, 30°, 45°, 60°, and 75° were simulated and analyzed. Because of the large size difference between model and fracture, propagation state of hydraulic fracture near natural fracture was analyzed in detail. Figure 5 was a partial enlargement of overall model. Figure 5a–f are 0°, 15°, 30°, 45°, 60°, and 75° of the dip angle of the bedding plane when fracture propagation, and we can obtain that shale bedding has certain influence on HFPD. When the bedding plane angle was small than 45°, HFPD trends along bedding plane, which rH was the main influencing factors and mainly extend along propagation direction of rH. When the bedding plane angle was more than 45°, HFPD mainly extend along the bedding plane direction. 3.2 Influence of natural fracture on HFPD in shale Based on the model of Sect. 3.1, a natural fracture was set at the 0.5 m at the tip of the perforation. Natural

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Fig. 5 The HFPD under different bedding plane dip angles Table 1 The parameters of numerical model of hydraulic fracturing Elastic modulus (GPa) E1

22.6

Minimum horizontal principal stress (MPa)

20

E2

16.9

Maximum horizontal principal stress (MPa)

30

G1

6

Initial pore pressure (MPa)

10

G2

10

Tensile strength (MPa)

2

0.25 0.37

Specific weight of wetting liquid (kN/m3) Injection rate of fracturing fluid (m3/s)

9.8 9.6 9 10-6

Initial porosity

0.0365

Critical energy release rate (kN/m)

214

Fluid viscosity (kPa s)

1 9 10-6

Permeability coefficient (m/s)

1.418 9 10-9

Simulation duration (s)

100

Shear modulus (GPa)

Poisson ratio l13 l23

3

Filtration coefficient {m /(kPa s)}

-11

5.879 9 10

fracture was used by weak unit substitution with a length of 1 m and a thickness of 0.05 m. Modulus of elasticity of natural fracture was 1/10 of shale matrix, and the tensile strength and the critical energy release rate are 1/100 of shale matrix. The remaining parameters are the same as the shale matrix. The dip angle

(b) between fracture surface and rH were taken at 45°, 60°, and 75° respectively. The natural fracture and numerical parameters of hydraulic fracturing were showed in Table 2. Under the same horizontal stress difference, simulation results were shown in Figs. 6, 7 and 8.

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Geomech. Geophys. Geo-energ. Geo-resour. Table 2 The parameters of natural fracture and numerical model of hydraulic fracturing Elastic modulus (GPa) Matrix

22.6

Natural fracture

2.26

Minimum horizontal principal stress (MPa)

20

Maximum horizontal principal stress (MPa)

20

Initial pore pressure (MPa)

10

Critical energy release rate (kN/m) Matrix

214

Natural fracture

2.14

Tensile strength (MPa) Matrix

2

Natural fracture

0.02

Filtration coefficient {m3/(kPa s)}

5.879 9 10-11

Poisson ratio

0.25

Specific weight of wetting liquid (kN/m )

9.8

Fluid viscosity (kPas)

1 9 10-6

Injection rate of fracturing fluid (m3/s)

9.6 9 10-6

Initial porosity

0.0365

Simulation duration (s)

150

3

Permeability coefficient (m/s)

Fig. 6 The HFPD when the angle (b) between fracture surface and rH was 45°

Fig. 7 The HFPD when the angle (b) between fracture surface and rH was 60°

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-9

1.418 9 10

Geomech. Geophys. Geo-energ. Geo-resour. Fig. 8 The HFPD when the angle (b) between fracture surface and rH was 75°

Fig. 9 The diagram of angle variation when the hydraulic fracture intersected the nature fracture

Angle of hydraulic fracture though the natural fracture/°

In the initial stage of hydraulic fracturing propagation, the HFPD was extend along the rH direction. As hydraulic fracture propagated to the vicinity of the natural fracture, that HFPD deflected and approximately intersected when the hydraulic fracture intersects the natural fracture. Under the action of fracturing fluid pressure, hydraulic fracture propagations into the natural fracture that the direction was approximately perpendicular to the natural fracture, when the hydraulic fracture propagations forward for some distance, the HFPD was deflected again, and there was a tendency of parallel natural fracture. Eventually continue to penetrate the natural fracture from the other with the rH. Under the same horizontal stress difference, when the angle between natural fracture and rH increases, HFPD tends to deviate from rH before it penetrates natural fracture, and tends to be perpendicular to natural fracture and HFPD easily extended vertically to the natural fracture in natural fracture.

3.2.1 Influence of horizontal stress difference on HFPD in shale Based on the 3.2 model, that horizontal stress difference was 0, 5 and 10 MPa, when the rH was 20, 25 and 30 MPa. The angle at which the hydraulic fracture intersected the natural fracture and the angle of the fracture through the intersection of the natural fracture can be obtained, as shown in Figs. 9 and 10. As shown in Figs. 9 and 10, the smaller the horizonal stress difference was, the easier hydraulic fracture tended to be perpendicular to natural fracture and deviated from the rH before penetrating into the natural fracture, the easier HFPD tend to parallel to natural fracture in natural fracture. The greater the horizonal stress difference was, the greater the angle of the fracture through the natural fracture was, the more difficult it was to turn parallel to the natural fracture propagation, and it was easy to penetrate the natural fracture in the direction of rH.

130 125

0Mpa

5Mpa

10Mpa

120 115 110 105 100 95 90 H

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60

Angle of hydraulic fracture intersection natural fracture /°

Fig. 10 The diagram of angle variation when hydraulic fracture though the nature fracture

50

0Mpa

5Mpa

10Mpa

40 30 20 10 0 H

4 Conclusion

References

Based on the strain energy density factor theory,This study considered hydraulic condition factor to analysis the relationships of pw, fracture dip, and confining pressure ratio on hydraulic fracture propagation angle. We can obtain that pw had a great fluence on the HFPD. Fracture always propagated the original line when it is parallel to the rH. We used the extended finite element method(XFEM) to analysis the influences of bedding plane, natural fracture and horizontal stress difference on fracture propagation characteristics. When the angle between bedding plane and rH is small, HFPD extend along the rH. Conversely, HFPD extend along the bedding plane. When the angle between natural fracture and rH increases, HFPD tends to deviate from rH before it penetrates natural fracture, and tends to be perpendicular to natural fracture and HFPD easily extend vertically to the natural fracture in natural fracture. The smaller horizontal stress difference is, the easier hydraulic fracture tends to be perpendicular to natural fracture, the easier HFPD tend to parallel to natural fracture in natural fracture. The greater the horizonal stress difference is, and it is easy to penetrate the natural fracture in the direction of rH.

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Acknowledgements The authors gratefully acknowledge the supported by Open Fund (PLN201717) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), National Science and Technology of the Ministry of Science and Technology of China (2017ZX05037001) and Sichuan Youth Science and Technology and Technology Innovation Research Team Program (2017TD0013).

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