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. 2022 Dec 30;8(1):1206–1219. doi: 10.1021/acsomega.2c06627

Analysis of the Induced Stress Fields Around Hydraulic Fractures Considering the Influence of Natural Fractures and Bedding Planes

Rui Huang †,*, Qun Lei †, DingWei Weng †, JunBin Chen ‡, HongBo Liang †
PMCID: PMC9835785  PMID: 36643490

Abstract

graphic file with name ao2c06627_0018.jpg

Natural fractures (NFs) and bedding planes (BPs) are well developed in shale reservoirs. The propagation of hydraulic fractures (HFs) and the opening of NFs and BPs can produce induced stress fields (ISFs) within the fracturing process, causing interference to the in situ stress field. Aiming at the “stress shadow” effect among HFs in horizontal wells, the calculation models of HFs, BPs, and NFs for induced stress distributions are established based on displacement discontinuity theory, which can quantitatively characterize the composite ISF of the three under different connecting states. In addition, the interference coefficient of stress intensity factor (ICSIF) is introduced to quantitatively evaluate the interference degree of the composite ISF to the propagation of HFs. The results show that: (1) the ISF forms a “tensile stress concentration zone” near the fracture surface to promote the HFs opening and a “compressive stress concentration zone” at the fracture tips to suppress the propagation of HFs; (2) the ISF forms an elliptical effective swept area around the fracture, which is affected by the propagation height of HFs, while NFs or BPs generate local disturbances to the ISF; (3) the in situ stress reverses in the swept area, and the stress reversal interval is related to the in situ stress difference, fracture propagation height, Poisson’s ratio, fracture net pressure, and fracture spacing; (4) the reasonable fracture spacing and fracture propagation height of horizontal wells can be determined by the ICSIF. The study can provide theoretical guidance for optimizing the fracture spacing and promoting the uniform propagation of multiple fractures in staged fracturing of horizontal wells.

1. Introduction

Shale reservoirs are usually characterized by low matrix permeability, and the conventional idea of reservoir reconstruction is to expand the drainage area and increase the seepage channel by artificially increasing the fracture density.1−5 During the propagation of hydraulic fractures (HFs), the fracture surface is subject to tensile deformation with the impact of fracture net pressure, resulting in the stress shadow effect (SSE).6−10 Within the scope of SSE, the in situ stress field is interfered to a certain extent, resulting in the adverse consequence of uneven propagation of multiple HFs.11−13 In addition, it is easy to form new induced breaking points in the main fracture if the fluid supply is continued, thus extending the branch fractures and increasing the complexity of the fracture network.14−16

Scholars have systematically studied the distribution law of induced stress field (ISF) through theoretical analysis. Daneshy17 obtained the distribution of SSE under different closed states of fractures. When the two horizontal principal stresses are quite close, the fracture deformation caused by SSE is the largest. Qiao et al.18 established the calculation model of induced stress considering the attenuation of fracture net pressure according to Sneddon formula and proposed that the horizontal principal stress difference and the fracture net pressure are the main controlling factors affecting the stress reorientation.

Numerical simulation is the mainstream method to study the SSE and the competitive propagation law of multiple HFs in horizontal wells. Ahmed et al.19 conducted numerical simulation for Eagle Ford Reservoir and proposed that SSE and lateral heterogeneity are important factors affecting oil production. Morrill and Miskimins20 simulated the distribution pattern of stress field around HFs in horizontal wells and suggested weakening the SSE by optimizing perforation spacing. Guan et al.21 used the finite element method to explore the distribution law of ISF and quantitatively characterized the control range of ISF within the process of hydraulic fracturing.

The non-uniform propagation pattern of multiple HFs with the SSE can be intuitively captured by a physical model experiment. Zhou et al.22 analyzed the fracture geometry of shale outcrops with different perforation intervals by triaxial fracturing experiments. Chen et al.23 designed an induced stress triaxial test device to evaluate the effects of fracture geometric parameters, fracture spacing, and pumping on the ISF.

Although predecessors have revealed the influence of SSE on HFs from theoretical analysis, numerical simulation, and physical model experiment, there are still several points needing to be considered in depth. Shale is essentially heterogeneous, which develops a large number of bedding planes (BPs) and natural fractures (NFs). Therefore, the propagation path of HFs is not only affected by the stress field but also related to the distribution characteristics of BPs and NFs.24 HFs separate part of net pressure to open NFs in the process of propagation, so the HFs may deflect along the direction of NFs. In addition, the propagation direction of HFs will also be deflected due to the significant difference in the stress field and rock physical properties on the upper and lower sides of the BPs. In general, the distribution of stress field determines the basic trend of HFs, while the connecting states of HF–BP or HF–NF affect the local trend of HFs.25 Hence, it is necessary to establish quantitative models of NFs and BPs to reveal the distribution law of the composite ISF of the BP–HF–NF system.

In view of the SSE among multiple fractures in horizontal wells, not only is the ISF calculation model of HFs based on the displacement discontinuity theory established but the distribution law of ISF around single HF or multiple HFs is also quantitatively characterized. In addition, the ICSIF is introduced to quantitatively evaluate the interference degree of the composite ISF to the propagation of HFs. The results can provide theoretical guidance for optimizing the fracture spacing and promoting the uniform propagation of multiple fractures in staged fracturing of horizontal wells.

2. Computational Methods

2.1. ISF Representation

2.1.1. Quantitative Model of the SSE

When the displacement discontinuity method (DDM) is used to deal with the propagation problem of fractures, it is not necessary to densify the fracture boundary, but only to combine the two sides of the fracture into a displacement discontinuity element. As shown in Figure 1, a three-dimensional fracture is scattered into m units at equal intervals along the length direction. The length of each unit is a, and the central coordinate of unit i is (xi,yi,H).

Figure 1.

Figure 1

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Diagram of three-dimensional fracture along the length direction.

For the fracture line element model in the x–y plane, if the center of element i is taken as the center of the coordinate axis, the s–n coordinate axis is established, respectively, along the tangent and normal directions of element i. The displacement discontinuity can be expressed as the difference between the displacements at both ends of the element, which can be obtained as follows26

2.1.1. 1

where Ds and Dn are, respectively, tangential and normal displacement discontinuities of element i, m; us is the displacement of element i in s direction, m; un is the displacement of element i in n direction, m; 0– represents the lower surface of element i; 0+ represents the upper surface of element i.

The ISF generated by element j is shown as follows

2.1.1. 2

where σssij and σnn are, respectively, the induced stress components generated by element j to element i, MPa; Cssij, Csn, Cnsij, and Cnn are the influence factors of induced stress generated by element j, respectively.

The influence of the fracture height on the displacement and stress discontinuity is characterized by the fracture height correction coefficient Mij, which is presented as follows26,27

2.1.1. 3

where dij is the distance from element j to element i, m; Hj is the height of element j; m, α, β, and ω are correction factors.

The SSE among multiple HFs can be characterized by the distribution of ISF. Hence, taking the effect of other elements into account, the SSE at element i is shown as follows

2.1.1. 4

where σssij and σnn are, respectively, the induced stresses in s direction and n direction at element i, MPa.

Discrete eq 4 to obtain

2.1.1. 5

Equation 5 is a 2m × 2m matrix equation, which can be solved by the Gaussian elimination method. After obtaining the displacement discontinuity of each element, the distribution of the entire stress field around the fracture can be determined.

The distribution of fluid pressure in the fracture is shown as follows

2.1.1. 6

where pfw is the fluid pressure at fracture inlet, MPa; L is the half length of the fracture, m.

2.1.2. Analytical Model of Fracture Mechanics

The distribution law of ISF around HFs is usually restricted by a variety of geological and engineering factors. To simplify the complexity of the model, the assumptions are made as follows:

  • (1)

    The formation is homogeneous and isotropic, and the rock is linear elastic;

  • (2)

    The changes of mechanical properties caused by physical and chemical interaction between rock and fracturing fluid are neglected;

  • (3)

    The temperature-induced stress caused by heat exchange between rock and fracturing fluid cannot be taken into consideration;

  • (4)

    All HFs are plane symmetrical, and the propagation pattern of HFs meets the PKN model.

As shown in Figure 2, the HFs are characterized by parallel propagation. The vertical direction of the first fracture surface is taken as the x axis, the length direction of the fracture is taken as the y axis, the height direction of the fracture is taken as the z axis, the height of the fracture is 2Hi, and the fracture spacing is di–1. Under the plane strain state, the range of ISF is mainly controlled by the height of fracture, so the x–z plane can be taken as the principal plane to analyze the distribution law of ISF.

Figure 2.

Figure 2

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Geometric model of ISF of multiple fractures.

The induced stress components at a certain point P(x, y, z) are as follows28,29

2.1.2. 7
2.1.2. 8

where σx, σy, and σz are induced stresses in x, y, and z directions, respectively, MPa; pi is the net pressure of fracture i, MPa; ri is the distance from the target point to the center of fracture i, m; ri1 is the distance from the target point to the bottom of fracture i, m; ri2 is the distance from the target point to the top of fracture i, m; θi is the angle that the target point deviates from the center of fracture i, rad; θi1 is the angle that the target point deviates from the top of fracture i, rad; and θi2 is the angle that the target point deviates from the bottom of fracture i, rad.

2.1.3. Model Validation

The Sneddon model is a classical model of fracture mechanics with many basic assumptions, which is generally applicable to homogeneous reservoirs, and the fractures must be plane symmetrical. The model based on DDM has few assumptions, which is not only applicable to homogeneous reservoirs but also to heterogeneous reservoirs, and the geometry of fractures can be complex.

The ISFs of Sneddon model and DDM model are calculated, respectively, by using the geological parameters of X shale oil horizontal well in Longdong, Changqing Oilfield, China (Table 1). Taking the width direction and the length direction of the fracture as examples, the calculation results of Sneddon model and DDM model are shown in Figure 3. The induced stress distribution curves of the two models along the coordinate axis are almost coincident, indicating that the DDM model can accurately characterize the distribution law of the ISF around HFs.

Table 1. Relative Parameters of the Quantitative Model of ISF.
parameters value
horizontal maximum principal stress σH/MPa 50
horizontal minimum principal stress σh/MPa 45
fracture net pressure pfw/MPa –10
Poisson’s ratio ν 0.25
porosity ϕ/% 0.05
Young’s modulus E/MPa 23,000
NF height h0/h 0.4
NF length 2a/h 0.25
NF approach angle α/rad 0.25π
BP height h0/h 0.4
BP opening length 2b/h 0.5
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Figure 3.

Figure 3

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Comparison of induced stress of two models along the coordinate axis. (a) Induced stress on the x axis. (b) Induced stress on the y axis.

2.2. Quantitative Models of NF and BP

2.2.1. ISFs of NF and BP

As is shown in Figure 4, HFs can form the following three typical connecting states after contacting with NFs or BPs in the process of propagation:30−32

  • (a)

    The NF (BP) is not opened: Although the fracturing fluid has sufficient energy to provide high net pressure for the propagation of HF (BP), the higher closure pressure of NF (BP) forces HF to directly penetrate NF (BP) without opening NF, in which case the fracturing fluid cannot flow into NF (BP);

  • (b)

    T-shape composite fracture: The energy of fracturing fluid is too weak to provide enough fracture net pressure, so the HF stops extending along the direction of fracture height, and the NF (BP) is effectively opened, in which case all fracturing fluid flows into the NF (BP);

  • (c)

    X-shape composite fracture: Owing to the high fracture net pressure of HF and low closure pressure of NF (BP), not only can the HF penetrate the NF (BP), but it can also open the NF (BP) effectively. Under this circumstance, part of fracturing fluid flows into the NF (BP), and the other part continues to extend along the initial path of HF.

Figure 4.

Figure 4

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Diagram of connecting states of NF–HF and BP–HF. (a) NF–HF connecting states. (b) BP–HF connecting states.

2.2.1.1. ISF Model Based on Displacement Discontinuity Method

The ISF generated by the NF (BP) is shown as follows

2.2.1.1. 9

Affected by the fracture net pressure, the stress boundary conditions of element i are as follows

2.2.1.1. 10
2.2.1.2. Analytical Model of Fracture Mechanics

According to Sneddon equation, the ISF of NF(BP) can be obtained as follows

2.2.1.2. 11
2.2.1.2. 12

The conversion relationship between x’ and y’ coordinate system and x–z coordinate system is presented as follows

2.2.1.2. 13

where a is the half length of NF, m; α is the angle between NF and the horizontal wellbore, rad.

2.2.2. Composite Induced Stress of NF–HF

The composite ISF formed in the process of propagation is the superposition of ISF of HFs and that of NFs. The induced stress distribution patterns of the three types of connectivity between HFs and NFs are as follows:

  • (a)

    The NF is not opened

Since the NF is not opened, the fracturing fluid cannot flow into the NF, and the NF cannot produce ISF. The ISF is completely generated by HF, and “tension is positive and pressure is negative”(Figure 5). The induced stress in the x direction can form a “tensile stress concentration zone” near the fracture surface, which promotes the radial opening of the HF; the induced stress in the y direction can form a “compressive stress concentration zone” near the fracture tip, which inhibits the axial extension of the HF. Therefore, the effect of induced stress generated by the fracture net pressure on the HF itself is to “inhibit its length but increase its width”. With the increasing distance from the fracture tip or fracture surface, the induced stress gradually decreases to 0 MPa, indicating that the ISF can form an elliptical effective swept area around the HF, which has the character of longitudinal symmetry.

  • (b)

    T-shaped fracture of NF–HF

Figure 5.

Figure 5

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Distribution patterns of composite ISF with NF not opened. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

As shown in Figure 6, the HF stops extending axially and the fracturing fluid completely flows into the NF. The ISF at the tip of HF is affected by the ISF of NF, resulting in the local deformation. The induced compressive stress concentration area at the fracture tip shrinks significantly, and the NF–HF composite ISF is no longer symmetrically distributed. Compared with the condition that the NF is not opened, the effective swept area of the NF–HF composite ISF is distinctly reduced.

  • (c)

    X-shaped fracture of NF–HF

Figure 6.

Figure 6

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Distribution patterns of composite ISF of T-shaped fracture. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

As shown in Figure 7, the fracturing fluid promotes the opening of NF while maintaining the propagation of HF. Due to the local disturbance caused by the ISF of NF, the NF–HF composite ISF is no longer symmetrically distributed. Distortion occurs in the local area close to the NF, but the effective swept area has almost no significant change compared with the condition that the NF is not opened.

  • (d)

    Multiple types of fractures of NF–HF

Figure 7.

Figure 7

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Distribution patterns of composite ISF of X-shaped fracture. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

The induced stress distribution patterns of three equally spaced NF–HF fractures are shown in Figure 8. The effective swept area of ISF of X-shaped fracture of NF–HF is almost the same as that of NF when NF is not opened, while the effective swept area of ISF of T-shaped fracture of NF–HF is significantly reduced. The two cases indicate that the effective swept area of composite ISF is mainly determined by the height of HF, while the ISF of small-scale NF only causes local disturbance in the surrounding limited space. In other words, the ISF of NF has no significant impact on the entire effective swept area.

Figure 8.

Figure 8

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Distribution patterns of composite ISF of multiple types of fractures. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

2.2.3. Composite Induced Stress of BP–HF

The composite ISF is the superposition of ISF of HFs and that of BPs. The induced stress distribution patterns of the three types of connectivity between HFs and BPs are as follows:

  • (a)

    The BP is not opened

As the BP is not opened, the fracturing fluid cannot flow into it (Figure 9). The ISF is only generated by HF, which is the same as the distribution pattern of SF–HF composite ISF when the NF is not opened (Figure 5).

  • (b)

    T-shaped fracture of BP–HF

Figure 9.

Figure 9

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Distribution patterns of composite ISF with BP not opened. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

As shown in Figure 10, the HF stops extending axially, and the fracturing fluid completely flows into the BP. The ISF at the tip of HF is significantly affected by the ISF of the BP, and the compressive stress concentration area at the fracture tip is distinctly reduced.

  • (c)

    X-shaped fracture of BP–HF

Figure 10.

Figure 10

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Distribution patterns of composite ISF of T-shaped fracture. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

As shown in Figure 11, because of the local disturbance formed by the ISF of BP, the BP–HF composite ISF is no longer symmetrically distributed.

  • (d)

    Multiple types of fractures of BP–HF

Figure 11.

Figure 11

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Distribution patterns of composite ISF of X-shaped fracture. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

The induced stresses distribution patterns of three equally spaced BP–HF fractures are shown in Figure 12. The effective swept area of ISF of T-shaped fracture of BP–HF is significantly reduced compared with that of X-shaped fracture, which indicates that the effective swept area of composite ISF is mainly determined by the height of HF, while the ISF of BP brings about local disturbance in the limited space.

Figure 12.

Figure 12

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Distribution patterns of composite ISF of multiple types of fractures. (a) Distribution of σx. (b) Distribution of σy. (c) Distribution of (σx – σy).

3. Results and Discussion

3.1. In Situ Stress Reversal Mechanism

In the propagation process of HF, if the induced stress difference around the fracture exceeds the original horizontal principal stress difference, local stress reversal behavior will occur. Therefore, the propagation pattern of HF may undergo special changes, and the HF may no longer extend along the initial direction.31

The criterion for stress reversing is shown as follows

3.1. 14

3.1.1. Characterization of SRI

Taking a certain point P(x,0) on the x axis, the relations of geometric parameters are as follows

3.1.1. 15

Hence, the induced stress difference on the x axis can be presented as follows

3.1.1. 16

Under the condition that the fracture net pressure is −10 MPa, the difference of in situ principal stress is 5 MPa, and Poisson’s ratio is 0.25, the induced stress distribution on the x axis around the fracture is obtained according to the established displacement discontinuity model. Based on the relation curve between the vertical displacement from the fracture surface and the induced stress difference (Figure 13), the stress reversal interval (SRI) can be formed under the effect of ISF, which is approximately shown as follows

3.1.1. 17
Figure 13.

Figure 13

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Distributions of vertical displacement from fracture surface with different induced stress differences.

3.1.2. Analysis of Influencing Factors

The SRI around HF is jointly affected by geological factors (in situ principal stress difference and rock Poisson’s ratio), engineering factors (fracture net pressure), and fracture geometric parameters (fracture height and fracture spacing).21

3.1.2.1. Influence of In Situ Principal Stress Difference

As shown in Figure 14a, under the condition that the fracture net pressure is −10 MPa and Poisson’s ratio is 0.25, the relationship between the SRI on the vertical direction of fracture surface and the in situ principal stress difference is approximately logarithmic. When the in situ principal stress difference is 6.35 MPa, the SRI is the smallest and the reversal area is 0.5πh2. If the in situ principal stress difference exceeds the critical value of 6.35 MPa, the stress reversal behavior will not occur around the fracture.

Figure 14.

Figure 14

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Distributions of induced stress difference with various influencing factors. (a) Influence of in situ principal stress difference. (b) Influence of Poisson’s ratio. (c) Influence of fracture net pressure. (d) Influence of fracture height. (e) Influence of fracture spacing.

3.1.2.2. Influence of Rock Poisson’s Ratio

As shown in Figure 14b, when the fracture net pressure is −10 MPa and the in situ principal stress difference is 5 MPa, the stress reverses around the fracture under the effect of induced stress with Poisson’s ratio increasing from 0.1 to 0.3, and the reversal area decreases from 1.38πh2 to 0.86πh2. Hence, the SRI is negatively related to Poisson’s ratio.

3.1.2.3. Influence of Fracture Net Pressure

As shown in Figure 14c, when the in situ principal stress difference is 5 MPa and Poisson’s ratio is 0.25, with the fracture net pressure increasing from −10 to −8 MPa, the stress reverses around the fracture under the effect of induced stress, and the reversal area decreases from πh2 to 0.65πh2. When the fracture net pressure drops to −7 MPa, the induced stress difference of the fracture is always lower than the original principal stress difference, so the stress will not reverse around the fracture.

3.1.2.4. Influence of Fracture Height

As shown in Figure 14d, when the in situ principal stress difference is 5 MPa, Poisson’s ratio is 0.25, and the fracture net pressure is −10 MPa; with the half fracture height less than 0.75h, the induced stress difference is always lower than the in situ principal stress difference. When the half fracture height increases from h to 1.5h, the stress reverses around the fracture under the effect of induced stress, and the reversal area increases from πh2 to 3.23πh2.

3.1.2.5. Influence of Fracture Spacing

As shown in Figure 14e, when the in situ principal stress difference is 5 MPa, Poisson’s ratio is 0.25, and the fracture net pressure is −10 MPa, with the fracture spacing increases from 2h to 4h, the stress reverses around the fracture under the effect of SSE, and the reversal area increases from πh2 to 2πh2. However, if the fracture spacing is larger than 4h, the stress reversal area drops from 2πh2 to 1.15πh2 (see Table 2).

Table 2. Relationship Between SRI and Various Influencing Factors.
number (σH – σh)/MPa ν pnet/MPa stress reversal interval stress reversal area
1 3 0.25 –10 [−1.58h,1.58h] 1.58πh2
2 4     [−1.42h,1.42h] 1.42πh2
3 5     [−h,h] πh2
4 6     [−0.75h,0.75h] 0.75πh2
5 7       0
6 5 0.1 –10 [−1.38h,1.38h] 1.38πh2
7   0.15   [−1.25h,1.25h] 1.25πh2
8   0.2   [−1.12h,1.12h] 1.12πh2
9   0.25   [−h,h] πh2
10   0.3   [−0.86h,0.86h] 0.86πh2
11 5 0.25 –6   0
12     –7   0
13     –8 [−0.65h,0.65h] 0.65πh2
14     –9 [−0.82h,0.82h] 0.82πh2
15     –10 [−h,h] πh2
number (σH – σh)/MPa ν h′/h stress reversal interval stress reversal area
16 5 0.25 0.5   0
17     0.75   0
18     1 [−h,h] πh2
19     1.25 [−1.75h,1.75h] 2.19πh2
20     1.5 [−2.15h,2.15h] 3.23πh2
number (σH – σh)/MPa ν d/h stress reversal interval stress reversal area
21 5 0.25 2 [0,2h] πh2
22     3 [0,3h] 1.5πh2
23     4 [0,4h] 2πh2
24     5 [0,1.2h]U[3.8h,5h] 1.2πh2
25     6 [0h,1.15h]U[4.85h,6h] 1.15πh2
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3.2. Reasonable Fracture Spacing in Horizontal Wells

For tight shale reservoirs, the conventional concept of hydraulic fracturing is to obtain the maximum stimulated reservoir volume (SRV) by artificially increasing the fracture density. When the fracture spacing exceeds the critical sweep length of SSE, the interaction of superimposed ISF among fractures can be eliminated. Therefore, it is necessary to optimize the fracture spacing to achieve the best effect of reservoir reconstruction.

3.2.1. Effect of Fracture Spacing

The implicit equation of reasonable fracture spacing which can eliminate the SSE is

3.2.1. 18

The distribution pattern of ISF generated by five unevenly spaced fractures is shown in Figure 15a. The ISF generated by the fracture net pressure can form a certain range of influence area around the fracture. When the fracture spacing exceeds the effective swept area of the original fracture ISF, the adjacent fractures can extend away from the control of the original fracture ISF, thus realizing free extension; when adjacent fractures are arranged within the effective swept area of the original fracture ISF, the propagation of adjacent fractures will be inevitably affected by the SSE of the original fracture.

Figure 15.

Figure 15

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Distributions of induced stress of five unevenly spaced fractures. (a) Distribution of (σx – σy). (b) Induced stress distribution on the x axis.

The induced stress distribution curve of five unevenly spaced fractures on the x axis is shown in Figure 15b. Due to the SSE, the distributions of superimposed ISF around multiple fractures are significantly different, which can be roughly shown as the induced stresses in both the x direction and y direction decrease gradually with the increasing distance from the middle fracture. In the case of small fracture spacing, the disturbance effect of fracture 3 is the strongest, and stress concentration areas can be formed on both sides of fracture 3; in the case of large fracture spacing, the ISF on the side close to fracture 1 (or fracture 5) of fracture 2 (or fracture 4) is significantly smaller than that on the side close to fracture 3, indicating that the fracture spacing is the key factor to determining the SSE under the condition that the height and net pressure of multiple fractures are equal.

3.2.2. Definition of ICSIF

Under the influence of SSE, the stress intensity factor of fracture i is shown as follows

3.2.2. 19

where Kfi is the stress intensity factor of fracture i; hi is half of the propagation height of fracture i; P̅fi is the average flow pressure in fracture i, MPa; σxj is the induced stress along the x direction of fracture j, MPa.

Within the influencing range of SSE, the lateral force on the fracture surface usually increases, but the stress intensity factor at the fracture tip always reduces. The interference coefficient of stress intensity factor (ICSIF) is introduced to characterize the influence of SSE on fracture propagation. Its physical meaning is the ratio of the stress intensity factor affected by the induced stress of adjacent fractures to the stress intensity factor of a single fracture without the interference of adjacent fractures.

3.2.2. 20

where ni is the ICSIF of fracture i; Koi is the stress intensity factor of fracture i without interference from adjacent fractures; hoi is half of the propagation height of fracture i without the interference of adjacent cracks; Δxij is the linear distance from fracture i to fracture j, m; hj is half of the height of fracture j, m.

If ni exceeds 0, the ISF inhibits the propagation of fracture i; if ni is less than 0, the ISF promotes the extension of fracture i; if ni equals 0, the ISF has no effect on the propagation of fracture i.

Taking three equally spaced fractures as an example, assuming that the fracture spacing is d and the heights of fracture 1 and fracture 3 are equal, the ICSIF is calculated by eq 21 to quantify the interference degree of induced stress of two adjacent fractures on the middle fracture.

3.2.2. 21

Because of the SSE, the propagation height of the intermediate fracture is restrained. The relationship between ICSIF and fracture spacing is shown in Figure 16a. The restriction regions with different ranges are formed around the intermediate fractures with different heights. The influence of the ISF generated by adjacent fractures on the middle fractures distinctly decreases with the increasing fracture spacing. When the fracture spacing is too small, the tips of the two adjacent fractures attract or repel each other in the process of propagation, resulting in the phenomenon of “Fracture hit.”29,31

Figure 16.

Figure 16

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Relation curves between SSE effect and fracture spacing of three evenly spaced fractures. (a) ICSIFs with different fracture spacings. (b) Propagation heights with different fracture spacings.

Taking the curve of h2 = 0.75h as an example, the ICSIF is always greater than 0 within 4.25 times of the half fracture height from the adjacent fracture, which indicates that the ISF of the two adjacent fractures forms a restriction region of about 2.125 times of the fracture height for the middle fracture. If the middle fracture is arranged in the restriction region, the difficulty of the fracture propagation will increase; if the middle fracture is arranged outside the restriction region, the fracture propagation will not be affected by the ISF.

As is shown in Figure 16b, under the influence of SSE, the relationship between the propagation height of middle fracture and the fracture spacing is as follows

3.2.2. 22

If the fracture spacing and the height of adjacent fracture are known, the propagation height of middle fracture can be obtained by using the eq 22.

4. Conclusions

  • (1)

    The effect of induced stress generated by the fracture net pressure on the HF itself is to “inhibit its length but increase its width”. A “tensile stress concentration zone” is formed near the HF surface to promote the fracture to open; a “compressive stress concentration zone” is formed at the HF tips to inhibit the fracture from extending. The SSE on the middle fracture is to “restrain its length and width,” which is specifically manifested in the increase of the radial bearing pressure and the inhibition of the axial extension length of the middle fracture;

  • (2)

    The ISF forms an elliptical effective swept area around the fracture. Under the condition that NF or BP is effectively opened, the effective swept area generated by NF–HF or BP–HF composite ISF is mainly determined by the propagation height of HF, while small-scale NF or BP only causes local disturbance in the surrounding limited space and has no significant impact on the entire effective swept area;

  • (3)

    The in situ stress reverses in the swept area of ISF, and the main controlling factors affecting the SRI around the fracture include geological factors (in situ principal stress difference and rock Poisson’s ratio), engineering factors (fracture net pressure), and fracture geometric parameters (fracture propagation height and fracture spacing);

  • (4)

    Under the condition that the fracture net pressure and the fracture propagation height are constant, the fracture spacing is the key factor to determine the SSE around the fracture. The ICSIF can be used to quantitatively characterize the influence of SSE on fracture propagation, and then the reasonable fracture spacing and fracture propagation height of horizontal wells can be determined.

5. Limitations and Prospects

In order to simulate the HF–NF–BP composite induced stress field, it is necessary to assume the length and height of NF and the opening length of BP in advance, which need to be improved in the future researches. In addition, the relevant data in the paper are derived from the geological parameters of X shale oil horizontal well in Longdong, Changqing Oilfield, China (Table 1), and the random arrangement of NFs with different sizes and angles cannot be fully considered in the numerical simulation process, which is a difficulty to be overcome in the subsequent research studies.

Acknowledgments

This research was supported by “Research on Key Technologies and Equipment for Reservoir Reconstruction” (2021DJ45), the technology development project of China National Petroleum Corporation.

Glossary

Abbreviations

NF

natural fracture

HF

hydraulic fracture

BP

bedding plane

SSE

stress shadow effect

ISF

induced stress field

SRI

stress reversal interval

DDM

displacement discontinuity method

ICSIF

interference coefficient of stress intensity factor

SRV

stimulated reservoir volume

The authors declare no competing financial interest.

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