An energy based analytical method to predict the influence of natural fractures on hydraulic fracture propagation

Accepted Manuscript An energy based analytical method to predict the influence of natural fractures on hydraulic fracture propagation Yao Yao, Wenhua Wang, Leon M. Keer PII: DOI: Reference:

S0013-7944(17)30864-0 https://doi.org/10.1016/j.engfracmech.2017.11.020 EFM 5758

To appear in:

Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

17 August 2017 9 November 2017 12 November 2017

Please cite this article as: Yao, Y., Wang, W., Keer, L.M., An energy based analytical method to predict the influence of natural fractures on hydraulic fracture propagation, Engineering Fracture Mechanics (2017), doi: https://doi.org/ 10.1016/j.engfracmech.2017.11.020

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An energy based analytical method to predict the influence of natural fractures on hydraulic fracture propagation Yao Yaoa*, Wenhua Wanga, Leon M. Keer b a.

School of Mechanics and Civil Engineering, Northwestern Polytechnical

University, Xi’an 710072, People’s Republic of China b.

Department of Civil and Environmental Engineering, Northwestern University,

2145 Sheridan Rd., Evanston, IL 60208, USA * Tel.: +86 29 88495935; E-mail address: [email protected]; [email protected] Abstract Hydraulic fracturing is a widely applied stimulation method to enhance the productivity of unconventional resources. The hydraulic fracturing operation in naturally fractured reservoirs is complex, and the fractures can intersect a natural interface such as a bedding plane. The hydraulic fracture may either cross or be arrested by slippage without dilation, and the fracture plane can be opened upon arriving the interface. In the current study, a theoretical approach to predict the fracture extension encountering a natural fracture under far field stresses is developed, based on the Griffith stability criterion. The critical fluid pressure required to cross the interface, open the natural fracture, or make slippage take place is obtained. New criteria to separate opening zone, arrest and crossing zone are proposed based on stress field difference. The theoretical predictions are compared with experimental data and show reasonable accuracy. Key Words: hydraulic fracture; analytical method; natural fracture; energy 1

Nomenclature

c

Half-length of a line crack

c0

Cohesion of the interface

c

Length of crack tip moving ahead Modulus of elasticity Modulus of elasticity for plane strain condition

E E' G0

Critical energy release rate of rock matrix

Gin

Critical energy release rate of interface filler

GIC

Critical energy release rate

K IC

Critical stress intensity factor

Q

Heat generated

w x

Width of line crack

W1

Elastic energy

W0

W0' W0'' W1

Strain energy increased without far field stress Strain energy increased crossing NF with far field stress Strain energy increased as NF opens with far field stress Elastic energy increased without far field stress

W1'

Elastic energy increased with far field stress

W1''

Elastic energy increased as NF opens with far field stress

Wp

Work done by constant pressure without far field stress

Wp' Wp''

Work done by constant pressure crossing NF with far field stress Work done by constant pressure as NF opens with far field stress



Surface energy

0

Surface energy of rock matrix 2

 in

Surface energy of interface filler



Thickness of infinite domain



Ratio of  in and  0



Intersection angle



Friction coefficient



Poisson’s ratio

H

Maximum horizontal in-situ stress

h

Minimum horizontal in-situ stress

n

Normal stress of NF

t

Tensile strength



Shear stress

p

Constant pressure in crack

pn

Net pressure

phf

Critical fluid pressure crossing NF

pnf

Critical fluid pressure as NF opens

psp

Critical fluid pressure as slippage takes place

v

Crack volume increased

vhf vnf

Crack volume increased with far field stress Crack volume increased as NF opens with far field stress

1. Introduction Hydraulic fracturing treatment plays an essential role in unconventional shale gas development. The fracturing fluid is pumped into the reservoir at high pressure, providing a path for oil or natural gas towards the production well. Over the past decades, several models were proposed to simulate the nucleation and propagation of 3

hydraulic fractures, such as the KGD and PKN models (Geertsma and De Klerk, 1969; Nordgren 1962). With the development of computational technology, numerical models were employed to simulate fractures with more complex geometries based on linear elastic fracture mechanics, which generally provides reasonable predictions for hard rocks (Settari and Cleary 1986; Shet and Chandra, 2002; Dean and Schmidt 2009; Mokryakov 2011; Yao2012; Yao et al. 2015). In the naturally fractured oil or gas reservoirs, the assumption that hydraulic fracture is a straight and bi-wing planar feature is untenable because of the existence of natural fractures, faults, bedding planes and stress contrasts. Hydraulic fracture propagation can be affected by the natural fractures in a shale gas reservoir. The interaction between hydraulic and natural fractures is a complex process. Experimental analysis (Blanton 1982; Zhou et al. 2008; Song et al.2014) shows three main modes when the interaction between hydraulic fracture and natural fracture occurs. The first mode is that the hydraulic fracture crosses the natural fracture directly. The second mode is when the natural fracture opens and the hydraulic fracture propagates along the natural fracture. The third mode denotes that the hydraulic fracture is arrested before reaching the interface and slippage takes place. Some effective criteria for this competition were developed to predict which mode will dominate the fracture behavior, based on linear-elastic fracture mechanics (Blanton 1982; Renshaw and Pollard 1985; Warpinski and Teufel 1987; Gu et al. 2011). The interaction between hydraulic and natural fractures could induce a hydraulic 4

fracture network and enhance the complexity of the problem. The interaction is affected by the approach angle, in-situ stresses, rock mechanical properties, properties of the natural fractures and the treatment parameters, including fracturing fluid properties, injection rate and others. It is essential to determine whether a hydraulic fracture crosses or is captured by natural fractures, since this behavior controls the geometry of the resulting fracture network. Large amounts of numerical, theoretical, and experimental research has been performed on the interaction between hydraulic and natural fractures (Norman et al. 1963; Blanton 1982; Renshaw and Pollard 1985; Warpinski and Teufel 1987; Beugelsdijk et al. 2000; Potluri et al. 2005; Zhang and Jeffrey 2006; Zhang et al.2007; Zhou et al. 2008; Chuprakov et al. 2011; Hou et al. 2014; Li et al. 2014). Most studies focus on the mechanical interaction when a hydraulic fracture reaches a pre-existing fracture, where the fluid flow in the hydraulic and natural fractures is investigated. Blanton (1982) developed a method considering the coefficient of friction to predict whether a hydraulic fracture will propagate across the interface or show pressure induced slip under different conditions. Experiments on both Devonian shale and hydrostone indicate that the morphology of hydraulic fractures is strongly influenced by natural fractures. The experiments on hydrostone in particular show that hydraulic fractures tend to cross pre-existing fractures under higher differential stresses and larger angles of approach. Under intermediate or low differential stresses with approach angles along the pre-existing direction, the hydraulic fractures could open the pre-existing fracture and divert the fracturing fluid or arrest the hydraulic 5

fracture propagation. Based on the linear-elastic fracture mechanics solution for stresses near the fracture tip, Renshaw and Pollard (1995) proposed another approach to predict the fracture propagation direction when a frictional interface is orthogonal to the approaching fracture. However, the intersection angle between a hydraulic fracture and natural fracture can range between 0 and 90°in the field. The Renshaw and Pollard criterion has been extended to predict intersection at nonorthogonal angles by Gu and Weng (2010). On the other hand, Lamont and Jessen (1963), and Daneshy (1974) have demonstrated that the propagating hydraulic fracture in a fractured reservoir could either cross or turn into the natural fracture. In some cases, the fracture will turn into the natural fracture for a short distance and then breakout again to propagate in a mechanically more favorable direction, depending on the orientation of natural fracture relative to the stress field. Warpinski and Teufel (1987), and Zhou et al. (2008) observed three types of interactions between hydraulic and natural fractures: namely crossing, opening and dilating the natural fracture. The hydraulic fracture can be arrested by shear slippage of the natural fracture without dilation when fluid flows along the natural fracture. Most of the experimental and field studies on the mechanism of hydraulic fracture propagation (Blanton 1986, Zhou et al. 2008) have shown that the in-situ stresses, horizontal stress difference, angle of approach, interfacial friction coefficient, fluid injection rate and fracturing fluid viscosity are the key parameters affecting hydraulic fracture propagation. Warpinski and Teufel (1987) developed a method that governs 6

the arresting mode based on differential stresses. However, the surface energy that generates the new facture is not considered, which ignores the variation of joint strength characteristics when the joint growth situation changes. Several analytical methods have been developed to predict the interaction mechanism of induced or natural fractures (Gale, et al. 2007; Dahi-Taleghani and Olson, 2009; Olson et al. 2012; Wang, et al. 2013). They investigated the effects of different parameters, including in-situ stresses, approach angle, rock mechanical properties, hydraulic fracture treatment parameters and the properties of natural fractures such as cohesion and friction angle of the interface. Fracture development on a weak interface ahead of a fluid-driven crack has been simulated to investigate the effects of delaminating and shear zone on the interface (Galybin et al. 2014), and the secondary crack may be induced because of friction along the natural fracture (Goldstein et al. 2015). However, the surface energy of the pre-existing interface was usually ignored. In the current study, the critical fluid pressures to cross the interface, open the natural fracture and allow slippage to take place are predicted, and a new criterion to describe the fracture propagation direction is proposed. The analytical prediction is compared with experimental data and shows good agreement.

2. Critical fluid pressure for induced fracture and opening natural fracture It is well accepted that a fluid lag exists between the fracture tip and the fluid front inside a hydraulic fracture from laboratory and field observations (Medlin and Masse 7

1984; van Dam et al. 2000). When the fracture tip reaches the interface of natural fracture, the fluid front remains behind due to fluid lag. Although the effects of fluid flow are limited, the natural fracture is under the influence of the stress field around the fracture tip. Gu and Weng (2012) developed a criterion to predicate crossing type interaction considering the fluid lag effect. When the fracture tip cannot cross the interface and the fluid front reaches the natural fracture, the stress concentration at the fracture tip will vanish and fluid pressure at the intersection point rises. The natural fracture may open or slippage will take place. A new approach to predict differential stress and approach angle is proposed in the process when the fluid front reaches the interface. The effect of fluid lag is ignored in the current study. It is assumed that the initial interaction with the natural fracture is blunted and arrested and the fracture fluid can penetrate into the intersection. The subsequent derivation is based on the effect of fluid pressure with fracture propagation. Griffith’s analysis of crack stability theory (Griffith 1920; Valko and Economides 1995) assume a slice of thickness

δ from an infinite domain in the absence of

far-field stress, containing a three dimensional extension of a “line crack” with half-length c, as shown in Figure 1.

Figure 1 Propagation of a linear crack

8

By assuming the linear crack is of elliptical shape, the width is given by equation (1). If the crack tip moves ahead by c , the increased volume can be calculated by equation (2) (Valko and Economides. 1995). w x 

4p 2 c  x2 0  x  c ' E 2 pcc v  E'

(1) (2)

Supposing that the crack is opened by a constant pressure p, the work done by the inner fracturing fluid pressure is the product of pressure p and the increased volume

v , and which is defined as Wp . As the crack propagates, energy is absorbed by the deformation of the rock matrix. This part of energy is stored in the medium as strain energy, which is denoted as W0 . The other part of the entire work Wp is to create new surfaces. Griffith postulated that if the specific surface energy  characterizes the energy consumption while a unit area of new surface is created and is a material property, then the energy required to create new surfaces is 2c . If there is surplus energy available, it will be dissipated as heat Q. The energy balance equation could be written as: Wp  W0  2c  Q

(3)

Based on the Griffith stability criterion (Valko and Economides. 1995), the crack is stable if:

Wp  W0  2c

(4)

When the far-field stress is not zero, the elastic energy W1 is required to be defined. It represents the product of crack volume and far-field stress perpendicular to the propagation direction. If the fracture propagation direction is perpendicular to the 9

minimum horizontal principal stress as shown in Figure 2, the elastic energy is changed and W1 is: W1  v h

(5)

Figure 2 Schematic of a hydraulic fracture that propagates perpendicular to the minimum horizontal principal stress

However, the existence of far-field stress makes the configuration more complex. To apply the Griffith stability theory with far-field stress, the simulation should be decomposed into two parts. Thus, an associated problem is created, as shown in Figure 3 (Valko and Economides. 1995). Figure 3b shows a scenario similar to the Griffith stability problem except for the difference of inner pressure, and the increased crack volume that can be obtained. In Figure 3c, it is assumed that  h exists in the far field and  h is not moved to the crack surface. Figure 3c represents that the elastic energy is accumulated by overcoming the far-field stress with increasing crack volume.

10

Figure 3 Schematic of configuration with far-field stress  h When the crack tip moves ahead by c , the changed volume is expressed by equation (2), here p   h takes the place of p as follows: vhf 

2  p   h   cc E'

(6)

From equation (6): Wp'  pvhf 

W  ' 0

2 p  p   h   cc E'

 p   h  vhf 2

  p   h  cc

(7)

2



E'

W1'  vhf  h

(8) (9)

The generated heat is small and can be neglected. Thus, the new stability criterion can be postulated as:

Wp'  W0'  W1'  2 0c

(10)

The critical energy release rate GIC is related to the stress intensity factors through Irwin’s relation: 2 K IC  GIC E '



' 2 where E  E 1  v



for plane strain conditions,  11

(11) is the Poisson’s ratio and

GIC  2 0 according to the fracture mechanics.

From equation (10) and equation (11), the critical fluid pressure required to drive the crack tip moving ahead can be calculated by: phf   h 

K IC c

(12)

The propagation of hydraulic fracture is assumed as planar Mode I. According to the fracture toughness criterion (Irwin, 1957), if the stress intensity factor is greater than the critical value K IC , the fracture will propagate. Assuming the load on the fracture face in a bi-directional compression plane is phf , the critical stress intensity factor (Westgaard 1939, Griffith 1920) is: K IC   p  phf



c

(13)

From equation (13), the critical pressure can be obtained. Then, the Griffith stability criterion can be extended to predict when the hydraulic fracture will propagate along the natural fracture direction.

Figure 4 Schematic of hydraulic fracture encountering a natural fracture 12

When the hydraulic fracture encounters the natural fracture as shown in Figure 4, the hydraulic fracture may open the natural fracture and propagate along the weak interlayer, and the width will change correspondingly. Assuming that the crack tip moves ahead by c , if it propagates along the natural fracture, the changed volume can be determined by: 2  p   n   cc E'

vnf 

(14)

The shear and normal stress acting on the plane of the natural fracture can be obtained from the 2D stress resolution (Jaeger, Cook et al. 2007):

n 

H h 2





 H  h

 H h 2

2

cos 2

sin 2

(15) (16)

Then, Wp''  pvnf 

W  '' 0

2 p  p   n   cc E'

 p   n  vnf 2

  p   n  cc

(17)

2



E'

W1''  vnf  n

(18) (19)

From equation (10), the stability criterion can be obtained:

Wp''  W0''  W1''  2 inc

(20)

From the above equations, the critical pressure required to open the natural fracture and move the fracture ahead can be determined as: pnf   n 

K IC  c

(21)

where η denotes  in  0 and equals to Gin / G0 ,  in is the surface energy of the interface 13

filler,  0 is the surface energy of rock mass matrix. If the bedding plane consists of the same rock as the mass matrix, it is assumed that the ratio of surface energy equals to the ratio of tensile strength of bedding plane and rock mass. It should be noted that tensile strength could be obtained from experiments and is easier to be determined, compared with the surface energy.

3. Validation of the critical fluid pressure Hydraulic fracture propagates along the direction with least required energy, and if

pnf is smaller than phf , then the natural fracture will open. On the contrary, the hydraulic fracture may cross the interface. By comparing equation (12) and equation (21), the crossing criterion can be obtained, as expressed in Figure 5. The curved line means that different fluid pressure is required to open natural fracture at various approaching angles. Various curves are displayed with respect to different values of the ratio  . The dashed line represents the required fluid pressure to propagate along the maximum horizontal in-situ stress. The curve above the dashed line represents that the fluid pressure of opening a natural fracture is larger than that of hydraulic fracture, which means that crossing will take place. Conversely, the curve below the dashed line represents that the fluid pressure required to open natural fracture is smaller than that of the hydraulic fracture, which means the natural fracture will open. The applied principal stresses are  H  10MPa ,  h  8MPa . The calculated fracture toughness is

0.59MPa m1 2 and the initial crack length is 0.06m.

14

Figure 5 the required fluid pressures with respect to different η

In Zhou et al.’s study (2008), different types and thickness of papers were cast into the block as pre-fractures. The block is made of cement and sand with a compressive strength of 28.34MPa. The angle in each block between the hydraulic fracture and the pre-fracture were varied systematically. The interaction angles were 30°, 60°and 90°, respectively. The ratio η can be regarded as zero for the surface energy of paper, which is much smaller than that of rock. Other related parameters are

K IC  0.59MPa m1 2 and c  0.06m . The comparison is displayed in Table 1.

15

Table 1 Comparison between Zhou et al.’s experiments (2008) and the developed approach



Our result

Experiment result Zhou (2008)

0.89 0.89

Crossing Crossing

Crossing Crossing

-

0.89

Crossing

Crossing

0 0

-

0.89 0.89

Crossing

Crossing

-5

0

+

0.89

Crossing Dilated

Dilated Dilated

-8

-5

0

+

0.89

Dilated

Dilated

30 90

-13 -8

-3 -3

0 0

-

0.89 0.38

Crossing Crossing

Arrested Crossing

90

-8

-5

0

-

0.38

Crossing

Crossing

60 60

-10 -8

-3 -3

0 0

-

0.38 0.38

Crossing Crossing

Crossing Dilated

30

-10

-3

0

-

0.38

Dilated

Arrested

30

-8

-3

0

+

0.38

Dilated

Dilated



phf  pnf



-5 -3

0 0

-

-10

-3

0

60 60

-13 -8

-3 -5

30

-10

30

H

h

(MPa)

(MPa)

90 90

-10 -10

60



In the experiments performed by Gu et al. (2011), samples were cut at specified angles 45°, 75°and 90°. Then samples were reassembled using epoxy. The block samples are made of Colton sandstone with a tensile strength of 4.054MPa . The fracture toughness is 0.7MPa m1 2 , and the initial crack length is 0.076m . The comparison is displayed in Table 2. Table 2 Comparison between Gu et al.’s experiments (2011) and the developed approach



Our result

Experiment result Zhou (2008)

0.615

Crossing

Crossing

+

0.615

No Crossing

No Crossing

0

-

0.615

Crossing

Crossing

-6.89

0

+

0.615

No Crossing

No Crossing

-17.24

-6.89

0

-

0.615

Crossing

No Crossing

-8.27

-6.89

0

+

0.615

No Crossing

No Crossing



phf  pnf



-6.89

0

-

-7.58

-6.89

0

75

-17.24

-6.89

75

-8.27

45 45

H

h

(MPa)

(MPa)

90

-13.78

90



16

To check applicability of the proposed method, the analytical results are compared with experiments by Zhou et al. (2008) and Gu et al. (2011). In Table 1 and Table 2, the bold cells show the cases that the criterion did not match the experimental results. Owing to the existence of plastic zones ahead of the crack tip, the actual fluid pressure to cross an interface is relatively larger, and hence the values are in the acceptable range. In general, the theoretical predictions show good agreement compared with the experimental data.

4. Opening and arresting criteria on differential stresses. When propagating towards the fracture interface, the hydraulic fracture may neither cross the interface nor open the natural fracture. There are two patterns: The first case denotes that the hydraulic fracture is arrested before encountering the interface. The second case is when slippage occurs and water conductivity path forms, but hydraulic fracture re-initiates later on the other side of the interface as the inner fluid pressure increases. The first case is not considered in the current study. For the second case, some theoretical and numerical methods were developed. If the hydraulic fracture is linked to the weak layer and pressure in the weak layer cannot open the natural fracture, slippage may occur (Blanton 1982, Zhou et al. 2008):

    n  p   c0

(22)

The critical value of the fracturing fluid pressure is: psp   n 

  c0 

where  is the coefficient of friction and c0 denotes cohesion of the interface. 17

(23)

If a natural fracture arrests an approaching fracture, the crack tip will be blunted and the stress singularity will diminish as the hydraulic fracture propagates into natural fracture; then fracture propagation stops and the fluid enters into the weak layer. As the pressure increases, slippage will occur if psp is smaller than pnf and phf . Thus, the following arresting criterion by slippage can be developed:

 K IC c  H h  sin  cos    sin 2  c0 

(24)

Warpinski and Teufel (1987) considered the Coulomb failure criterion for slippage along the natural fracture and proposed a relationship that governs the arresting mode based on the differential stresses:

 H h 

c0   pn sin  cos    sin 2 

(25)

Figure 6 Regions of arresting mode based on Warpinski and Teufel’s criterion for four different net pressures; pn =0.1, 0.7, 1.0, 1.1MPa, respectively 18

In equation (25), the arresting of hydraulic fracture is dependent on the net pressure, hydraulic fracture length, approach angle, friction coefficient and cohesion of the interface. From Figure 6, the scope of arresting mode extends as net pressure increases. This is especially the case when the right side of equation (25) is zero, and all the zones that belong to the curve above the X-axis could induce the arresting mode. However, the natural fracture may open or the hydraulic fracture may cross the interface if the net pressure is high enough. In fact, whether slippage takes place depends on the property of natural fracture such as friction coefficient and cohesion of the interface. The jointing crevasse grows as long as the fluid pressure is larger than the critical fluid pressure psp , which means, the potential of slippage has been determined for a special weak layer. The role of fluid pressure is to achieve the potential. As the pressure at the intersection continues to increase, the opening and crossing modes may initiate, and re-initiation can occur later in the slippage zone or at the end of the natural fracture. It should be noted that re-initiation is not considered in the current study. When the fracturing fluid meets the interface, the natural fracture will open if the fluid pressure pnf is less than the pressure phf required to drive the crack tip to move along the initial direction with increasing fluid pressure. The opening criterion in terms of differential stress can be expressed as:

 H h 

K IC 1    c sin 2 

(26)

Equation 26 depends on the hydraulic fracture length, intensity factor of rock mass matrix, approach angle and the ratio of  in /  0 . Figure 7 shows comparison of 19

the proposed opening criterion to the experimental results (Blanton 1986) with respect to various ratios of  , where the rock mass matrix is Devonian Shale and the weak mixture is hydrostone. The value  is determined to be 0.04 in Equation 26. Fracture toughness of Devonian shale is 1.57MPa m1 2 and the initial crack length is 0.06m. In Figure 7, with increasing approach angle, the differential stress decreases. The curve of opening criterion can describe the opening zones accurately.

Figure 7 Comparison of the proposed opening criterion to the experimental results (Blanton 1986)

To further prove the validity of equation (26), the analytical prediction is compared with another experimental study by Blanton (1982). The rock mass matrix is Devonian Shale and weak mixture is hydrostone in the experiments. Fracture toughness of Devonian shale is 1.57MPa m1 2 and the initial crack length is 0.06m. 20

The parameter  is 0.04. The figure indicates a good agreement of theoretical predictions to the experimental results. As shown in Figure 8, a larger  represents a more difficult opening mechanism, and the opening zone decreases as  increases.

Figure 8 Comparison of the proposed opening criterion to experimental results from Blanton (1982)

In the developed model, the tendency of variation of opening with surface energy is different from Blanton’s work. In Figure 9, from the experimental analysis of Blanton (1982), the opening mechanism is easier when the surface energy of interface filler increases, which is not realistic. More energy is required to open the natural fracture if the surface energy is larger. Thus the opening of a natural fracture with larger surface energy requires lower differential stress.

21

Figure 9 Opening criterion of Blanton (1982) for G=35, 70 and 105 N / m , from left to right

Figure 10 Configuration of hydraulic fracture encountering an interface filler for approach angle equals 90°

22

In the particular case (approach angel equals 90°), as shown in Figure 10, the interface opens easier as the G of interface filler decreases under certain differential stress. Thus, when G of the interface filler increases, the differential stress reduces if the natural fracture opens, and the opening zone is reduced. The opening zone will vanish if the interface filler properties are the same as mass matrix, and therefore the opening criterion in equation (26) is reasonable. In Figure 11, according to equation (26), fracture simulation of the purple point for   0.05 should cross, simulation of the red point for   0.05 should not cross and the simulation of red point for

  0.5 should cross.

Figure 11 Change of differential stresses with the angle of approach

Numerical simulation is performed to verify the proposed method by using the commercial finite element software ABAQUS. The finite element model is plane 23

strain and the size is 105 105 m, the fluid injection point A is in the middle of the edge and vertical to the X axis. The displacement of U1 of the left edge and displacement of U1, U2 of the other edges are constrained. The element type of the model is CPE4P. The hydraulic fracture model is displayed in Figure 12. Parameters of corresponding elements are adjusted to represent the natural fracture according to different values of

.

The change of values is shown in Table 3. Mechanical

parameters of the rock matrix are obtained from the shallow shale in Longmaxi, South China (Li et al. 2014). Permeability of the rock is set as 5 109 m2 , and leak-off is

5.8 1010 m / (KPa s) . The water based fluid parameters are given in Table 4.

Figure 12 Hydraulic fracture model using XFEM

24

Table 3 Mechanical parameters of shale adopted in the numerical simulation



E ( GPa )

t ( MPa )

Porosity

GIC

K IC 12

( MPa m

(N/m)

)

Rock

14.06

0.367

11.67

1.20

89

0.33

NF

14.06

0.367

 11.67

 1.2

 89

0.33

Table 4 Fluid parameters in the numerical simulation Injection rate

Gap flow

3

( KPa s )

8 105

1106

(m /s)

Unit weight

Injection time (s)

3

( KN / m ) 7.8

240

Table 5 Loading condition in the numerical simulation



v

H

h

( MPa )

( MPa )

( MPa )

Figure 13

0.05

16

16

12

Figure 14

0.5

16

16

12

Figure 15

0.05

16

20

12

It is assumed that the highlighted elements represent the natural fracture in Figure 13, 14, and 15, and the approach angle is set to be 90°to ignore the effects of the arrest zone. The critical stress difference is 4.8 MPa when  is 0.05 and 1.811 MPa when  is 0.5. In the numerical analysis, stress differences are 4 MPa and 8 MPa, respectively. The simulation results are presented below.

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a. Crack tip before NF

b. Crack tip in NF

Figure 13 Hydraulic fracture process no crossing NF when stress difference is 4 MPa and η is 0.05

a. Crack tip before NF

c. Crack tip cross NF

b. Crack tip close to NF

d. Crack tip far from NF

Figure 14 Hydraulic fracture process crossing NF when stress difference is 4 MPa and η is 0.5

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a. Crack tip before NF

c. Crack tip cross NF

b. Crack tip close to NF

d. Crack tip far from NF

Figure 15 Hydraulic fracture process crossing NF when stress difference is 8 MPa and η is 0.05 When the stress difference is 8 MPa, the crack tip can cross natural fracture. Stress distribution at the crack tip shows a butterfly pattern and symmetric when crack is far from the natural fracture. When crack tip is close to natural fracture, the stress distribution is disturbed by the natural fracture, which is influenced by the stress shadow affect. When the stress difference is 4 MPa, the crack tip cannot cross the natural fracture. In the numerical analysis, elements of weaker parameters are assumed as the natural fracture. When the crack tip intersects the weaker elements, the hydraulic fracture propagates along the maximum principal stress direction. When the crack tip comes from the weaker elements under a stress difference of 4 MPa, the crack tip arrests at the interface because hydraulic fracture propagates along the 27

maximum principal stress direction in any element. However, at the right interface, the fracture should propagate along natural fracture, which could lead to convergence problems and is consistent with the theoretical prediction: opening of the natural fracture. The results indicate that the opening zone decreases as the value of η increases.

Figure 16 Regions of different interaction modes

From equation (24) and equation (26), Figure 16 can be obtained, which shows regions of different interaction modes with respect to differential stress. In Figure 16, the ratio of surface and approach angles dominate the interaction mode, in which case the approach angle plays a more important role. It is noted that hydraulic fractures cross the pre-fractures only at high horizontal differential stress at approach angles of 28

60º or larger. Hydraulic fractures open the pre-existing fractures only at low horizontal differential stress or low approach angles because the fluid pressure in the hydraulic fracture is sufficient to open the pre-fractures. On the other hand, hydraulic fractures will be arrested by shear slippage of the pre-fracture only at high differential stress and approach angles range from 30°to 60°.

5. Conclusion Natural fracture plays a significant role in hydraulic fracture propagation and pressure response. An energy based theoretical method is developed to predict the fluid pressures crossing the interface and opening the natural fracture, and the corresponding opening and arresting criteria are proposed. Good agreement is obtained between the analytical predictions and experimental results. The analysis illustrates that mechanical properties (including surface energy, the friction coefficient and cohesion) of the interface and approach angle have a strong influence on the interaction mode, and the interaction mode is sensitive to the approach angle. From the numerical analysis, crossing is easier to occur when the approach angle is close to 90°under high differential stresses. When the intersection angle decreases from 90°, the interface slip is more likely to occur under high differential stresses at medium angle ranges from 30°to 60°. The opening mode will be retained with small differential stresses. At a lower intersection angle range, opening of the natural fracture will occur. By comparing with experimental results, the proposed method can predict the complex fracture network and the corresponding fluid pressure with 29

reasonable accuracy.

Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 11572249, 11772257).

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Highlights  An energy based theoretical method is developed to predict the fluid pressures crossing the interface and opening the natural fracture, and the corresponding opening and arresting criteria are proposed.  Good agreement is obtained between the analytical predictions and experimental results.  The analysis illustrates that mechanical properties (including surface energy,

the friction coefficient and cohesion) of the interface and approach angle have a strong influence on the interaction mode, and the interaction mode is sensitive to the approach angle.

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