Numerical investigation on the effect of well interference on hydraulic fracture propagation in shale formation

Journal Pre-proofs Numerical investigation on the effect of well interference on hydraulic fracture propagation in shale formation Heng Zheng, Chunsheng Pu, Ersi Xu, Chao Sun PII: DOI: Reference:

S0013-7944(20)30075-8 https://doi.org/10.1016/j.engfracmech.2020.106932 EFM 106932

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Engineering Fracture Mechanics

Received Date: Revised Date: Accepted Date:

15 January 2020 11 February 2020 11 February 2020

Please cite this article as: Zheng, H., Pu, C., Xu, E., Sun, C., Numerical investigation on the effect of well interference on hydraulic fracture propagation in shale formation, Engineering Fracture Mechanics (2020), doi: https://doi.org/10.1016/j.engfracmech.2020.106932

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Numerical investigation on the effect of well interference on hydraulic fracture propagation in shale formation Heng Zheng1*

Chunsheng Pu1*

Ersi Xu2

Chao Sun1

1 School of Petroleum Engineering, China University of Petroleum(East China), Qingdao, Shan dong 266580, China 2 Shale Gas Research Institute, PetroChina Southwest Oil & Gas field Company, Chengdu, Sichuan 610051, China

Correspondence to: Heng Zheng([email protected]) ;Chunsheng Pu ([email protected]) Abstract: Due to the low permeability of shale reservoir , the multiple well fracturing technologies are widely used for the development of shale gas. In this paper , a numerical model coupled rock deformation , fracturing fluid flow to investigate the effect of well interference on fracture dynamic propagation in multiple well fracturing. The simulation results indicate that asymmetric fracture propagations occur in multi-well fracturing , and the lateral growths of interior fractures are suppressed due to the intense interwell stress interference. With the increase well space , the effect of inter-well stress interference decreases gradually. The case studies also indicate that , for successful fracturing treatment , a possible lower limit of well spacing should be considered in design , to avoid a sharp reduction of effectiveness in treatment. Meanwhile , the optimized well space obtained from zipper fracturing is tighter than that from conventional well factory hydraulic fracturing. Compared with the conventional well factory hydraulic fracturing , the zipper fracturing has a better performance in improving the stimulated reservoir volume and decrease the effect of inter-well stress interference , which is favorable to enhance hydrocarbon production. Keywords: multiple well fracturing; well interference; hydraulic fracturing; numerical simulation; XFEM

1 Introduction

Unconventional oil and gas reservoirs can not achieve economic productivity without hydraulic fracturing because of their extremely low porosity and permeability[1-3]. Compared with conventional hydraulic fracturing technology, multi-stage hydraulic fracturing technology with horizontal wells, due to its high cost , also restricts the efficient development of unconventional oil and gas resources[4-6]. In order to reduce the cost of unconventional oil and gas development and improve production efficiency , simultaneous fracturing and zipper fracturing technology have developed rapidly[7-10]. Because this technology can implement hydraulic fracturing for many horizontal wells at the same time or in a short time , the cost of hydraulic fracturing is effectively reduced. However , due to the well interference effect in multi-well fracturing process , it will have a negative impact on hydraulic fracture propagation. At present , the research on fracture dynamic propagation mainly focuses on the influence of inter-fracture interference on fracture dynamic propagation[4, 5, 7, 11-18]. Therefore , it is of great significance to study the effect of inter-well interference on dynamic fracture propagation for well factory fracturing optimization.

The numerical simulation technology is an important technical means to study the dynamic propagation of hydraulic fracture in the process of hydraulic fracturing. At present , the research on fracture dynamic propagation mainly focus on the stress interaction between multiple fractures from single well , the inter-well interference on dynamic fracture propagation is not taken into consideration. Based on the displacement discontinuity method (DDM) , Wu and Olson[19, 20] established a coupled fracturing fluid flow model to study the effect of stress shadow on fracture propagation. Xie [21] established a hydraulic fracturing model of multi-fracture propagation , and the effect of stress shadow on fracture propagation was analyzed. Castonguay[22] analyzed the fracture dynamic propagation in isotropic formations under high stress difference. The simulation indicated that the stress shadow effect has a significant influence on the fracture dynamic propagation. In the process of simultaneous multi-fracture propagation , the middle fracture propagation is obviously restrained by the edge fracture , which made the extension length of the middle fracture obviously smaller than that of the edge fracture. Based on the discrete element method (DEM) , Hosseini[23] analyzed the interaction between hydraulic fractures and natural fractures under different approaching angles. The results show that approaching angles , the hydraulic fractures are more likely to be captured by natural fractures , and shear slip failure occurs on the natural fracture surface. In addition , the cement in natural fractures also has an important influence on natural fractures and hydraulic fractures[15]. Gao[24]analyzed the influence of young's modulus of reservoir rock , cement strength of natural fracture and pumping parameters on the formation of complex fracture network by establishing a hydraulic fracturing model. Based on the cohesive element method (CZM) , Zhu and Guo[25] analyzed the influence of stress shadow on the multiple fractures propagation , and the cluster space was optimized based on this method. Dahi[26] analyzed the relationship between natural fractures and hydraulic fractures. From the review of the previous researches , it can be concluded that the effect of inter-well interference on fracture dynamic propagation has not been analyzed at present. In this paper , the aim of our research is to investigate the simultaneous growth of fractures in multiple well case by a fracturing model coupled rock deformation and fracturing flow with fracture. And the effect of interwell interference on fracture dynamic propagation is analyzed. Meanwhile , the effect of well space on fracture dynamic propagation is also analyzed.

2 Methodology

The crack propagation is described by the Heaviside step function and the two-dimensional linear elastic progressive crack tip displacement field. Relative to the basic finite element mesh , the location of fracture discontinuities can be arbitrary , and fracture propagation simulation can be carried out without changing with the advance of fractures. For a displacement vector function u , the unit enrichment partition is approximately. 𝑁

4

u = ∑𝑖 = 1𝑁𝑖(𝑥)[𝑢𝑖 + 𝐻(𝑥)𝑎𝑖 + ∑𝛼 = 1𝐹𝛼(𝑥)𝑏𝛼𝑖]

(1)

where 𝑁𝑖(𝑥) is the normal nodal shape function; 𝑢𝑖 is the nodal displacement vector; 𝑎𝑖 is the product of the nodal enriched degree of freedom vector[27]; 𝐻(𝑥) is the Heaviside function; 𝑏𝛼𝑖is the product of the nodal enriched degree of freedom vector, 𝐹𝛼(𝑥) is the fracture-tip functions associated elastic asymptotic. The first term on the right-hand side is applicable to all the nodes in the model; the second term is valid for nodes whose shape function support is cut by the fracture

interior; and the third term is used only for nodes whose shape function support is cut by the fracture tip. The asymptotic fracture-tip functions 𝐹𝛼(𝑥) are given by 𝜃

𝜃

𝜃

𝜃

𝐹𝛼(𝑥) = [ 𝑟𝑠𝑖𝑛2, 𝑟𝑐𝑜𝑠2, 𝑟𝑠𝑖𝑛2𝑠𝑖𝑛 𝜃, 𝑟𝑐𝑜𝑠2𝑠𝑖𝑛𝜃]

(2)

Where (r,θ) is a polar coordinate system with its origin at the fracture tip and θ = 0 is tangent to the fracture at the tip. 𝜃

These functions span the asymptotic fracture-tip function of elasto-statics , and 𝑟𝑠𝑖𝑛2 takes into account the discontinuity across the fracture face. The use of asymptotic fracture-tip functions is not restricted to fracture modeling in an isotropic elastic material. Level set method is used to track mobile interfaces and describe interfaces in the domain. In this method , the interface is represented as a zero-order set of functions with one dimension higher than the dimension of the interface, and it is evolved by solving the hyperbolic conservation law. Considering that a domain Ω divided into two domains Ω𝐴 and Ω𝐵. The interface between these two domains is expressed in Γ𝑑. Generally speaking , the interface Γ𝑑 between open and closed interfaces is distinguishable (Fig.1). which is related to different materials in the domain. An open interface only partially cuts off the domain , that is , at least one end of the interface is in the domain ， such as in a cracked domain. The most common level set function is the signed distance function ，which is defined to represent the location of the interface. φ(𝑥) = ‖𝑥 ― 𝑥 ∗ ‖𝑠𝑖𝑔𝑛(𝐧Γ𝑑 ∙ (𝑥 ― 𝑥 ∗ )) (3) ∗ Where 𝑥 is the closest point projection of 𝑥 onto the discontinuity𝐧Γ𝑑 ，𝐧Γ𝑑 is the normal vector to the interface at point𝑥 ∗ . ‖‖denotes the Euclidean norm ， ‖𝑥 ― 𝑥 ∗ ‖ specifies the distance of point 𝑥 to the discontinuity Γ𝑑 . It can be seen from equation (4) that the symbols on both sides of the closed interface are di fferent. By this definition ，discontinuity can be implicitly expressed as the zero-level contour of the level set function.

{

> 0 ， 𝑖𝑓 𝑥 ∈ Ω𝐴 φ(𝑥) = = 0 ， 𝑖𝑓 𝑥 ∈ Γ𝑑 < 0 ， 𝑖𝑓 𝑥 ∈ Ω𝑏

(4)

Figure 2 Weak discontinuity and strong discontinuity problems: (a) Weak discontinuity interface; (b) Strong discontinuity crack interface described by level set function[28]

When the displacement field on one side of the crack is completely different from that on the other

side of the crack , the displacement discontinuity will occur. In this model , strong discontinuity kinematics can be defined based on the Heaviside function.

{

―1 𝑖𝑓𝜑(𝑥) < 0 H(x) = 1 𝑖𝑓𝜑(𝑥) > 0

(5)

2.1 Modeling fractures with the CZM

The fracture propagation criterion based on LEFM assumes that there is a fracture process zone near the fracture tip. Compared with the fracture size , the material behaves very little in the inelastic state. Fracture propagation occurs when the stress intensity factor exceeds the material strength. Nevertheless , the adequacy of this assumption is questioned because fracture propagation in quasibrittle and ductile materials will lead to important plastic deformation around the fracture tip due to shared failure. In addition , even for fragile materials , the fracture course area is able to be concentrated at a single point , and initial cracks need to exist to make LEFM applicable. The bond zone model is a simple model suitable for processing zones and materials that fail due to crack propagation and coalescence. The constitutive behavior of the viscous region is defined by the traction-separation relationship. The elastic behavior is represented by an elastic constitutive matrix , which links the normal stress and shear stress with the normal and shear separation of crack elements.

{} [

]{ }

𝑡𝑛 𝐾𝑛𝑛 0 0 𝛿𝑛 t = 𝑡𝑠 = 0 𝐾𝑠𝑠 0 𝛿𝑠 = 𝐾𝛿 𝑡𝑡 0 0 𝐾𝑡𝑡 𝛿𝑡

(6)

The nominal traction stress vector t consists of the following components: 𝑡𝑛 ，𝑡𝑠 ，and 𝑡𝑡 ，which represent the normal and the two shear tractions ，respectively. The corresponding separations are denoted by 𝛿𝑛 ，𝛿𝑠 and 𝛿𝑡. Damage modeling can be used to simulate the degradation and ultimate failure of strengthening elements. The failure mechanism consists of two parts: damage initiation criterion and damage evolution law. In this law (Fig. 2) , the element does not undergo damage under pure compression unless the traction reaches the cohesive strength 𝑇0 or the separation reaches the displacement of damage𝛿0. When the δ exceeds 𝛿0 , the traction will reduce linearly to 0 as the displacement increases , and the cohesive element is completely damaged.

Figure 2 Traction-Separation law for XFEM[29]

f=

〈𝑡𝑛〉 2

𝑡𝑠 2

𝑡𝑡 2

𝑡0𝑠

𝑡0𝑡

{ } +{ } +{ } 𝑡0𝑛

(7)

where 𝑡𝑛is the nominal stress , Pa; 𝑡𝑠is the first shear stress ，Pa; 𝑡𝑡 is the second shear stress ， Pa. 0 0 0 𝑡𝑛 is the nominal peak value stress , Pa; 𝑡𝑠 is the first shear peak stress , Pa and 𝑡𝑡 is the second shears tress , Pa. The Macaulay bracket < > in the normal direction signifies that pure compression does not initiate damage[28]. 2.2 Damage evaluation

The damage evolution law describes the rate at which the material stiffness is degraded once the corresponding initiation criterion is reached. In this model ， the damage law is described as follows. 𝑡𝑛=

{

(1 ― 𝐷)𝑡𝑛 ，𝑡𝑛 ≥ 0 𝑡𝑛,𝑡𝑛 < 0

(8)

𝑡𝑠 = (1 ― 𝐷)𝑡𝑠 (9) 𝑡𝑡 = (1 ― 𝐷)𝑡𝑡 (10) where D is the overall damage in the material , capturing the combined effects of all the active mechanisms. The initial value of D is 0. If the damage evolution is modeled , D monotonously evolves from 0 to 1 after the damage starts and further loads. The parameters𝑡𝑛 ，𝑡𝑠 and 𝑡𝑡 are the stress components predicted by the elastic traction separation behavior of the current strain without damage. 0 𝛿𝑓𝑚(𝛿𝑚𝑎𝑥 𝑚 ― 𝛿𝑚)

D = 𝛿𝑚𝑎𝑥(𝛿𝑓 𝑚

Where 𝛿𝑚 =

𝑚

(11)

― 𝛿0𝑚)

< 𝛿𝑛 > 2 + 𝛿2𝑠 + 𝛿2𝑡

2.3 Fluid flow within the fractures

In the model ， the leakage of fracturing fluid from fracture to matrix is considered. The Biot theory[30, 31] is employed in an updated Lagrangian framework. The governing equations of a deformable porous medium is presented for the solid–fluid mixture in the framework of Biot theory. And the fluid flow in the fracture and the proppant transport within fracture is controlled by UMAT subroutine. ∇ ∙ σ ― ρ𝑢 +𝜌𝑏 = 0 (12) 1

∇ ∙ [𝑘𝑓( ―∇𝑝 ― 𝜌𝑓𝑢 + 𝜌𝑓𝑏)] +𝛼∇ ∙ 𝑢 + 𝑄𝑝 = 0

(13)

In order to model the fluid flow through the discontinuity ，the continuity equation for the fluid flow within the fracture can be written according to Eq.(13). 1

∇ ∙ 𝑤 + 𝛼∇ ∙ 𝑢 + 𝑘𝑓𝑝 = 0

(14)

For a viscous fluid flow with Newtonian rheology ，the fracture permeability is estimated using the cubic law as 1 𝑤2

𝑘𝑓 = 𝛽12𝜇𝑓

(15)

And the Carter leak-off model is introduced to describe the normal flow of hydraulic fracturing. The proposed model calculates the leak-off of the fracturing fluid through the fracture faces into the

surrounding medium on the basis of the fully coupled poroelastic analysis. The normal flow rates at the top and bottom sides of the cohesive element are defined as

{𝑞𝑞 == 𝑐𝑐 (𝑝(𝑝 ―― 𝑝𝑝 )) 𝑡

𝑡

𝑖

𝑡

𝑏

𝑏

𝑖

𝑏

(16)

According to the fluid mass balance , a part of that injected fluid at a certain period fills the fracture and the rest will be lost to the rock matrix as shown in the following equation. ∂𝑑 ∂𝑡

+ ∇ ∙ 𝑞 + (𝑞𝑡 + 𝑞𝑏) = 𝑞𝑖𝑛𝑗𝛿(𝑥,𝑦)

(17)

Then the control equation of rock matrix involves coupling fluid flow and rock deformation as follows. 𝐸

(

𝑣

)

𝜎𝑖𝑗 ― 𝜎0𝑖𝑗 = 1 + 𝑣 𝜀𝑖𝑗 + 1 ― 2𝑣𝜀𝑘𝑘𝛿𝑖𝑗 ―𝛼(𝑝𝑤 ― 𝑝0𝑤)𝛿𝑖𝑗

(18)

3 Simulation and Results 3.1 Model Description

In order to understand the effect of well interference on fracture propagation in hydraulic fracturing , the hydraulic fracture propagation model with two horizontal wells is established which coupled the fracture surface deformation and leak off of fracture fluid within fracture. In this simulation , the well space between the two horizontal wells is 90m , the fracture space is 30m (Fig.3). The parameters of this model is shown as Table. 1. In the research , the fracture propagation dynamics under two different fracturing modes of conventional well factory hydraulic fracturing and zipper fracturing are compared. In conventional hydraulic fracturing , two fractures in well 2 are fractured first , then turned to well 1. The fracturing sequence is ①→②→③→④. In zipper fracturing mode , the fracturing sequence is ①→③→②→④. Due to the research area is only part of the whole formation , the boundary of the study area is fixed boundary.

Figure 3

Diagram of well factory fracturing model

Table .1 Input key parameters of the numerical model

Parameters

Values

Young’s Modulus (GPa) Poission’s ratio Permeability , (10-3μm2) Porosity , (%) Normal nominal stress , (MPa) 1st shear nominal stress , (MPa) 2st shear nominal stress , (MPa)

23.2 0.23 0.032 2.1 3.4 3.0 3.0 1

Strength of natural fracture , (MPa ∙ 𝑚2)

2.0

Maximum horizontal principle stress , (MPa) Minimum horizontal principle stress , (MPa) Vertical Stress ，(MPa) Fracturing fluid viscosity,(mPa.s) Injection Rate , (m3/min.m)

32.7 29.8 40.1 1.0 8.0

3.2 Results

Fig. 4 is the fracture propagation under conventional well factory hydraulic fracturing mode. In this simulation , the Fracture 1 and Fracture 2 of Well 2 are fractured firstly , then turned to Well 1. According to the simulation results , the Fracture 1 propagates in a straight line along the direction of maximum horizontal principal stress until the injection is completed , then the Fracture 2 is injected. In the process of Fracture 2 propagation , an obvious deflection occurred at the tip of Fracture 2. When the Fracture 3 of Well 1 is injected , the upper half of Fracture 3 can extend freely along a straight line , but propagation length of the lower half decreases significantly. When the Fracture 4 is injected , an obvious deflection occurred during the propagation , and the propagation length is also limited. After the whole simulation , the propagation length of the four fractures are 86.4m , 74.5m , 66.2m and 62.7m. It can be found that the propagation length of Fracture 3 and Fracture 4 are shorter than that of Fracture 1 and Fracture 2 , and an obvious deflection can be found at the tip of the fractures. The main reason accounts for this phenomenon is that the propagation of Fracture 1 and Fracture 2 alters the original distribution of the minimum horizontal principal stress , thus making the following fractures deflected when propagates. Meantime , the Fracture 4 also affected by the induced stress field from Fracture 3 in addition to Fracture 1 and Fracture 2 , thus making the propagation length of Fracture 4 reduced furtherly. It can be seen that the total length of the two hydraulic fractures formed in Well 2 is obviously longer than that formed in Well 1. The difference of the total length of the fractures is 30.2m , and the total length of the fractures decreases by 18.98%. The main reason for this phenomenon is that after the initiation and expansion of the two fractures in Well 2 , the induced stress field greatly increases the extension resistance of the lower half branch of Fracture 3 and Fracture 4 , which makes the extension length of the lower half branch of Fracture 3 and Fracture 4 significantly less than that of the upper half branch. In addition to the induced stress field from Fracture 1 and Fracture 2 , Fracture 4 is also affected by the induced stress field formed during the propagation of Fracture 3 , which further reduces the extension length

of Fracture 4. Based on the extension length of fracture 1 , the extension length of Fracture 2 is reduced by 11.93% , the Fracture 3 reduced by 21.75% and Fracture 4 reduced by 25.89%.

Figure 4 Fracture dynamic propagation in conventional well factory hydraulic fracturing mode

Fig. 5 is the fracture propagation under zipper hydraulic fracturing mode. The simulation indicates that the Fracture 1 propagates along the straight line because it is not affected by other fractures , then Fracture 3 is injected. During the propagation of Fracture 3 , the upper half propagates along a straight line without interference from other fractures , but the lower half of Fracture 3 has slight deflection due to the induced stress field formed by Fracture 1. Then the Fracture 2 is injected. During the propagation , the Fracture 2 propagates along the straight line , only a slight deflection occurred at the tip of the fracture. Finally the Fracture 4 is injected. It can be found that asymmetric propagation still exists in Fracture 4 , but the propagation length of lower branch which close to Well 2 increases obviously. After the whole simulation , the extension length of the four fractures are 84.6m , 77.5m , 72.5m and 70.4m , respectively. In addition , the extension length difference of Fracture 2 , Fracture 3 and Fracture 4 is relatively small , but there is still a significant gap when compared with Fracture 1. The total propagation length difference between Well 1 and Well 2 is 19.2m. Taking the Fracture 1 as the baseline , the extension length of Fracture 2 is reduced by 8.39% , that of Fracture 3 by 14.30% , and that of Fracture 4 by 16.78%.

Figure 5 Table 2

Fracture dynamic propagation in zipper hydraulic fracturing mode Fracture dynamic propagation under different fracturing modes

conventional well

zipper hydraulic

factory

fracturing

Fracture 1

84.6

84.6

0.0

Fracture 2

74.5

79.5

6.71%

Fracture 3

66.2

77.5

17.07%

Fracture 4

62.7

72.4

15.47%

Total length

288.0

314.0

9.02%

Fracture

Amplification

Table. 2 is the comparison of fracture propagation length under conventional well factory fracturing and zipper fracturing. It can be seen that the total length of hydraulic fracture extension is 288.0m under conventional well factory hydraulic fracturing mode , while under zipper fracturing mode , the total length of hydraulic fracture extension is 325.0m and the total length of fracture extension is increased by 9.02%. By comparing the length of Fracture 2 , 3 and 4 , it is found that the extension length of Fracture 2 increased by 6.71% , Fracture 3 increased by 17.02% and Fracture 4 increased by 15.47%. This shows that that zipper fracturing technology can effectively alleviate the interference effect of adjacent wells fracture.

4 Discussion

In order to further discover the influence of well space on fracture dynamic propagation under different fracturing modes , the well space with 100 m , 110 m , 120 m , 130m , 140 m and 150 m are analyzed. Firstly , the different well space under conventional well factory hydraulic fracturing

mode are discussed. From the simulation results (Fig. 6) , it can be seen that the propagation length of Fracture 1 and Fracture 2 are not changed , but the Fracture 3 and Fracture 4 increase greatly with the increase of well space. When the well space are 110m and 120m , the total length of four fractures are 311.6m and 324.1m , in which the Fracture 3 increases by 15.11% , and Fracture 4 increases by 18.34% when the well space is 110m. When the well space is 120m , the Fracture 3 increases by 30.14% , and the Fracture 4 increases by 34.17%. When the well space is 130m , the total length increases by 14.19% compared with the well space is 100m , in which the Fracture 3 increases by 33.41% , while the Fracture 4 increases by 36.17%. When the well space is over 140 m , the fracture length becomes stable , and the total length changes slightly with the increase of well space. From the simulation , it can be concluded that the fracture length of Fracture 1 and Fracture 2 are kept stable , while the Fracture 1 and Fracture 2 experience great increase. The main reason is that fracture space is the main reason affecting the fracture propagation length of Fracture 1 and Fracture 2 , while well space is the main reason affecting fracture propagation length of Fracture 3 and Fracture 4. With the increase of well space , the effect of well interference is reduced continuously , so the length of Fracture 3 and Fracture 4 increase with the increase of well space.

Figure 6

Fracture propagation in conventional well factory hydraulic fracturing with different well space

To further quantify the variation of fracture extension length under different well space. Taking the well space of 100m as the baseline , the variation of fractures under different well space is quantified. From Fig. 7 and Table. 3 , it can be seen that with the increase of well space , the total length of fracture extension continues to increase , but the increase rate gradually decreases. When the well space reaches 140 m , the total length of fracture extension increases by only 1.36% compared with 130m. This indicates that when the well space exceeds 140 m , the effect of well interference in adjacent wells is greatly reduced , and the hydraulic fracture between wells can propagate freely. Therefore , the reasonable well space for conventional well factory hydraulic fracturing should be between 140 m and 150 m. Table 3

Well Space/m

Variation of fracture extension length with different well space

Total length /m

Increase in total length

Difference of increase

of fracture extension/%

amplitude/%

100

288.0

0.00

-

110

311.6

7.99

7.99

120

324.1

12.50

4.01

130

331.2

14.93

2.19

140

335.7

16.32

1.36

150

337.2

17.01

0.48

20

Total length of all Fractures Increase of fracture Length

340

15

320

10

300

5

280 100

110

120

130

140

Increase of fracture Length/%

Total length of all Fractures/m

360

0 150

Well Space/m

Figure 7

Variation of fracture extension length in conventional well factory hydraulic fracturing with different well space

Fig. 8 is the fracture propagation in zipper hydraulic fracturing with different well space. From the dynamic simulation , it can be found that Fracture 1 extends along a straight line , but the lower branch of Fracture 3 is seriously limited when the well space is less than 130m , only the upper branch can propagate freely. Meantime , an obvious deflection can be found at the tip of Fracture 3. Then Fracture 2 is injected. Affected by the induced stress field generated by Fracture 1 and Fracture 3 , the upper branch of Fracture 2 is limited , and an obvious deflection can be found at the tip of Fracture 2. The fracture length of Fracture 4 is limited most seriously when the well space is less than 130m. With the increase of well space , the remaining three fractures have an obvious increase in the fracture length except for Fracture 1. When the well space is 130m , the inhibited degree of Fracture 2 and Fracture 4 in the process of propagation is obviously reduced and the free expansion is basically realized. When the well space is greater than 130m , the simulation results show that the influence of well space on fracture dynamic propagation is greatly reduced.

Figure 8 Table 4

Fracture propagation in zipper hydraulic fracturing with different well space

Variation of fracture extension length with different well space in zipper fracturing model

Well Space/m

Total length /m

Increase in total length of

Difference of increase

fracture extension/%

amplitude/%

100

314.0

0.00

-

110

335.8

6.94

6.94

120

347.2

10.57

3.39

130

352.8

12.36

1.61

140

354.9

13.03

0.60

150

355.4

13.18

0.14

In order to further discover the influence of well space on fracture dynamic propagation under different fracturing modes , the well space with 100 m , 110 m , 120 m , 130m ,140 m and 150 m are analyzed (Table.4). Taking the well space with 100m as the baseline , and the results are shown as Fig. 6 and Table. 5. From the Fig. 9 , it can be seen that with the increase of well space , the total length of hydraulic fracture increases gradually , but the increase of fracture slows down gradually. When the well space is 110 m , the total length of fracture propagation increases by 6.94% compared with 100 m , when the well space increases to 120 m , the total length of fracture increases by 10.57%. When the well space exceeds 130m , the total length of fracture hardly increases. When the well space is 140 m , the total length of fracture extension increases by 13.03%. Compared with the fracture propagation with well space is 130m , the fracture extension length increases by only 2.1 m , and the relative increase is only 0.60%. This shows that when the well space exceeds 130m , the influence of well space on the fracture dynamic propagation is greatly reduced , and the

hydraulic fracture can expand freely. In this case , the optimum well spacing is between 130m and 140 m.

360

20

350

Increase of fracture Length

16

340 12 330 8 320 4

310

300 100

110

120

130

140

Increase of fracture Length/%

Total length of all Fractures/m

Total length of all Fractures

0 150

Well Space/m

Figure 9 Variation of fracture extension length with different well space in zipper hydraulic fracturing model

Compared with conventional well factory fracturing and zipper fracturing , it is found that under the same well space , the fracture propagation length obtained by zipper fracturing is significantly longer than that obtained by conventional well factory hydraulic fracturing. According to the Table. 5 , it can be seen that the fracture propagation length increases obviously when zipper fracturing model is adopted , but the increase amplitude decreases gradually with the increase of well space. When the well space is 100 m , the fracture extension length of zipper fracturing is 9.02% longer than that of conventional well factory fracturing. When the well space is increased to 120 m , the difference of fracture length between zipper fracturing and conventional well factory fracturing decreases to 7.13%. With the increase of well space , when the well spacing is 130m , the difference of fracture length between them is 6.52% , and when the well space is 150 m , the difference of fracture length continues to decrease to 5.40%. This indicates that with the increase of well space , the influence of well interference on fracture dynamic propagation decreases gradually. When well spacing is reduced to 140 m , the effect of fracturing mode on fracture dynamic propagation is minimized. At the same time , the optimum well spacing obtained by zipper fracturing is smaller than that obtained by conventional well factory mode , which indicates that zipper fracturing mode can better reduce the influence of well interference on fracture dynamic propagation. Table 5

Comparison of fracture propagation between conventional well factory fracturing mode and zipper fracturing mode

Conventional well Well Space/m

Zipper fracturing /m

Increase amplitude/%

factory fracturing /m 100

288.0

314.0

9.02

110

311.6

335.8

7.77

120

324.1

347.2

7.13

130

331.2

352.8

6.52

140

335.7

354.9

5.72

150

337.2

355.4

5.40

5 Conclusion

Based on the established numerical simulation model of multi-hydraulic fracture propagation , the multi-hydraulic fracture propagation under well factory fracturing is simulated and studied. During the research , the effects of fracturing mode and well space on the dynamic propagation of hydraulic fractures are simulated and analyzed , and the following conclusions are drawn. (1) Well interference is the main factor affecting the fracture dynamic propagation under well factory fracturing. Comparing with the conventional well factory fracturing and zipper fracturing , it is found that zipper fracturing can significantly reduce the well interference on fracture dynamic propagation , achieving uniform extension of hydraulic fractures and increase the stimulated reservoir volume by hydraulic fracturing. (2) With the increase of well space , the influence of well interference on the fracture dynamic propagation is gradually weakened. Compared with conventional well factory fracturing and zipper fracturing , the optimal well space obtained by zipper fracturing is smaller , which can effectively increase the stimulated reservoir volume by hydraulic fracturing.

Notation list

𝑑 Fracture opening E

Elasticity modulus

f

Surface force vector

Gc Cohesive energy g

Gravitational acceleration vector

J Formation volume change rate , dimensionless Kn Initial cohesive stiffness kt Tangential permeability k nw

Permeability matrix Void ratio

∇p Pressure gradient pi Midface pressure

pt Pore pressures on the top element surface pb Pore pressures on the bottom element surface pw Pore pressure 𝑝0𝑤 initial pore pressure q Mass flow rate along the fracture 𝑞𝑖𝑛𝑖 Injection rate at a known location denoted by the Dirac delta function, δ(x,y) 𝑞𝑡 Normal flow rates into the top surfaces 𝑞𝑏 Normal flow rates into the bottom surfaces 𝑐𝑡 Fluid leak-off coefficients at the top surfaces 𝑐𝑏 Fluid leak-off coefficients at the bottom surfaces 𝑇𝑚𝑎𝑥 Maximum tensile strength 𝑡𝑛,𝑡𝑠,𝑡𝑡 Stress components predicted by the elastic traction-separation behavior for the strains without damage 𝑣𝑤 Fluid velocity X

Spatial vector

𝜎 Effective stress tensor 𝜎𝑖𝑗 Components of the total stress 𝜎0𝑖𝑗 Components of the initial stress 𝜀𝑖𝑗 Components of the strain tensor 𝛿𝑖𝑗 Kronecker's delta function 𝛿𝑚0 Displacements when the traction reaches Tmax 𝛿𝑚𝑓 Displacements when the element is completely damaged 𝛿𝑣 Virtual velocity vector 𝛿𝜀 Virtual strain rate tensor γ Body force matrix α

Biot's constant

υ Poisson's ratio μ Fluid viscosity 𝜌𝑤 Fracturing fluid density 𝜎𝐻 The maximum horizontal principle stress 𝜎ℎ The minimum horizontal principle stress 𝐾𝑠 The stress difference coefficient 𝐾𝑐𝑠 The composite stress factor

Acknowledgement The authors express their appreciation to the National Science and Technology Major Project (No. 2009ZX05009), a project of the China Natural Science Foundation (50774091).

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Numerical investigation on the effect of well interference on hydraulic fracture propagation in shale formation Highlight: (1) A hydraulic fracturing model coupled rock deformation and fluid flow is established; (2) The effect of well interference on fracture dynamic propagation is investigated. (3) The effect of fracturing sequence on fracture dynamic propagation is investigated.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: