Propagation behavior of hydraulic fracture across the coal-rock interface under different interfacial friction coefficients and a new prediction model

Journal of Natural Gas Science and Engineering 68 (2019) 102894

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Propagation behavior of hydraulic fracture across the coal-rock interface under different interfacial friction coefficients and a new prediction model

T

Yulong Jianga,b, Haojie Liana,b, Vinh Phu Nguyenc, Weiguo Lianga,b,∗ a

College of Mining Engineering, Taiyuan University of Technology, Taiyuan, 030024, China Key Laboratory of In-situ Property Improving Mining of Ministry of Education, Taiyuan University of Technology, Taiyuan, 030024, China c Department of Civil Engineering, Monash University, Clayton, Victoria, 3800, Australia b

ARTICLE INFO

ABSTRACT

Keywords: Hydraulic fracturing Coal-rock interfaces Interface friction Fracture propagation behavior Prediction model

Indirect fracturing from rocks to coal is a promising technology of coalbed methane exploitation in soft and lowpermeability coal seams. Central to this technology is the prediction of the propagation path of the fluid-driven fractures at the coal-rock interfaces, which requires a deep understanding of the role of the friction properties of coal-rock interfaces. In this paper, we performed hydraulic fracturing experiments using coal-rock blocks with interfaces that were not lubricated, lubricated by oil grease or Vaseline. The results show that the fracture behaviors at different interfaces depend mostly on the vertical stress and the interfacial friction coefficients. Only when the vertical stress reaches a certain threshold, can the fractures directly penetrate the interfaces, where the threshold value increases gradually as the interfacial friction coefficient decreases. Moreover, the abrupt change in the interfacial friction causes hydraulic fractures to deflect at the interface. The injection pressure evolution curves are significantly different due to different fractures behaviors at the interfaces. When hydraulic fractures penetrate the interface, the injection pressure evolution curve shows a significant secondary rise. A distinct prediction model of hydraulic fractures across the coal-rock interface that considers the interfacial friction, the stress state and the intersection angle between the hydraulic fracture and the coal-rock interface was established. This model shows high accuracy in predicting the propagation behavior of hydraulic fracture at different interfaces in hydraulic fracturing.

1. Introduction As an important source of clean energy, coalbed methane has drawn tremendous attention in many countries around the world. Due to the extremely low permeability of coals, hydraulic fracturing has become widely used to enhance the permeability of coal reservoirs for efficient coalbed methane exploitation (Ren et al., 2014; Li et al., 2014; Meng et al., 2011). The recent advances in fracturing technology include ScCO2 fracturing (Yang et al., 2018; Zhang et al., 2017), LN2 fracturing (Cai et al., 2016) and cyclic injection fracturing (Zhou et al., 2017; Patel et al., 2017). Although the aforementioned fracturing techniques show superior performance in hard coal reservoirs, they encounter substantial difficulties in soft coal seams. The elastic modulus in soft coal reservoirs is small, the Poisson ratio is great, and the permeability is always less than 0.1 × 10−15 m2 (Zhang et al., 2018). Many technical problems arise in the direct hydraulic fracturing of soft and low-permeability coal seams, such as (1) the fracture propagation is limited, the expansion range of pressure relief is small, and the coalbed methane

∗

production in the well is small, and decays rapidly, and (2) drilling failure is common in gas well construction, which influences the fracturing effect. To solve these problems, “indirect fracturing” technology was proposed by Olsen et al. (2003, 2007). In this method, the well is drilled in the roof or floor of the soft coal seam, and the hydraulic fractures initiate in the rock layer and penetrate the rock-coal interfaces to form complicated fracture networks in the coals seam. The key to this technology is that hydraulic fracture can penetrate into the coal seam from the roof or floor, followed by efficient propagation into the coal seam, and many have tried to better understand the effect of the reservoir's mechanical properties and interfacial properties on the propagation behaviors of hydraulic fractures within the past few decades. Based on linear elastic fracture mechanics, many researchers have pointed out that the behavior of hydraulic fractures at the coal-rock interface is mainly determined by the material properties of the coal and surrounding rock and the stress state (Zhang and Jeffrey, 2006a; Simonson et al., 1978; Renshaw and Pollard, 1995; Settaru, 1988). Hanson et al. (Hanson et al., 1980a, 1980b) and Biot et al. (1983) found

Corresponding author. College of Mining Engineering, Taiyuan University of Technology, Taiyuan, 030024, China. E-mail address: [email protected] (W. Liang).

https://doi.org/10.1016/j.jngse.2019.05.007 Received 19 February 2019; Received in revised form 30 April 2019; Accepted 22 May 2019 Available online 25 May 2019 1875-5100/ © 2019 Published by Elsevier B.V.

Journal of Natural Gas Science and Engineering 68 (2019) 102894

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that the difference between the material properties at the two sides of the interface has significant influence on the stress intensity factor at the fracture tip. Daneshy (1978) and Guo et al. (2017) argue that the large bonding strength of the interface helps fluid-driven fracture penetrate the interface. Parab and Chen (2014) and Zhang and Jeffrey (2006b) demonstrated that the thickness of the interface plays a crucial role in determining the fracture propagation path as the interface is penetrated; specifically, when the thickness of the interface reaches a certain threshold, the fractures can not only penetrate the interface but also branch to form complicated fracture networks. In reality, increasing the bond strength and thickness of the interface can effectively improve the shear strength of the interface and prevent fractures from arresting at the interface (Teufel and Clark, 1981; Zhou et al., 2008; Fu et al., 2016). Many studies on the influence of the interface angle and in situ stress state on fracture propagation have also been extensively conducted based on tri-axial hydraulic fracturing tests (Dehghan et al., 2015; Tan et al., 2017). Based on field test data, Thiercelin et al. (1989) and Rahim and Holditch (1994) found that the viscosity and the injection rate of the fracturing fluid play a dominant role in the initiation and propagation of vertical fractures. Regarding the development of numerical simulation technology, Nguyen et al. (2017) proposed the algorithm for computational modeling of a fluid-driven fracture propagating in a permeable porous medium using zero-thickness flow cohesive interface elements, and the cases of both continuous and discontinuous pressure fields penetrating the fractures were implemented. Paggi and Reinoso (2017) applied the phase field approach to brittle fracture and cohesive zone models, revealing the competition between fracture penetration and fracture deflection at an interface and explaining the complex fracture patterns observed in layered materials. However, there has been no reported work on the experimental investigation of the effect of coal-rock interfacial friction on the hydraulic fracture behavior of these systems, and the critical condition for hydraulic fractures to penetrate the different coal-rock interfaces have not yet been given. As for models that can be used to predict the behavior of hydraulic fractures penetrating coal-rock interfaces, several models have been established to solve the stress field at the hydraulic fracture tip, and the corresponding prediction model can be obtained when the maximum principal stress at the hydraulic fracture tip reaches the tensile strength of the rock at the other side of the interface, including the models presented in Renshaw-Pollard (1995), Zhou et al. (2008), Blanton (1982), Warpinski et al. (1982), Gu et al. (2012), Llanos et al. (2017) and Zhao et al. (2018). However, these models are presented in complicated forms and lack distinct expressions, and programming calculations are needed in several of the models, which creates difficulty when trying to extend the models to new applications. In view of the above gaps, in this paper, tri-axial hydraulic fracturing tests using coal-rock blocks with different frictional interfaces are undertaken; a systematic investigation of the influences of the interfacial friction properties on fracture behavior and the critical stress conditions for fracture propagation and penetration of different coalrock interfaces are studied; and the effects of interfacial friction on fracture deflection are further discussed. Moreover, a new theoretical model is proposed for better description and prediction of the fracture behavior at different coal-rock interfaces. The results of this work will develop our understanding of the fracture behavior at different coalrock interfaces subject to hydraulic fracturing and will provide practical guidelines in indirect fracturing. The remainder of this paper is organized as follows. Section 2 introduces the procedure used for coal-rock block preparation and the experimental conditions. Section 3 presents and discusses the experimental results. A new model is proposed and verified in Section 4, followed by the conclusions in Section 5.

Fig. 1. HADSZ-IV tri-axial testing machine.

2. Experiments The experiment consists of two parts: (1) Static friction tests were conducted on coal-rock interfaces to determine the static friction coefficient of different interfaces. (2) Hydraulic fracturing tests were performed sequentially on coal-rock blocks with different interfaces, and the fracture propagation behaviors at the different coal-rock interfaces were studied. 2.1. Test device 2.1.1. Static friction test The static friction tests were performed on a HADSZ-IV tri-axial testing machine (Fig. 1). This testing machine was hydraulically loaded with a maximum load of 1000 kN and loading rate precision of 0.01 kN/ s. High-precision displacement sensors were attached at both sides (left and right) of the shear box (Fig. 4) to measure the relative displacement of the upper/lower surfaces. 2.1.2. Hydraulic fracturing tests The hydraulic fracturing tests were conducted in a TCHFSM-I largescale tri-axial fracturing simulation testing machine. The sample placed in the testing machine was hydraulically loaded by five high-precision cylinders, and a servo control valve was used to ensure constant loading. The maximum loading of this testing machine was 3000 kN and the loading rate precision was 0.01 kN/s. The size of the specimen was limited to 400 mm × 400 mm × 400 mm. In addition, this testing machine was equipped with a high-precision constant flow pump with a maximum flow rate of 200 mL/min. A high-precision sensor was used to monitor the pressure evolution in real-time, with a monitoring frequency of 10−3 s. The schematic diagram of the hydraulic fracturing test is shown in Fig. 2. 2.2. Test sample Anthracite coal samples were taken from the #3 coal seam in Zhaozhuang Coal Mine, Qinshui Coalfield. The rock samples were simulated by using cement mortar. The basic mechanical parameters of coal and cement mortar are listed in Table 1. Coal-rock blocks with different interfaces were prepared for hydraulic fracturing tests according to the following steps (Fig. 3). (1) Cuboid coal samples measuring 100 mm × 100 mm × 50 mm were machined along the bedding-perpendicular direction. Stirred cement mortar was placed in a 100 mm × 100 mm × 100 mm cubic mold and allowed to cure for 28 days, after which point the samples were cut into 100 mm × 100 mm × 50 mm cuboid samples. To avoid confusing existing fractures with hydraulic fractures, only coal and cement samples without visible initial fractures were 2

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Fig. 2. Schematic diagram of the hydraulic fracturing test.

inner diameter of 3 mm and length of 200 mm was placed in the borehole and then sealed by using a high-strength and high-temperature resistant sealant. The sealing depth was 20 mm leaving a naked fracturing distance of 5 mm. (4) The cement mortar samples were placed in a cool place for 48 h to allow the sealant, fracturing tube and borehole wall to bond thoroughly. (5) Each cement mortar sample was stacked on a coal sample. To create different interfaces, we lubricated the interfaces using either foodgrade lubricant, oil grease, butter or Vaseline. The coal-rock blocks without interfacial lubrication were also used in the experiments. This set of coal-rock blocks constitutes the full test set, with different interfaces for hydraulic fracturing tests (Fig. 3). The coalrock blocks with different interfaces prepared for the static friction tests were prepared using only step (1) and step (5). All of the prepared coal-rock blocks were placed in a cool room with a constant temperature of 20 °C for 48 h to ensure that the coal, cement and lubricant were well coupled.

Table 1 The basic mechanical parameters of coal and cement mortar. Material properties

Coal

Cement

Porosity (%) Permeability K (10−15m2) Elastic modulus E (GPa) Poisson ratio v Tensile strength T0 (MPa)

8.90 0.014–0.20 3.48 0.23 1.69 0.20

7.90 0.0039 6.58 0.19 4.56 0.98

Fracture toughness KIC (MPa m1/2 )

selected for the preparation of the coal-rock blocks intended for testing. (2) A bore hole of Φ 6 mm × 25 mm was drilled in each cuboid cement mortar sample measuring 100 mm × 100 mm × 50 mm. The borehole was cleaned by using an acetone solution and then was allowed to air dry. (3) A high-strength fracturing tube with an outer diameter of 4 mm,

Fig. 3. Preparation of coal-rock blocks with different interfaces.

3

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Fig. 4. The static friction test.

2.3. Testing procedure and scheme

tube (Kim and Abass, 1991).

2.3.1. Static friction test In the static friction test, direct shear tests were performed on the coal-rock blocks with different interfaces (not lubricated or lubricated with food-grade lubricant, oil grease, butter or Vaseline). In the static friction tests, the coal-rock block was placed in a shear box (Fig. 4), and then the gap between the block and box was filled with fine sand (0.15–0.18 mm) to guarantee that the block would be uniformly loaded. Next, the shear box was placed inside the HADSZ-IV tri-axial testing machine, and a normal stress (F) was applied to predetermined values. Then, the shear stress (W) was gradually applied to both sides of the shear box until relative slip occurred in the coal-rock interface, and the critical shear stress was recorded. In this way, the critical shear stress values when the relative slip occurred at the interface at normal stresses of 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15 MPa were measured. The shear tests were repeated in triplicate for each sample type, and then, the relationship between normal stress and shear stress was plotted. The slopes of the curves were determined to tabulate the friction coefficient of the coal-rock interface. The test scheme is shown in Table 2.

3. Results In this section, the results of the static friction test and the hydraulic fracturing tests using the coal-rock blocks with different frictional interfaces are presented. Specifically, the fracture behaviors at the coalrock interfaces, the critical stress conditions for fractures to penetrate the different coal-rock interfaces and the effect of interfacial friction on the fractures propagation path are discussed. 3.1. The friction coefficient of different interfaces Fig. 5 shows the relationship between the critical shear stress and normal stress when relative slip occurs at the interface. As seen in Fig. 5, with the growth of normal stress, the critical shear stress increased linearly. The relationship between the shear stress and the normal stress was linearly fitted, allowing the friction coefficient of the interfaces to be obtained by calculating the slope of the curve (Table 3). The different static friction coefficients reflect the unique effects of each lubricant. According to Table 3, the five interfaces can be ordered from the largest friction coefficients to the smallest as: no lubrication and lubricated with food-grade lubricant, oil grease, butter or Vaseline. The Vaseline-lubricated sample exhibited the best lubrication effect on the interface and it was able to significantly reduce the interfacial friction, whereas the friction was quite large for the interface without any lubricant. When the interface was not lubricated, the interfacial friction was mainly due to the interaction between particles at the interface. When the interface was coated with low-viscosity lubricant (food-grade lubricant, oil grease), the lubricant was able to penetrate into the coalrock interface, reducing the interfacial friction. When the interface was lubricated with high-viscosity lubricant (butter, Vaseline), the filtration of the lubricant at the interface was minimal; as a result, the lubrication layer that formed was able to significantly reduce the interfacial friction. Consequently, three of five lubricants with significant frictional

2.3.2. Hydraulic fracturing tests In the hydraulic fracturing tests, the prepared coal-rock blocks were first placed in a five-sided, sealed rubber sleeve to guarantee that the load on the surface of each block was distributed uniformly. Second, the coal-rock block was placed into the fracturing chamber and tri-axial stresses were applied to the blocks through pressure transmission plates and hydraulic cylinders. The surfaces of both the pressure transmission plates and rubber sleeves were coated with lubricants to minimize shear stress during loading. In the loading procedure, to avoid non-uniform loading and shear failure of the blocks, the confining stresses on each coal-rock block were applied in accordance with the following steps. First the stresses ( h , H and v ) in the three directions were all uniformly loaded to the predetermined minimum horizontal principal stress ( h ). After 10 min of constant loading, both the vertical stress ( v ) and the maximum horizontal principal stress ( H ) were loaded to the maximum horizontal principal stress. After another 10 min of constant loading, the vertical stress ( v ) was loaded to the target values. Afterward, the confining stresses were held constant for 30 min, ensuring that a uniform stress state was established around the fracturing Table 2 The static friction test scheme. Specimen

Interfaces properties

Normal stress/MPa

#1, #2, #3 #4, #5, #6

Not lubricated Lubricated by food-grade lube Lubricated by oil grease Lubricated by butter Lubricated by Vaseline

5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15

#7, #8, #9 #10, #11, #12 #13, #14, #15

Fig. 5. The relationship between the maximum shear stress and normal stress at different interfaces in the shear tests. 4

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respectively. For the coal-rock blocks without lubricant, hydraulic fractures were observed in the cement mortar of blocks #20 (6 MPa), #22 (7 MPa) and #24 (8 MPa), but no fractures were observed in the coal, suggesting that in this stress state, the fluid-driven fractures did not penetrate the interfaces, but instead propagated along the interfaces. When the vertical stress was increased, it can be seen that in block #28 (9 MPa), the hydraulic fractures were observed in the coal. The hydraulic fracture behavior changed with the magnitude vertical stress. The friction coefficient of the coal-rock blocks lubricated with oil grease was lower than the friction coefficient without lubricant. Comparing the fracture behavior of blocks #10, #9, #8 and #5 at different vertical stresses, we found that only the hydraulic fractures in block #5 (11 MPa) directly penetrated into the interface without changing direction, whereas the fractures in blocks #10 (6 MPa), #9 (7 MPa) and #8 (8 MPa) were only observed in the cement mortar, suggesting that the hydraulic fractures failed to penetrate into the coalrock interface but instead propagated along the interface. The friction coefficient of the coal-rock blocks lubricated with Vaseline was the smallest of the experimental set. Comparing to the fracture behaviors of blocks #30, #31, #33, #36 and #40 at different vertical stresses, the hydraulic fractures in blocks #30 (11 MPa), #31 (12 MPa), #33 (13 MPa) and #36 (14 MPa) failed to penetrate the coalrock interface. When the vertical stress was increased further in block #28 (15 MPa), the fractures initiated in the cement mortar penetrated directly into the interface without changing direction. Fig. 7 shows a statistical diagram of the fracture behavior in the different coal-rock interfaces in hydraulic fracturing tests. As illustrated, when the vertical stress exceeds the critical threshold, the hydraulic fractures penetrate the interface. Otherwise, the hydraulic fractures deflect along the interface. The smaller the interfacial friction coefficient, the greater the stress threshold required for the hydraulic fractures to penetrate the coal-rock interface, indicating that increasing the interfacial friction coefficient can make the hydraulic fractures penetrate the coal-rock interface at low stresses. In summary, the vertical stress plays a critical role in the propagation behaviors of the hydraulic fractures at the interface. Specifically, for the same interfacial friction coefficient, the hydraulic fractures

Table 3 The linear fitting results of the friction coefficients for the different interfaces. Interfaces properties

Fitting results

R2

Static friction coefficients

Not lubricated Lubricated by food-grade lube Lubricated by oil grease Lubricated by butter Lubricated with Vaseline

y = 0.7200x + 0.1402 y = 0.6149x + 0.0518

0.9779 0.9553

0.7200 0.6149

y = 0.4976x + 0.0871 y = 0.3569x + 0.1235 y = 0.2557x + 0.3757

0.9756 0.9827 0.9828

0.4976 0.3569 0.2557

coefficient differences (no lubricant, lubricated with oil grease or Vaseline) were selected for the hydraulic fracturing tests of the coal-rock blocks with different interfaces to compare and analyze the fluid-driven fracture propagation behaviors of the at different interfacial conditions. The hydraulic fracturing test scheme is shown in Table 4. 3.2. Hydraulic fracturing results at different interfaces After hydraulic fracturing, a magenta solution was coated onto the surface of each coal-rock block for enhanced fracture visualization. Photos of the coal-rock blocks after fracturing are shown in Fig. 6. In general, there are three types of fracture behavior at the interfaces: (1) diverting and propagating along the interface, (2) penetrating into the interface directly without changing in direction, and (3) propagating along the interface and then penetrating into the interface. 3.2.1. Fractures behavior at different interfaces In Fig. 6, the interfaces of blocks #20, #22, #24 and #28 were not lubricated, and the vertical stresses in these blocks during the hydraulic fracturing tests were 6 MPa, 7 MPa, 8 MPa and 9 MPa, respectively. The interfaces of blocks #30, #31, #33, #36 and #40 were lubricated with Vaseline, and the vertical stresses in these blocks during the hydraulic fracturing tests were 11 MPa, 12 MPa, 13 MPa, 14 MPa and 15 MPa, respectively. The interfaces of blocks #10, #9, #8, and #5 were lubricated with oil grease and the vertical stresses in these blocks during the hydraulic fracturing tests were 6 MPa, 7 MPa, 8 MPa and 11 MPa, Table 4 The hydraulic fracturing test scheme. Specimens

Interfaces

Tri-axial stress/MPa

#1 #2 #3 #4 #5 #6 #7

Lubricated by oil grease

3 3 3 3 3 3 3

5 5 5 5 5 5 5

3 3 3

h

#8 #9 #10

Specimens

Interfaces

Tri-axial stress/MPa

11 11 11 11 11 11 8

#22 #23 #24 #25 #26 #27 #28

Not lubricated

3 3 3 3 3 3 3

5 5 5 5 5 5 5

7 7 8 8 9 9 9

5 5 5

8 7 6

#29 #30 #31

Lubricated by Vaseline

3 3 3

5 5 5

11 11 12

H

v

h

H

v

#11 #12 #13 #14 #15 #16 #17 #18 #19

The central area (20 mm × 100 mm) was lubricated by Vaseline, and the residual areas (40 mm × 100 mm) were not lubricated

3 3 3 3 3 3 3 3 3

5 5 5 5 5 5 5 5 5

17 17 17 16 16 16 15 15 15

#32 #33 #34 #35 #36 #37 #38 #39 #40

3 3 3 3 3 3 3 3 3

5 5 5 5 5 5 5 5 5

12 13 13 14 14 15 15 15 15

#20 #21

Not lubricated

3 3

5 5

6 6

#41 #42

3 3

5 5

15 15

5

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Fig. 6. Hydraulic fractures in the coal-rock blocks treated with different interfacial lubricant conditions.

penetrate the interface and enter the coal only when the vertical stress reaches a critical threshold. The greater the difference between the vertical stress and the minimum horizontal principal stress, the more

favorably the fractures penetrate the interface. At the different interfaces, the critical vertical stress threshold required for hydraulic fractures penetration into the interface gradually increases with the 6

Journal of Natural Gas Science and Engineering 68 (2019) 102894

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indicates that the variation of the friction coefficient along the interface changes the propagation path of the hydraulic fractures due in part to the fact that in the tri-axial stress state, stress concentration occurs in the region where the friction coefficients change abruptly, thereby determining the deflection behavior of the hydraulic fractures. 3.2.3. Injection pressure evolution Fig. 9 shows the injection pressure evolution of the coal-rock blocks in fracturing. Despite the differences in the coal-rock blocks, the injection pressure evolution curves of these specimens are similar. These curves can be divided into four phases: a slow increase, a sharp rise, a sharp decline and a region of stability. At the initial stage of fracturing, the injection pressure increases slowly due to infiltration within the naked fracturing distance (5 mm) at the bottom of the fracturing tube. As the amount of injected fluid is increased, the pressure in the fracturing tube increases sharply after the water injection volume exceeds the overall infiltration volume. Once the injection pressure reaches the tensile strength of the cement, fractures initiate and propagate, forming stable seepage channels, and afterward, the injection pressure decreases with small fluctuations. The injection pressure evolution curves exhibit distinct behaviors depending on whether the hydraulic fractures behave penetrate the interface or deflect along the interface. When the hydraulic fractures enter the coal by penetrating the interface, the injection pressure first decreases and then increases, whereas when the fractures are prevented from penetrating the interface, no such changes appeared in the injection pressure evolution curves. For example, Fig. 9a and b are the injection pressure evolution curves of coal-rock blocks with no lubricated interfaces. In block #28 (Fig. 9b), the notation phase a-b indicates that the pressure in the fracturing tube gradually increases as the injection fluid (fresh water) is continuously pumped into the specimen. When the pressure reaches point b, the critical fracturing pressure is reached, and the fracture starts to initiate and propagate. Phase b-c indicates that the hydraulic fractures propagate continuously within the cement block, where point c denotes when the fracture has arrived at the interface. When this event happens, the injection pressure declines sharply. Phase c-d indicates that the fractures are propagating along the interface. Because there is no cohesive stress at the interface, the fractures propagate along the interface, resulting in a further decrease of the injection pressure. Phase d-e indicates that when the fractures have propagated to a certain position, they can no longer propagate along the interface but instead form a closed space due to the interfacial friction and confining pressure, yielding a second increase in the injection pressure curve. Finally, at point e, the fracturing pressure in the coal is reached, and the fractures enter the coal penetrating through the interface. The curve after point e indicates that the fractures continue to grow within the coal, eventually forming complete

Fig. 7. The statistical diagram of the fracture behaviors at different coal-rock interfaces.

decrease of the interfacial friction coefficient. When indirect fracturing technology is adopted in in-situ applications, reservoirs with significant stress differences should be selected to guarantee that the hydraulic fractures can penetrate the coal-rock interface and form effective fracture network, thereby improving the permeability of the reservoir. 3.2.2. Effect of interfacial friction on fracture deflection To get deeper insight into the influence of interfacial friction on the fracture propagation path, we conducted hydraulic fracturing tests using coal-rock blocks with interfaces treated in the following way. The interface was 100 mm × 100 mm, and the central area (20 mm × 100 mm) of which was coated with Vaseline, leaving two residual areas (each 40 mm × 100 mm) that were not lubricated. The minimum and maximum horizontal principal stresses of the coal-rock block in hydraulic fracturing tests were 3 MPa and 5 MPa, and the vertical stresses were 15 MPa, 16 MPa and 17 MPa, respectively, which were greater than or equal to the critical vertical stress threshold for fractures penetrating the coal-rock interfaces lubricated with Vaseline. Fig. 8 shows the photos of the blocks before and after the hydraulic fracturing tests. As seen, at the vertical stress of 15 MPa, the fractures first propagated within the cement, and then, upon encountering the Vaseline-lubricated interface, the fractures deflected along the interface. After a certain distance of propagation, the fractures penetrated into the coal through the non-lubricated region of the interface, forming a through-connected fracture. Even though the vertical stress was increased (16 MPa, 17 MPa) to values that were larger than the critical vertical stress threshold for fractures penetrating into the coal-rock interface lubricated by Vaseline (15 MPa), the fractures still did not penetrate the coal at the Vaseline-lubricated interface. This observation

Fig. 8. The coal-rock blocks with different interfaces before and after hydraulic fracturing. 7

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Fig. 9. The injection pressure evolution curves of the samples with non-lubricated interfaces (a, b) and interfaces lubricated with either Vaseline (c, d, e) or oil grease (f, g).

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fracturing seepage channels. Similarly, the hydraulic fractures in blocks #40 (Fig. 9e) and #5 (Fig. 9g) also penetrate the interface, and the injection pressure evolution of the two blocks is consistent with that of block #28 (Fig. 9b). The injection pressure evolution curves of blocks #20, #22, #24, #30, #31, #33, #36, #8, #9 and #10 in Fig. 9 did not show the second rise, which implies that the hydraulic fractures initiated in the cement but failed to enter the coal and instead propagated along the interface. As it can be seen, the critical fracture initiation pressure varies slightly for each coal-rock block, even for samples subject to the same stress state. This observation can be attributed to defects in the bare fracturing holes which are not totally the same. Additionally, the critical vertical stress threshold required for the hydraulic fractures to penetrate the interface is independent of the critical fracture initiation pressure and, hence, is only related to the vertical stress. Specifically, when the vertical stress exceeds the critical vertical stress threshold, the hydraulic fractures penetrate the interface; otherwise, the hydraulic fractures propagate along the coal-rock interface.

=

KIC 2 r

=

KIC 2 r

=

KIC 2 r

xx yy xy

sin 2 sin

3 2

+

v

cos 2 1 + sin 2 sin

3 2

+

h

cos 2 1

cos 2 sin 2 cos

3 2

(1)

where xx , xx and xy are the stress components at the fracture tip (MPa), KIC is the stress intensity factor at the fracture tip (MPa·m1/2), r and are the polar coordinates of the fracture tip, v is the vertical stress (MPa), and h is the minimum horizontal principal stress (MPa). When the intersection angle between a hydraulic fracture and the coal-rock interface is = , the relationship between the stress components of the coal-rock interface and the stress field at the fracture tip is r

xx + yy

=

2 xx + yy 2

= r

=

xy

xx

+

yy

cos(2 ) +

xy

sin(2 )

yy

cos(2 )

xy

sin(2 )

2 xx

2 xx

cos(2 )

yy

sin(2 )

2

(2)

where r , and r are the stress components of any point at the coalrock interface (MPa). Substituting Eq. (1) into Eq. (2), the stress components at the coalrock interface can be obtained as:

4. New model for cracks across interfaces Of the existing models that describes the fracture behavior at the coal-rock interface, the impact of fluid injection pressure has not yet taken into consideration in neither the R&P model (1995) nor the extended model (Zhou et al., 2008; Gu et al., 2012; Llanos et al., 2017; Xu et al., 2018). The calculations in these models are difficult, and these models are not conducive for direct use in in-situ applications. In order to better describe the fracture behavior at the coal-rock interface in indirect fracturing, a prediction model considering the interfacial friction, stress state and intersection angle between a hydraulic fracture and an interface is established in this section.

r

=K

K sin 2 sin

v+ h

+

2

v

+

h

2

+ r

v

h

2

= K sin 2 cos

v

h

cos(2 ) + K sin 2 cos

3 2

sin(2 )

3 2

sin(2 )

cos(2 )

= K + K sin 2 sin v+ h 2

3 2

3 2

cos 2

K sin 2 cos

cos(2 ) 3 2

cos(2 ) + K sin 2 sin

3 2

sin(2 )

sin(2 )

2

(3)

cos 2 . where K = During hydraulic fracturing, the fracturing fluid is continuously injected into the fracture. The hydraulic fracture will penetrate the interface and enter the coal if the coal-rock interface is not opened or has slipped and the maximum tensile stress at the fracture tip reaches the tensile strength of coal on the other side. Renshaw and Pollard (1995) demonstrated that plastic deformation occurs in a very small zone near the fracture tip, where the stress is equal to or less than the stress at the zone edge, and in general, the stress component of the fracture tip has a maximum value if = 0° (Xu et al., 2018). The conditions for hydraulic fractures to penetrate the coal-rock interface can be written as: KIC 2 r

4.1. Model establishment Under the external load, a hydraulic fracture with an internal uniform fluid distribution approaches the interface at any angle, . As a result of the external load, the minimum horizontal principal stress ( h ) is vertical to the hydraulic fracture, and the vertical stress ( v ) is parallel to the hydraulic fracture, in Fig. 10, where p is the fluid pressure, l is the half length of the hydraulic fracture, H is the half length of the interface, and Q0 is the fluid injection rate. The fluid is assumed to be incompressible, and the fluid filtration at the fracture tip is assumed to be minimal. Based on linear elastic fracture mechanics, the stress state at the fracture tip, which is affected by the interaction between the far stress field and the fracture, can be written as (Renshaw and Pollard, 1995),

= 0o

=

t

<

0

+ µ(

p< r

(4)

p)

where t is the tensile strength (MPa), 0 is the cohesion at the interface (MPa), α is the intersection angle between a hydraulic fracture and the coal-rock interface, and μ is the friction coefficient. Substituting Eq. (3) into Eq. (4), we obtain the prediction model of the hydraulic fracture penetrating the coal-rock interface under the action of internal stress ( pnet = p h ) and external stress:

p

h

< (

p

h

<

(cos

2

+A

h)

t 0

(cos

( t

where 9

)

+A

h )(C + D )

B +

v 2

h sin 2

µ

)

B +

3 A = cos 2 sin 2 sin 2 3 C = cos 2 sin 2 cos 2

Fig. 10. Schematic diagram of the intersection between a hydraulic fracture and the coal-rock interface.

2

v

h

2

v

h

2

v

h

v

2

+(

h

2

t

cos 2

h)

cos 2

cos 2 , B = cos 2 sin 2 cos cos 2 , D = cos 2 sin 2 sin

3 2 3 2

sin 2 sin 2

(5)

is the cohesion, µ is the friction coefficient of the interface, and

Journal of Natural Gas Science and Engineering 68 (2019) 102894

Y. Jiang, et al.

pnet is the net pressure within the hydraulic fracture. To find the value of pnet , according to Detournay (2004), the unified form of the theoretical solution to a single fracture is transformed into:

l (t ) = L (t ) [ (t )] pnet (x , t ) = (t ) E

( , t)

with the critical model curve. Both Wang's model and our model can better predict the behavior of the fractures penetrating the interface under different interfacial friction conditions relative to Zhou's model. The prediction model established in this paper shows more accuracy when the friction coefficient is less than 0.72, and Wang's model is more suitable to describe the behavior of fractures when the interfacial friction coefficient is equal to or greater than 0.72. In summary, the interaction between fractures and the coal-rock interface is complicated, and the behavior of the fracture is usually affected by many factors. In this paper, it is assumed that the fluid pressure in the fracture is uniformly distributed. However, in fact, there is a certain infiltration pressure along the propagation path due to the frictional resistance of the interface at the fracture tip. In general, the prediction model for hydraulic fracture to penetrate the interface can efficiently describe the experimental behavior of hydraulic fracture at different interfaces. Compared to the criteria proposed by Renshaw and Pollard (1995), Zhou et al. (2008), Blanton (1982), Warpinski et al. (1982), Gu et al. (2012), Llanos et al. (2017) and Zhao et al. (2018), the prediction criterion established in this paper is of explicit expression, and is easy to calculate and significantly more suitable for use in in-situ engineering projects.

(6)

where l (t ) is the half-length of the hydraulic fracture, pnet denotes the net pressure within the hydraulic fracture, which depends only on the x coordinate, (t ) denotes a small dimensionless parameter that guarantees the variation range of from zero to infinity, L is the length scale, and represent the dimensionless fracture half-length and net pressure, respectively, and (t ) is the dimensionless evolution parameter. In the toughness-dominated regime, the classic zero-viscosity solution is determined by taking the first term, as shown by Garagash (2000) and Garagash and Detournay (2002). The related dimensionless parameters are expressed as: 8 2

K =

KIC, E =

2

1/3

4

K

(t ) =

E 1

4

E Q0 t

,

2

=

2/3

,

1/3

=

(7)

8

5. Conclusions

where E is defined as the plane-strain elastic modulus, denotes Poisson's ratio, KIC is the rock fracture toughness, and Q0 represents the injection rate. From Eq. (6) and Eq. (7), the net pressure in a hydraulic fractures can be obtained as:

( ) E K

l (t ) = p

h

2/3

In this paper, the laboratory hydraulic fracturing tests on coal-rock blocks with different lubricated interfaces were conducted, and the fracture behaviors at different interfaces and the injection pressure evolution were analyzed. A new prediction model for the hydraulic fractures penetrating the interface was established. The conclusions are summarized as follows:

(Q0 t )2/3

= pnet ( , t ) = E

( )

( ) K E

4/3

( Q0 t )

1/3

= KIC [l (t ) ]

1/2

(1) The vertical stress and interfacial friction coefficient significantly affect the behavior of fracture penetrating the interface. For coalrock blocks with the same friction coefficients at the same horizontal principal stress (3 and 5 MPa), the greater the difference between the vertical stress and the minimum horizontal principal stress, the more likely that the fracture will penetrate the coal-rock interface. For non-lubricated coal-rock blocks (µ =0.7200) and coal-rock blocks lubricated by either oil grease ( µ =0.4976) or Vaseline ( µ =0.2557), the critical vertical stress thresholds for hydraulic fractures to penetrate the interface were 6 MPa, 11 MPa and 15 MPa, respectively. As the friction coefficient of the interface decreases, the critical vertical stress required for the hydraulic fractures to penetrate the interface increases. Additionally, the abrupt changes in interfacial friction cause hydraulic fractures to deflect at the interface. (2) The injection pressure evolution curves are significantly different due to differences in the behavior of fracture penetrating the interfaces. When hydraulic fractures penetrate the interface, the injection pressure evolution curve shows a significant secondary rise, whereas when hydraulic fractures propagate along the interface, there is no secondary rise in the injection pressure evolution curve. (3) A distinct prediction model of the hydraulic fracture behavior penetrating the coal-rock interface that considers the interfacial friction, stress state and intersection angle between a hydraulic fracture and the coal-rock interface was established. This model was validated with experimental results and compared with other models. This model can efficiently describe the behavior of hydraulic fracture at different coal-rock interfaces.

(8)

Moreover, upon the substitution of Eq. (8) into Eq. (5), an improved prediction model for the penetration of a hydraulic fracture through the interface in the expression of the ground stress difference and the intersection angle can be obtained as follows: v

h

>

v

h

>

2KIC [l

] 1/2

2KIC [l

(

2( t 1

h) cos

2

+A

B

)

cos 2

] 1/2 + 2( t 1

h ) (C + D ) / µ

cos

cos 2

/µ

+ sin 2

2

A+B

2 0/ µ

(9)

In our tests, the intersection angle between a hydraulic fracture and the coal-rock interface is α = 90°, and the cohesion at the interface is given by 0 = 0 . Therefore, Eq. (9) can be simplified as, v

h

> KIC [l ]

1/2

v

h

> KIC [l ]

1/2

+

2 4

(

t

h)

2 4

(

t

h)

( ) 1

µ

µ

(10)

4.2. Model validation To verify the accuracy and applicability of the theoretical prediction model for a hydraulic fracture penetrating the interface, the experimental data in this work and the results of other models proposed in previous papers were compared to the model. The verification results are shown in Table 5. Fig. 11 shows the experimental data and the critical model curves, in which the solid line denotes the critical model curve for the hydraulic fracture penetrating the interface, as proposed in this work, the dot and dash line denotes the critical model curve proposed by Wang et al. (2018), and the dotted line denotes the critical model curve proposed by Zhou et al. (2008). As seen in Fig. 11 and Table 5, most of the experimental results (14 points) are in good agreement with the critical model curve proposed in this paper, and only one experimental point did not completely coincide

Acknowledgements Support for this work was provided by the National Natural Science Foundation of China (No. 51874206; No. 51704204), Australian Research Council via project (DE160100577) and Science and Technology Major Project in Shanxi Province (MQ2016-01), which are 10

Journal of Natural Gas Science and Engineering 68 (2019) 102894

Y. Jiang, et al.

Table 5 Validation of the experimental results and the model ( = 90o ). Data This paper

Zhou

v /MPa

h /MPa

v

h /MPa

KIC

µ

Experimental results

Model in this paper

Wang's model (2018)

Zhou's model (2008)

6 7 8 9

3 3 3 3

3 4 5 6

0.98

0.7200

No crossing No crossing No crossing Crossing

No crossing No crossing Crossing Crossing

No crossing No crossing No crossing Crossing

No crossing No crossing Crossing Crossing

6 7 8 11

3 3 3 3

3 4 5 8

0.98

0.4976

No crossing No crossing No crossing Crossing

No crossing No crossing No crossing Crossing

No crossing No crossing No crossing Crossing

No crossing No crossing No crossing Crossing

11 12 13 14 15

3 3 3 3 3

8 9 10 11 12

0.98

0.2557

No crossing No crossing No crossing No crossing Crossing

No crossing No crossing No crossing No crossing Crossing

No crossing No crossing No crossing Crossing Crossing

Crossing Crossing Crossing Crossing Crossing

10 10

5 3

5 7

0.59

0.8900

Crossing Crossing

Crossing Crossing

Crossing Crossing

Crossing Crossing

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Fig. 11. The critical curve from the model and experimental data under different test conditions.

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