# An Analytical Model for Fracture Initiation from a Particular Radial Borehole in Hydraulic Fracturing Guided by Multiradial Boreholes

Authors: Yuxin Chen; Yunhong Ding; Chong Liang; Dawei Zhu; Yu Bai; Chunmei Zou
Journal: Geofluids
 (2021)

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## Abstract

Radial drilling-fracturing, the combination of the hydraulic fracturing and radial borehole, is a technology that can guide the hydraulic fractures to directionally propagate and efficiently develop low permeability reservoir. In this paper, an analytical model of two radial boreholes (a basic research unit) is established to predict fracture initiation pressure (FIP) from one particular radial borehole and the interference between radial boreholes when the hydraulic fracturing is guided by multi-radial boreholes. The model is based on the stress superposition principle and the maximum tensile stress criterion. The effects of in situ stress, wellbore pressure, and fracturing fluid percolation are considered. Then, sensitivity analysis is performed by examining the impact of the intersection angle between radial boreholes, the depth difference between radial boreholes, the radius of radial boreholes, Biot coefficient, and the number of radial boreholes. The results show that FIP declines with the increase of radial boreholes number and the decrease of intersection angle and depth difference between radial boreholes. Meanwhile, the increase of radial borehole number and the reduction of intersection angle and depth difference strengthen the interference between radial boreholes, which conduce to the formation of the fracture network connecting radial boreholes. Besides, FIP declines with the increase of radial borehole radius and the decrease of Biot coefficient. Large radius and low Biot coefficient can enlarge the influence range of additional stress field produced by radial boreholes, enhance the mutual interference between radial boreholes, and guide fracture growth between radial boreholes. In hydraulic fracturing design, in order to reduce FIP and strengthen the interference between radial boreholes, the optimization design can be carried out by lowering intersection angle, increasing radius and number of boreholes, and reducing the depth difference between boreholes when the conditions permit. The research clarifies the interference between radial boreholes and provides the theoretical basis for optimizing radial boreholes layout in hydraulic fracturing guided by multi-radial boreholes.


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## Body

## 1. Introduction

Radial drilling is a technology that is applied to accelerate the recovery of hydrocarbon reserves and reach reserves that are not economically exploitable with conventional completion techniques [1–3]. This technology, utilizing high-pressure water jet or mechanical drill bit, can forms several radial laterals, with a length between 10 and 100 m and borehole diameter between 25 and 50 mm [4, 5]. These radial laterals can penetrate the near well damage zone, augment wellbore-reservoir contact area, and connect the target reservoir area with a low producing degree.However, due to the small diameter, radial drilling applied alone often has an unsatisfactory stimulation result in the reservoir with low permeability or undeveloped natural fracture. Thus, radial drilling-fracturing, the combination of the hydraulic fracturing and radial borehole, is proposed [6]. This technology integrates the merits both of hydraulic fracturing and radial drilling, guiding the hydraulic fracture toward the target area with radial borehole and, meanwhile, enhancing the wellbore-reservoir contact area significantly [7–10].Up to now, many types of research have been conducted on the radial borehole layout patterns for different goals. As shown in Figure1, Guo et al. [11] presented a layout method that radial borehole rows are vertically distributed along the main wellbore and extend toward the target development zone. This method can control hydraulic fractures to propagate along with the orientation of radial borehole row and communicate the target zone effectively. Besides, tree-type hydraulic fracturing technology [12, 13] adopts another classic layout pattern. Radial boreholes, protruding toward different orientations, have a radial tree-type distribution. Under the action of ground stress and water pressure, hydraulic fractures, initiating in individual radial boreholes, can extend to the adjacent boreholes and form a widespread hydraulic fracture network which can holistically enlarge drainage area and recovery.Figure 1
The layout pattern of radial boreholes. (a) Hydraulic fracturing guided by vertical multi-radial boreholes. (b) Tree-type hydraulic fracturing.
(a)(b)Nevertheless, no matter which layout is taken, accurate prediction of fracture initiation pressure (FIP), a keystone parameter in the design and evaluation of hydraulic fracturing, is cardinal for a successful operation. Many pieces of researches on this issue have been conducted through experimental, analytical, and numerical methods. Ketterij et al. [14] experimentally demonstrated the influence of perforation azimuth on fracture initiation and propagation in case of a deviated wellbore. Hossain et al. [15] presented an analytical model to predict the fracture initiation pressure with and without perforations. Lhomme et al. [16] experimentally investigated the fracture initiation from an open-hole section. They found that fracture initiation is influenced by rock microstructure directly. Fallahzadeh et al. [17] developed a computer model of fracture initiation pressure for deviated-cased perforated wellbore and studied the near-wellbore stress distribution. Alekseenko et al. [18] established a boundary-element numerical model to stimulate fracture initiation from a perforated wellbore. Based on finite element method, Gong et al. [19] researched various parameters in radial drilling-fracturing and found that azimuth is most influenced on fracture initiation pressure, followed by length and diameter of the radial borehole. Liu et al. [20] established a generic model to predict fracture initiation from the radial borehole and analyzed the effect of preexisting weakness plane. Corresponding with three different in situ stress statues, they concluded three types of fracture initiation from radial borehole relatively, and the in situ stress regime and radial borehole orientation are most significant parameters influencing FIP for intact rock.Currently, most fracture initiation analytical models of radial drilling-fracturing are based on the case of a single radial borehole and fail to consider the interference between multi-radial boreholes. Guo et al. established a theoretical mechanical model of hydraulic fracture guided by vertical multi-radial boreholes. However, they had focused on the calculation of stress distribution around an open main wellbore and failed to consider cased main wellbore and the fracture initiation from radial borehole. Besides, in fact that radial boreholes considered in their model have the same azimuth, their model cannot meet the demand of fracture initiation study for hydraulic fracturing that is induced by multi-radial boreholes with different azimuths. Thus, an analytical model for fracture initiation from a particular radial borehole, considering the interference between multi-radial boreholes with different azimuths, is still blank in petroleum engineering.In this paper, we establish a theoretical mechanical model for fracture initiation from one particular radial borehole under the action of two radial boreholes with different azimuths (basic research unit) for the first time. Then, based on this model, a serial of sensitivity analysis is taken by investigating the effects of different parameters including the intersection angle between radial boreholes, the depth difference, radius of the radial borehole, Biot coefficient, and the number of radial boreholes.


## 2. Model Establishment

In this part, considering the stress caused by wellbore pressure, fracturing fluid percolation effect, and in situ stress, we develop a model of two radial boreholes with arbitrary azimuths. Radial boreholes are considered as small open holes intersecting with the vertical main wellbore and yield independent additional stress fields. Next, the stress distribution along the wall of one particular radial borehole is available by coordinate transformation and stress superposition. Then, fracture initiation pressure of particular radial borehole is predicted based on the maximum tensile stress criterion.


### 2.1. Model Assumptions and Coordinate System

The following assumptions are made during the derivation of the analytical model:(1)
Formation rock is a homogeneous, isotropic, and linear elastic porous medium(2)
Microcracks inside rock have no influence(3)
The effect caused by the physicochemical action between fracturing fluid and formation rock is neglected(4)
Radial boreholes do not influence in situ stressCoordinate systems established during derivation and the layout of radial boreholes are as shown in Figure2. The radial borehole 0 is taken as the research object, and the model derivation is aimed at establishing an analytical model which can calculate the stress distribution along radial borehole 0 (RB0) under the effect of radial borehole 1 (RB1). In the process of model derivation, tensile stress is negative, and pressure stress is positive. A rectangular coordinate system (x, y, z) is established in which the origin is located at the center of the wellbore cross-section, and x axis, y axis, and z axis are aligned with the maximum horizontal stress, the minimum horizontal stress, and the main wellbore axis, relatively. Borehole coordinate systems are established. z0 axis and z1 axis are along the axis of two radial boreholes separately, while y0 axis and y1 axis are along the axis of the main wellbore. Thus, rectangular coordinate systems of the three boreholes, main wellbore and two radial boreholes, are (x, y, z), (x0, y0, z0), and (x1, y1, z1), and their cylindrical coordinate systems are (r, θ, z), (r0, θ0, z0), and (r1, θ1, z1), respectively. L is the depth difference between two radial boreholes. β0 and β1, the angle between the maximum horizontal stress direction and the axis of the radial borehole, are the azimuth angles of radial borehole 0 and radial borehole 1, respectively. The intersection angle of radial boreholes, the angle between the projections of radial boreholes axes on the horizontal plane, is denoted with φ, which equals β1-β0. r0 is the distance between research point and the axis of radial borehole 0. Thus, the coordinate of research point where the stress is calculated is (r0, θ0, z0) in (r0, θ0, z0) coordinate systems.Figure 2
Schematic diagram of hydraulic fracturing guided by multi-radial boreholes.


### 2.2. Model Assumptions and Coordinate System

Due to the assumption that radial boreholes do not influence in situ stress, the in situ stress in rectangular coordinate system (x, y, z) is:
(1)σxx000σyy000σzz=σH000σh000σv,where σH is the maximum horizontal in situ stress, MPa; σh is the minimum horizontal in situ stress, MPa; σv is the vertical in situ stress, MPa; and σxx, σyy, and σzz are the normal stress components in the coordinate system (x, y, z), MPa.Combining with the existing formula, the stress distribution around the main wellbore caused by in situ stress can be attained in the coordinate system (r, θ, z):
(2)σrIn‐Situ=12σH+σh1−Re2r2+12σH−σh1+3Re4r4−4Re2r2cos2θ,σθIn‐Situ=12σH+σh1+Re2r2−12σH−σh1+3Re4r4cos2θ,σzIn‐Situ=σν−2vrcos2θσH−σhRe2r2,τrθIn‐Situ=12σH−σh1−3Re4r4+2Re2r2sin2θ,τθzIn‐Situ=0,τrzIn‐Situ=0,where σr, σθ, and σz are the normal stress components in the cylindrical coordinate system (r, θ, z), MPa; τrθ, τθz, and τrz are the shear stress components in the cylindrical coordinate system (r, θ, z), MPa; vr is the Poisson’s ratio of rock; Re is the exterior radius of the main wellbore, m; and r is the distance between the axis of the main wellbore and any research point in formation, m.In order to attain the stress distribution along the radial borehole 0, the stress components caused by in situ stress need a serial of transformations from the coordinate system (r, θ, z) into (x, y, z), (x0, y0, z0), and (r0, θ0, z0), successively.First owing to the overlapping origin of the two coordinate systems, the stress components caused by in situ stress can be converted from (r, θ, z) coordinate system into (x, y, z) coordinate system by the following formulas.
(3)σxτxyτzxτxyσyτyzτzxτyzσz=cosθ−sinθ0sinθcosθ0001σrτθrτzrτrθσθτzθτrzτθzσzcosθsinθ0−sinθcosθ0001,(4)σxIn‐Situ=x2x2+y212σH+σh1−Re2x2+y2+12σH−σh1+3Re4x2+y22−4Re2x2+y2x2−y2x2+y2+y2x2+y212σH+σh1+Re2x2+y2−12σH−σh1+3Re4x2+y22x2−y2x2+y2−2x2y2x2+y22σH−σh1−3Re4x2+y22+2Re2x2+y2=F,σyIn‐Situ=y2x2+y212σH+σh1−Re2x2+y2+12σH−σh1+3Re4x2+y22−4Re2x2+y2x2−y2x2+y2+x2x2+y212σH+σh1+Re2x2+y2−12σH−σh1+3Re4x2+y22x2−y2x2+y2+2x2y2x2+y22σH−σh1−3Re4x2+y22+2Re2x2+y2=K,σzIn‐Situ=σν−2νRe2σH−σhx2−y2x2+y22=Q,τxyIn‐Situ=2xyx2+y2x2−y2x2+y2σH−σh−xyx2+y2σH+σhRe2x2+y2=T,τyzIn‐Situ=0,τzxIn‐Situ=0.The expressions ofσxIn‐Situ,σyIP‐Situ,σzIP‐Situ,andτxyIP‐Situ are denoted with F, K, Q, and T, respectively, for the facility of derivation. Second, the coordinate transformation expression from (x, y, z) coordinate system into (x0, y0, z0) coordinate system can be written as follows.
(5)σx0τx0y0τz0x0τx0y0σy0τy0z0τz0x0τy0z0σz0=−sinβ0cosβ00001cosβ0sinβ00σxτxyτzxτxyσyτyzτzxτyzσz−sinβ00cosβ0cosβ00sinβ0010.Then, the stress component transformation from (x0, y0, z0) coordinate system into (r0, θ0, z0) coordinate system can be conducted as the formula.
(6)σr0τθ0r0τz0r0τr0θ0σθ0τz0θ0τr0z0τθ0z0σz0=cosθ0sinθ00−sinθ0cosθ00001σx0τx0y0τz0x0τx0y0σy0τy0z0τz0x0τy0z0σz0cosθ0−sinθ00sinθ0cosθ00001.Based on the above formulas, the final expression of stress components caused by in situ stress can be obtained in (r0, θ0, z0) coordinate system.
(7)σr0In‐Situ=cos2θ0sin2β0F+cos2β0K−2sinβ0cosβ0T+Qsin2θ0,σθ0In‐Situ=sin2θ0sin2β0F+cos2β0K−2sinβ0cosβ0T+Qcos2θ0,σz0In‐Situ=cos2β0F+sin2β0K+2sinβ0cosβ0T,τr0θ0In‐Situ=Q−sin2β0F−cos2β0K+2sinβ0cosβ0Tsinθ0cosθ0,τθ0z0In‐Situ=−sinθ0cos2β0−sin2β0T+sinβ0cosβ0K−F,τz0r0In‐Situ=cosθ0cos2β0−sin2β0T+sinβ0cosβ0K−F.The above some components are shown in the formula.(8)x=cosβ0z0−r0cosθ0tanβ0,y=r0cosθ0cosβ0+sinβ0z0−r0cosθ0tanβ0,z=r0sinθ0.


### 2.3. Stress Caused by Wellbore Pressure

When fracturing fluid is pumped into wellbore, the internal pressure,Pw, is acting radially on the wall of the wellbore and generates stress field around the wellbore. Unlike radial boreholes, the main wellbore is cased, and the internal pressure, Pw, only can be delivered partially to the rock due to the high rigidity of the casing. Thus, the main wellbore and radial boreholes need to be investigated separately.


#### 2.3.1. Stress Caused by Wellbore Pressure

Considering Young’s modulus of the casing is much higher than rock, and Young’s modulus of cement sheath and rock are in the same magnitude, the influence of cement sheath can be ignored, and the effect of casing should be considered. The stress generated by the internal pressure of the main wellbore can be written as follows in the cylindrical coordinate system (r, θ, z), considering the effect of casing [21].
(9)σrIP‐MB=Re2r2Pi,σθIP‐MB=−Re2r2Pi.Of which:(10)Pi=1+vcEc21−vcRe2−Ri2Ri2Pw/1+vrEr+1+vcEcRi2+1−2vcRe2Re2−Ri2,where vc is Poisson’s ratio of the casing; vr is Poisson’s ratio of rock; Er is Young’s modulus of rock, GPa; Ec is Young’s modulus of the casing, GPa; and Ri is the internal radius of the casing, m.Based on the coordinate transformation matrix (3), the stress expressions caused by main wellbore internal pressure can be available in (x, y, z) coordinate system.
(11)σxIP‐MB=Re2Pix2−y2x2+y22=M,σyIP‐MB=−Re2Pix2−y2x2+y22=−M,σzIP‐MB=0,τxyIP‐MB=2Re2Pixyx2+y22=N,τyzIP‐MB=0,τzxIP‐MB=0.The expressions ofσxIP‐MB,σyIP‐MB,andσxyIP‐MB are denoted with M, -M, and N separately for facilitating model derivation. Similar to in situ stress, the terms of the stress components caused by main wellbore internal pressure in the (r0, θ0, z0) coordinate system can be obtained through a serial of conversion according to formulas (5) and (6).
(12)σr0IP‐MB=cos2θ0sin2β0−cos2β0M−2sinβ0cosβ0N,σθ0IP‐MB=sin2θ0sin2β0−cos2β0M−2sinβ0cosβ0N,σz0IP‐MB=cos2β0−sin2β0M+2sinβ0cosβ0N,τr0θ0IP‐MB=−sinθ0cosθ0sin2β0−cos2β0M−2sinβ0cosβ0N,τθ0z0IP‐MB=−sinθ0cos2β0−sin2β0N−2sinβ0cosβ0M,τz0r0IP‐MB=cosθ0cos2β0−sin2β0N−2sinβ0cosβ0M.The above some components are shown in the formula.(13)x=cosβ0z0−r0cosθ0tanβ0,y=r0cosθ0cosβ0+sinβ0z0−r0cosθ0tanβ0,z=r0sinθ0.


#### 2.3.2. Stress Caused by Internal Pressure of Radial Borehole 0

Compared with the main borehole, the influence of casing is not considered in radial boreholes because radial boreholes are open boreholes. Thus, the stress components in (r0, θ0, z0) coordinate system caused by the internal pressure of radial borehole 0 can be attained directly.
(14)σr0IP‐RB0=R02r02Pw,σθ0IP‐RB0=−R02r02Pw,where R0 is the radius of radial borehole 0, m, and Pw is the internal pressure, MPa.


#### 2.3.3. Stress Caused by Internal Pressure of Radial Borehole 1

Similar to radial borehole 0, the stress caused by the internal pressure of radial borehole 1 can be obtained in the coordinate system (r1, θ1, z1).
(15)σr1IP‐RB1=R12r12Pw,σθ1IP‐RB1=−R12r12Pw,where R1 is the radius of radial borehole 1, m, and r1 is the distance between the axis of radial borehole 1 and any research point in formation, m.Next, the stress components are converted from (r1, θ1, z1) coordinate system into (x1, y1, z1) coordinate system with coordinate transformation matrix (3).
(16)σx1IP‐RB1=R12Pwx12−y12x12+y122=M1,σy1IP‐RB1=−R12Pwx12−y12x12+y122=−M1,σz1IP‐RB1=0,τx1y1IP‐RB1=2R12Pwx1y1x12+y122=N1,τy1z1IP‐RB1=0,τz1x1IP‐RB1=0.The expressions ofσx1IP‐RB1,σy1IP‐RB1,andσx1y1IP‐RB1 are written as M1, −M1, and N1, respectively. Continue to convert the stress components from (x1, y1, z1) coordinate system into (x0, y0, z0) coordinate system by the following coordinate transformation matrix (17).
(17)σx0τx0y0τz0x0τx0y0σy0τy0z0τz0x0τy0z0σz0=cosφ0sinφ010−sinφ0cosφσx1τx1y1τz1x1τx1y1σy1τy1z1τz1x1τy1z1σz1cosφ0−sinφ010sinφ0cosφ.Then, combining the expressions of stress components in (x0, y0, z0) coordinate system and formula (6), the final expressions of stress components caused by the internal pressure of radial borehole 1 can be achieved in (r0, θ0, z0) coordinate system.
(18)σr0IP‐RB1=cos2θ0cos2φ−sin2θ0M1+2sinθ0cosθ0cosφN1,σθ0IP‐RB1=sin2θ0cos2φ−cos2θ0M1−2sinθ0cosθ0cosφN1,σz0IP‐RB1=sin2φM1,τr0θ0IP‐RB1=−1+cos2φsinθ0cosθ0M1+cos2θ0−sin2θ0cosφN1,τθ0z0IP‐RB1=sinφcosφsinθ0M1−sinφcosθ0N1,τz0r0IP‐RB1=−sinφcosφcosθ0M1−sinφsinθ0N1.The above some components are shown in the formula.(19)x1=cosφr0cosθ0−z0tanφ,y1=r0sinθ0−L,z1=z0cosφ+sinφr0cosθ0−z0tanφ.


### 2.4. Stress Caused by Fracturing Fluid Percolation

The fracture fluid will permeate into formation and increase the pore fluid pressure around borehole under the high wellbore pressure during the hydraulic fracture operation. The fracturing fluid percolation of radial boreholes is taken into account, and the fracturing fluid percolation of the cased main wellbore can be neglected.


#### 2.4.1. Stress Caused by Fracturing Fluid Percolation of Radial Borehole 0

Assuming that fluid flow in formation is radial flow and follows Darcy’s law, the stress caused by fracturing fluid percolation of radial borehole 0 can be provided in (r0, θ0, z0) coordinate system according to former research [22].
(20)σr0FP‐RB0=A−B0r02,σθ0FP‐RB0=A+B0r02,σz0FP‐RB0=C.Of which:(21)A=δPw−Ppε1−2vr21−vr−ϕ,B0=δR02Pw−Ppε1−2vr21−vr,C=δε1−2vr1−vr−ϕPw−Pp,wherePp is the initial pore fluid pressure, MPa; δ is the permeability coefficient; ϕ is the porosity of formation; εis the Biot coefficient, ε=1−Cr/Cb; and Cr and Cb are the compression rates of the rock skeleton and volume, respectively.


#### 2.4.2. Stress Caused by Fracturing Fluid Percolation of Radial Borehole 1

The stress components yielded by fracturing fluid percolation of radial borehole 1 and expression of stress components can be obtained directly in the cylindrical coordinate system (r1, θ1, z1).
(22)σr1FP‐RB1=A−B1r12,σθ1FP‐RB1=A+B1r12,σz1FP‐RB1=C.Of which:(23)B1=δR12Pw−Ppε1−2vr21−vr.The stress components can be converted into (x1, y1, z1) coordinate system by coordinate transformation matrix (3), and the expressions of σx1FP‐RB1,σy1FP‐RB1,andσx1y1FP‐RB1 are denoted as A−G1, A+G1, and −H1, respectively, for the facility of the following derivation.
(24)σx1FP‐RB1=A−B1x12−y12x12+y122=A−G1,σy1FP‐RB1=A+B1x12−y12x12+y122=A+G1,σz1FP‐RB1=C,τx1y1FP‐RB1=−2B1x1y1x12+y122=−H1,τy1z1FP‐RB1=0,τz1x1FP‐RB1=0.Through a serial of transformations which are same with the treatment of stress generated by the internal pressure of radial borehole 1, the final expressions of stress components caused by fracturing fluid percolation of radial borehole 1 can be achieved in the coordinate system (r0, θ0, z0) based on formulas (17) and (6).
(25)σr0FP‐RB1=cos2θ0cos2φA−G1+sin2φC+sin2θ0A+G1−2cosφsinθ0cosθ0H1,σθ0FP‐RB1=sin2θ0cos2φA−G1+sin2φC+cos2θ0A+G1+2cosφsinθ0cosθ0H1,σz0FP‐RB1=sin2φA−G1+cos2φC,τr0θ0FP‐RB1=A+G1−cos2φA−G1+sin2φCsinθ0cosθ0−cosφcos2θ0−sin2θ0H1,τθ0z0FP‐RB1=−sinφcosφsinθ0C−A−G1+sinφcosθ0H1,τz0r0FP‐RB1=sinφcosφcosθ0C−A−G1+sinφsinθ0H1.The above some components are shown in the formula.(26)x1=cosφr0cosθ0−z0tanφ,y1=r0sinθ0−L,z1=z0cosφ+sinφr0cosθ0−z0tanφ.


### 2.5. Total Stress Components around the Radial Borehole 0

All stress components generated by in situ stress, internal pressure, and fracturing fluid percolation of all boreholes have been obtained separately and converted into (r0, θ0, z0) coordinate system through the above derivation. According to stress superposition principle, the total stress components around the radial borehole 0 can be obtained in (r0, θ0, z0) coordinate system as follow.
(27)σr0Total=σr0In‐Situ+σr0IP‐MB+σr0IP‐RB0+σr0FP‐RB0+σr0IP‐RB1+σr0FP‐RB1,σθ0Total=σθ0In‐Situ+σθ0IP‐MB+σθ0IP‐RB0+σθ0FP‐RB0+σθ0IP‐RB1+σθ0FP‐RB1,σz0Total=σz0In‐Situ+σz0IP‐MB+σz0IP‐RB0+σz0FP‐RB0+σz0IP‐RB1+σz0FP‐RB1,τr0θ0Total=σr0θ0In‐Situ+σr0θ0IP‐MB+σr0θ0IP‐RB0+σr0θ0FP‐RB0+σr0θ0IP‐RB1+σr0θ0FP‐RB1,τθ0z0Total=σθ0z0In‐Situ+σθ0z0IP‐MB+σθ0z0IP‐RB0+σθ0z0FP‐RB0+σθ0z0IP‐RB1+σθ0z0FP‐RB1,τz0r0Total=σz0r0In‐Situ+σz0r0IP‐MB+σz0r0IP‐RB0+σz0r0FP‐RB0+σz0r0IP‐RB1+σz0r0FP‐RB1.


### 2.6. Fracture Initiation from Radial Borehole 0

There are many fracture criteria used in predicting fracture initiation, such as maximum tensile stress criterion, maximum tensile strain criterion, Tresca criterion, and Von Mises criterion. The maximum tensile stress criterion is chosen as the prediction principle in this paper. The three principal stresses can be calculated by formula (28) according to elasticity mechanics theory.
(28)σ1=σr0Total,σ2=12σθ0Total+σz0Total+σθ0Total−σz0Total2+4τθ0z0Total2,σ3=12σθ0Total+σz0Total−σθ0Total−σz0Total2+4τθ0z0Total2.In terms of the porous medium, the total stress should be substituted by effective stress. For the convention that pressure stress is defined as positive, the maximum tensile stress of research point (r0, θ0, z0) can be written as formula (29).
(29)σtr0,θ0,z0=−σ3r0,θ0,z0−Pp.The condition of fracture initiation is given as:(30)σtr0,θ0,z0>σT,whereσT is the tensile strength of rock, MPa.


## 2.1. Model Assumptions and Coordinate System

The following assumptions are made during the derivation of the analytical model:(1)
Formation rock is a homogeneous, isotropic, and linear elastic porous medium(2)
Microcracks inside rock have no influence(3)
The effect caused by the physicochemical action between fracturing fluid and formation rock is neglected(4)
Radial boreholes do not influence in situ stressCoordinate systems established during derivation and the layout of radial boreholes are as shown in Figure2. The radial borehole 0 is taken as the research object, and the model derivation is aimed at establishing an analytical model which can calculate the stress distribution along radial borehole 0 (RB0) under the effect of radial borehole 1 (RB1). In the process of model derivation, tensile stress is negative, and pressure stress is positive. A rectangular coordinate system (x, y, z) is established in which the origin is located at the center of the wellbore cross-section, and x axis, y axis, and z axis are aligned with the maximum horizontal stress, the minimum horizontal stress, and the main wellbore axis, relatively. Borehole coordinate systems are established. z0 axis and z1 axis are along the axis of two radial boreholes separately, while y0 axis and y1 axis are along the axis of the main wellbore. Thus, rectangular coordinate systems of the three boreholes, main wellbore and two radial boreholes, are (x, y, z), (x0, y0, z0), and (x1, y1, z1), and their cylindrical coordinate systems are (r, θ, z), (r0, θ0, z0), and (r1, θ1, z1), respectively. L is the depth difference between two radial boreholes. β0 and β1, the angle between the maximum horizontal stress direction and the axis of the radial borehole, are the azimuth angles of radial borehole 0 and radial borehole 1, respectively. The intersection angle of radial boreholes, the angle between the projections of radial boreholes axes on the horizontal plane, is denoted with φ, which equals β1-β0. r0 is the distance between research point and the axis of radial borehole 0. Thus, the coordinate of research point where the stress is calculated is (r0, θ0, z0) in (r0, θ0, z0) coordinate systems.Figure 2
Schematic diagram of hydraulic fracturing guided by multi-radial boreholes.


## 2.2. Model Assumptions and Coordinate System

Due to the assumption that radial boreholes do not influence in situ stress, the in situ stress in rectangular coordinate system (x, y, z) is:
(1)σxx000σyy000σzz=σH000σh000σv,where σH is the maximum horizontal in situ stress, MPa; σh is the minimum horizontal in situ stress, MPa; σv is the vertical in situ stress, MPa; and σxx, σyy, and σzz are the normal stress components in the coordinate system (x, y, z), MPa.Combining with the existing formula, the stress distribution around the main wellbore caused by in situ stress can be attained in the coordinate system (r, θ, z):
(2)σrIn‐Situ=12σH+σh1−Re2r2+12σH−σh1+3Re4r4−4Re2r2cos2θ,σθIn‐Situ=12σH+σh1+Re2r2−12σH−σh1+3Re4r4cos2θ,σzIn‐Situ=σν−2vrcos2θσH−σhRe2r2,τrθIn‐Situ=12σH−σh1−3Re4r4+2Re2r2sin2θ,τθzIn‐Situ=0,τrzIn‐Situ=0,where σr, σθ, and σz are the normal stress components in the cylindrical coordinate system (r, θ, z), MPa; τrθ, τθz, and τrz are the shear stress components in the cylindrical coordinate system (r, θ, z), MPa; vr is the Poisson’s ratio of rock; Re is the exterior radius of the main wellbore, m; and r is the distance between the axis of the main wellbore and any research point in formation, m.In order to attain the stress distribution along the radial borehole 0, the stress components caused by in situ stress need a serial of transformations from the coordinate system (r, θ, z) into (x, y, z), (x0, y0, z0), and (r0, θ0, z0), successively.First owing to the overlapping origin of the two coordinate systems, the stress components caused by in situ stress can be converted from (r, θ, z) coordinate system into (x, y, z) coordinate system by the following formulas.
(3)σxτxyτzxτxyσyτyzτzxτyzσz=cosθ−sinθ0sinθcosθ0001σrτθrτzrτrθσθτzθτrzτθzσzcosθsinθ0−sinθcosθ0001,(4)σxIn‐Situ=x2x2+y212σH+σh1−Re2x2+y2+12σH−σh1+3Re4x2+y22−4Re2x2+y2x2−y2x2+y2+y2x2+y212σH+σh1+Re2x2+y2−12σH−σh1+3Re4x2+y22x2−y2x2+y2−2x2y2x2+y22σH−σh1−3Re4x2+y22+2Re2x2+y2=F,σyIn‐Situ=y2x2+y212σH+σh1−Re2x2+y2+12σH−σh1+3Re4x2+y22−4Re2x2+y2x2−y2x2+y2+x2x2+y212σH+σh1+Re2x2+y2−12σH−σh1+3Re4x2+y22x2−y2x2+y2+2x2y2x2+y22σH−σh1−3Re4x2+y22+2Re2x2+y2=K,σzIn‐Situ=σν−2νRe2σH−σhx2−y2x2+y22=Q,τxyIn‐Situ=2xyx2+y2x2−y2x2+y2σH−σh−xyx2+y2σH+σhRe2x2+y2=T,τyzIn‐Situ=0,τzxIn‐Situ=0.The expressions ofσxIn‐Situ,σyIP‐Situ,σzIP‐Situ,andτxyIP‐Situ are denoted with F, K, Q, and T, respectively, for the facility of derivation. Second, the coordinate transformation expression from (x, y, z) coordinate system into (x0, y0, z0) coordinate system can be written as follows.
(5)σx0τx0y0τz0x0τx0y0σy0τy0z0τz0x0τy0z0σz0=−sinβ0cosβ00001cosβ0sinβ00σxτxyτzxτxyσyτyzτzxτyzσz−sinβ00cosβ0cosβ00sinβ0010.Then, the stress component transformation from (x0, y0, z0) coordinate system into (r0, θ0, z0) coordinate system can be conducted as the formula.
(6)σr0τθ0r0τz0r0τr0θ0σθ0τz0θ0τr0z0τθ0z0σz0=cosθ0sinθ00−sinθ0cosθ00001σx0τx0y0τz0x0τx0y0σy0τy0z0τz0x0τy0z0σz0cosθ0−sinθ00sinθ0cosθ00001.Based on the above formulas, the final expression of stress components caused by in situ stress can be obtained in (r0, θ0, z0) coordinate system.
(7)σr0In‐Situ=cos2θ0sin2β0F+cos2β0K−2sinβ0cosβ0T+Qsin2θ0,σθ0In‐Situ=sin2θ0sin2β0F+cos2β0K−2sinβ0cosβ0T+Qcos2θ0,σz0In‐Situ=cos2β0F+sin2β0K+2sinβ0cosβ0T,τr0θ0In‐Situ=Q−sin2β0F−cos2β0K+2sinβ0cosβ0Tsinθ0cosθ0,τθ0z0In‐Situ=−sinθ0cos2β0−sin2β0T+sinβ0cosβ0K−F,τz0r0In‐Situ=cosθ0cos2β0−sin2β0T+sinβ0cosβ0K−F.The above some components are shown in the formula.(8)x=cosβ0z0−r0cosθ0tanβ0,y=r0cosθ0cosβ0+sinβ0z0−r0cosθ0tanβ0,z=r0sinθ0.


## 2.3. Stress Caused by Wellbore Pressure

When fracturing fluid is pumped into wellbore, the internal pressure,Pw, is acting radially on the wall of the wellbore and generates stress field around the wellbore. Unlike radial boreholes, the main wellbore is cased, and the internal pressure, Pw, only can be delivered partially to the rock due to the high rigidity of the casing. Thus, the main wellbore and radial boreholes need to be investigated separately.


### 2.3.1. Stress Caused by Wellbore Pressure

Considering Young’s modulus of the casing is much higher than rock, and Young’s modulus of cement sheath and rock are in the same magnitude, the influence of cement sheath can be ignored, and the effect of casing should be considered. The stress generated by the internal pressure of the main wellbore can be written as follows in the cylindrical coordinate system (r, θ, z), considering the effect of casing [21].
(9)σrIP‐MB=Re2r2Pi,σθIP‐MB=−Re2r2Pi.Of which:(10)Pi=1+vcEc21−vcRe2−Ri2Ri2Pw/1+vrEr+1+vcEcRi2+1−2vcRe2Re2−Ri2,where vc is Poisson’s ratio of the casing; vr is Poisson’s ratio of rock; Er is Young’s modulus of rock, GPa; Ec is Young’s modulus of the casing, GPa; and Ri is the internal radius of the casing, m.Based on the coordinate transformation matrix (3), the stress expressions caused by main wellbore internal pressure can be available in (x, y, z) coordinate system.
(11)σxIP‐MB=Re2Pix2−y2x2+y22=M,σyIP‐MB=−Re2Pix2−y2x2+y22=−M,σzIP‐MB=0,τxyIP‐MB=2Re2Pixyx2+y22=N,τyzIP‐MB=0,τzxIP‐MB=0.The expressions ofσxIP‐MB,σyIP‐MB,andσxyIP‐MB are denoted with M, -M, and N separately for facilitating model derivation. Similar to in situ stress, the terms of the stress components caused by main wellbore internal pressure in the (r0, θ0, z0) coordinate system can be obtained through a serial of conversion according to formulas (5) and (6).
(12)σr0IP‐MB=cos2θ0sin2β0−cos2β0M−2sinβ0cosβ0N,σθ0IP‐MB=sin2θ0sin2β0−cos2β0M−2sinβ0cosβ0N,σz0IP‐MB=cos2β0−sin2β0M+2sinβ0cosβ0N,τr0θ0IP‐MB=−sinθ0cosθ0sin2β0−cos2β0M−2sinβ0cosβ0N,τθ0z0IP‐MB=−sinθ0cos2β0−sin2β0N−2sinβ0cosβ0M,τz0r0IP‐MB=cosθ0cos2β0−sin2β0N−2sinβ0cosβ0M.The above some components are shown in the formula.(13)x=cosβ0z0−r0cosθ0tanβ0,y=r0cosθ0cosβ0+sinβ0z0−r0cosθ0tanβ0,z=r0sinθ0.


### 2.3.2. Stress Caused by Internal Pressure of Radial Borehole 0

Compared with the main borehole, the influence of casing is not considered in radial boreholes because radial boreholes are open boreholes. Thus, the stress components in (r0, θ0, z0) coordinate system caused by the internal pressure of radial borehole 0 can be attained directly.
(14)σr0IP‐RB0=R02r02Pw,σθ0IP‐RB0=−R02r02Pw,where R0 is the radius of radial borehole 0, m, and Pw is the internal pressure, MPa.


### 2.3.3. Stress Caused by Internal Pressure of Radial Borehole 1

Similar to radial borehole 0, the stress caused by the internal pressure of radial borehole 1 can be obtained in the coordinate system (r1, θ1, z1).
(15)σr1IP‐RB1=R12r12Pw,σθ1IP‐RB1=−R12r12Pw,where R1 is the radius of radial borehole 1, m, and r1 is the distance between the axis of radial borehole 1 and any research point in formation, m.Next, the stress components are converted from (r1, θ1, z1) coordinate system into (x1, y1, z1) coordinate system with coordinate transformation matrix (3).
(16)σx1IP‐RB1=R12Pwx12−y12x12+y122=M1,σy1IP‐RB1=−R12Pwx12−y12x12+y122=−M1,σz1IP‐RB1=0,τx1y1IP‐RB1=2R12Pwx1y1x12+y122=N1,τy1z1IP‐RB1=0,τz1x1IP‐RB1=0.The expressions ofσx1IP‐RB1,σy1IP‐RB1,andσx1y1IP‐RB1 are written as M1, −M1, and N1, respectively. Continue to convert the stress components from (x1, y1, z1) coordinate system into (x0, y0, z0) coordinate system by the following coordinate transformation matrix (17).
(17)σx0τx0y0τz0x0τx0y0σy0τy0z0τz0x0τy0z0σz0=cosφ0sinφ010−sinφ0cosφσx1τx1y1τz1x1τx1y1σy1τy1z1τz1x1τy1z1σz1cosφ0−sinφ010sinφ0cosφ.Then, combining the expressions of stress components in (x0, y0, z0) coordinate system and formula (6), the final expressions of stress components caused by the internal pressure of radial borehole 1 can be achieved in (r0, θ0, z0) coordinate system.
(18)σr0IP‐RB1=cos2θ0cos2φ−sin2θ0M1+2sinθ0cosθ0cosφN1,σθ0IP‐RB1=sin2θ0cos2φ−cos2θ0M1−2sinθ0cosθ0cosφN1,σz0IP‐RB1=sin2φM1,τr0θ0IP‐RB1=−1+cos2φsinθ0cosθ0M1+cos2θ0−sin2θ0cosφN1,τθ0z0IP‐RB1=sinφcosφsinθ0M1−sinφcosθ0N1,τz0r0IP‐RB1=−sinφcosφcosθ0M1−sinφsinθ0N1.The above some components are shown in the formula.(19)x1=cosφr0cosθ0−z0tanφ,y1=r0sinθ0−L,z1=z0cosφ+sinφr0cosθ0−z0tanφ.


## 2.3.1. Stress Caused by Wellbore Pressure

Considering Young’s modulus of the casing is much higher than rock, and Young’s modulus of cement sheath and rock are in the same magnitude, the influence of cement sheath can be ignored, and the effect of casing should be considered. The stress generated by the internal pressure of the main wellbore can be written as follows in the cylindrical coordinate system (r, θ, z), considering the effect of casing [21].
(9)σrIP‐MB=Re2r2Pi,σθIP‐MB=−Re2r2Pi.Of which:(10)Pi=1+vcEc21−vcRe2−Ri2Ri2Pw/1+vrEr+1+vcEcRi2+1−2vcRe2Re2−Ri2,where vc is Poisson’s ratio of the casing; vr is Poisson’s ratio of rock; Er is Young’s modulus of rock, GPa; Ec is Young’s modulus of the casing, GPa; and Ri is the internal radius of the casing, m.Based on the coordinate transformation matrix (3), the stress expressions caused by main wellbore internal pressure can be available in (x, y, z) coordinate system.
(11)σxIP‐MB=Re2Pix2−y2x2+y22=M,σyIP‐MB=−Re2Pix2−y2x2+y22=−M,σzIP‐MB=0,τxyIP‐MB=2Re2Pixyx2+y22=N,τyzIP‐MB=0,τzxIP‐MB=0.The expressions ofσxIP‐MB,σyIP‐MB,andσxyIP‐MB are denoted with M, -M, and N separately for facilitating model derivation. Similar to in situ stress, the terms of the stress components caused by main wellbore internal pressure in the (r0, θ0, z0) coordinate system can be obtained through a serial of conversion according to formulas (5) and (6).
(12)σr0IP‐MB=cos2θ0sin2β0−cos2β0M−2sinβ0cosβ0N,σθ0IP‐MB=sin2θ0sin2β0−cos2β0M−2sinβ0cosβ0N,σz0IP‐MB=cos2β0−sin2β0M+2sinβ0cosβ0N,τr0θ0IP‐MB=−sinθ0cosθ0sin2β0−cos2β0M−2sinβ0cosβ0N,τθ0z0IP‐MB=−sinθ0cos2β0−sin2β0N−2sinβ0cosβ0M,τz0r0IP‐MB=cosθ0cos2β0−sin2β0N−2sinβ0cosβ0M.The above some components are shown in the formula.(13)x=cosβ0z0−r0cosθ0tanβ0,y=r0cosθ0cosβ0+sinβ0z0−r0cosθ0tanβ0,z=r0sinθ0.


## 2.3.2. Stress Caused by Internal Pressure of Radial Borehole 0

Compared with the main borehole, the influence of casing is not considered in radial boreholes because radial boreholes are open boreholes. Thus, the stress components in (r0, θ0, z0) coordinate system caused by the internal pressure of radial borehole 0 can be attained directly.
(14)σr0IP‐RB0=R02r02Pw,σθ0IP‐RB0=−R02r02Pw,where R0 is the radius of radial borehole 0, m, and Pw is the internal pressure, MPa.


## 2.3.3. Stress Caused by Internal Pressure of Radial Borehole 1

Similar to radial borehole 0, the stress caused by the internal pressure of radial borehole 1 can be obtained in the coordinate system (r1, θ1, z1).
(15)σr1IP‐RB1=R12r12Pw,σθ1IP‐RB1=−R12r12Pw,where R1 is the radius of radial borehole 1, m, and r1 is the distance between the axis of radial borehole 1 and any research point in formation, m.Next, the stress components are converted from (r1, θ1, z1) coordinate system into (x1, y1, z1) coordinate system with coordinate transformation matrix (3).
(16)σx1IP‐RB1=R12Pwx12−y12x12+y122=M1,σy1IP‐RB1=−R12Pwx12−y12x12+y122=−M1,σz1IP‐RB1=0,τx1y1IP‐RB1=2R12Pwx1y1x12+y122=N1,τy1z1IP‐RB1=0,τz1x1IP‐RB1=0.The expressions ofσx1IP‐RB1,σy1IP‐RB1,andσx1y1IP‐RB1 are written as M1, −M1, and N1, respectively. Continue to convert the stress components from (x1, y1, z1) coordinate system into (x0, y0, z0) coordinate system by the following coordinate transformation matrix (17).
(17)σx0τx0y0τz0x0τx0y0σy0τy0z0τz0x0τy0z0σz0=cosφ0sinφ010−sinφ0cosφσx1τx1y1τz1x1τx1y1σy1τy1z1τz1x1τy1z1σz1cosφ0−sinφ010sinφ0cosφ.Then, combining the expressions of stress components in (x0, y0, z0) coordinate system and formula (6), the final expressions of stress components caused by the internal pressure of radial borehole 1 can be achieved in (r0, θ0, z0) coordinate system.
(18)σr0IP‐RB1=cos2θ0cos2φ−sin2θ0M1+2sinθ0cosθ0cosφN1,σθ0IP‐RB1=sin2θ0cos2φ−cos2θ0M1−2sinθ0cosθ0cosφN1,σz0IP‐RB1=sin2φM1,τr0θ0IP‐RB1=−1+cos2φsinθ0cosθ0M1+cos2θ0−sin2θ0cosφN1,τθ0z0IP‐RB1=sinφcosφsinθ0M1−sinφcosθ0N1,τz0r0IP‐RB1=−sinφcosφcosθ0M1−sinφsinθ0N1.The above some components are shown in the formula.(19)x1=cosφr0cosθ0−z0tanφ,y1=r0sinθ0−L,z1=z0cosφ+sinφr0cosθ0−z0tanφ.


## 2.4. Stress Caused by Fracturing Fluid Percolation

The fracture fluid will permeate into formation and increase the pore fluid pressure around borehole under the high wellbore pressure during the hydraulic fracture operation. The fracturing fluid percolation of radial boreholes is taken into account, and the fracturing fluid percolation of the cased main wellbore can be neglected.


### 2.4.1. Stress Caused by Fracturing Fluid Percolation of Radial Borehole 0

Assuming that fluid flow in formation is radial flow and follows Darcy’s law, the stress caused by fracturing fluid percolation of radial borehole 0 can be provided in (r0, θ0, z0) coordinate system according to former research [22].
(20)σr0FP‐RB0=A−B0r02,σθ0FP‐RB0=A+B0r02,σz0FP‐RB0=C.Of which:(21)A=δPw−Ppε1−2vr21−vr−ϕ,B0=δR02Pw−Ppε1−2vr21−vr,C=δε1−2vr1−vr−ϕPw−Pp,wherePp is the initial pore fluid pressure, MPa; δ is the permeability coefficient; ϕ is the porosity of formation; εis the Biot coefficient, ε=1−Cr/Cb; and Cr and Cb are the compression rates of the rock skeleton and volume, respectively.


### 2.4.2. Stress Caused by Fracturing Fluid Percolation of Radial Borehole 1

The stress components yielded by fracturing fluid percolation of radial borehole 1 and expression of stress components can be obtained directly in the cylindrical coordinate system (r1, θ1, z1).
(22)σr1FP‐RB1=A−B1r12,σθ1FP‐RB1=A+B1r12,σz1FP‐RB1=C.Of which:(23)B1=δR12Pw−Ppε1−2vr21−vr.The stress components can be converted into (x1, y1, z1) coordinate system by coordinate transformation matrix (3), and the expressions of σx1FP‐RB1,σy1FP‐RB1,andσx1y1FP‐RB1 are denoted as A−G1, A+G1, and −H1, respectively, for the facility of the following derivation.
(24)σx1FP‐RB1=A−B1x12−y12x12+y122=A−G1,σy1FP‐RB1=A+B1x12−y12x12+y122=A+G1,σz1FP‐RB1=C,τx1y1FP‐RB1=−2B1x1y1x12+y122=−H1,τy1z1FP‐RB1=0,τz1x1FP‐RB1=0.Through a serial of transformations which are same with the treatment of stress generated by the internal pressure of radial borehole 1, the final expressions of stress components caused by fracturing fluid percolation of radial borehole 1 can be achieved in the coordinate system (r0, θ0, z0) based on formulas (17) and (6).
(25)σr0FP‐RB1=cos2θ0cos2φA−G1+sin2φC+sin2θ0A+G1−2cosφsinθ0cosθ0H1,σθ0FP‐RB1=sin2θ0cos2φA−G1+sin2φC+cos2θ0A+G1+2cosφsinθ0cosθ0H1,σz0FP‐RB1=sin2φA−G1+cos2φC,τr0θ0FP‐RB1=A+G1−cos2φA−G1+sin2φCsinθ0cosθ0−cosφcos2θ0−sin2θ0H1,τθ0z0FP‐RB1=−sinφcosφsinθ0C−A−G1+sinφcosθ0H1,τz0r0FP‐RB1=sinφcosφcosθ0C−A−G1+sinφsinθ0H1.The above some components are shown in the formula.(26)x1=cosφr0cosθ0−z0tanφ,y1=r0sinθ0−L,z1=z0cosφ+sinφr0cosθ0−z0tanφ.


## 2.4.1. Stress Caused by Fracturing Fluid Percolation of Radial Borehole 0

Assuming that fluid flow in formation is radial flow and follows Darcy’s law, the stress caused by fracturing fluid percolation of radial borehole 0 can be provided in (r0, θ0, z0) coordinate system according to former research [22].
(20)σr0FP‐RB0=A−B0r02,σθ0FP‐RB0=A+B0r02,σz0FP‐RB0=C.Of which:(21)A=δPw−Ppε1−2vr21−vr−ϕ,B0=δR02Pw−Ppε1−2vr21−vr,C=δε1−2vr1−vr−ϕPw−Pp,wherePp is the initial pore fluid pressure, MPa; δ is the permeability coefficient; ϕ is the porosity of formation; εis the Biot coefficient, ε=1−Cr/Cb; and Cr and Cb are the compression rates of the rock skeleton and volume, respectively.


## 2.4.2. Stress Caused by Fracturing Fluid Percolation of Radial Borehole 1

The stress components yielded by fracturing fluid percolation of radial borehole 1 and expression of stress components can be obtained directly in the cylindrical coordinate system (r1, θ1, z1).
(22)σr1FP‐RB1=A−B1r12,σθ1FP‐RB1=A+B1r12,σz1FP‐RB1=C.Of which:(23)B1=δR12Pw−Ppε1−2vr21−vr.The stress components can be converted into (x1, y1, z1) coordinate system by coordinate transformation matrix (3), and the expressions of σx1FP‐RB1,σy1FP‐RB1,andσx1y1FP‐RB1 are denoted as A−G1, A+G1, and −H1, respectively, for the facility of the following derivation.
(24)σx1FP‐RB1=A−B1x12−y12x12+y122=A−G1,σy1FP‐RB1=A+B1x12−y12x12+y122=A+G1,σz1FP‐RB1=C,τx1y1FP‐RB1=−2B1x1y1x12+y122=−H1,τy1z1FP‐RB1=0,τz1x1FP‐RB1=0.Through a serial of transformations which are same with the treatment of stress generated by the internal pressure of radial borehole 1, the final expressions of stress components caused by fracturing fluid percolation of radial borehole 1 can be achieved in the coordinate system (r0, θ0, z0) based on formulas (17) and (6).
(25)σr0FP‐RB1=cos2θ0cos2φA−G1+sin2φC+sin2θ0A+G1−2cosφsinθ0cosθ0H1,σθ0FP‐RB1=sin2θ0cos2φA−G1+sin2φC+cos2θ0A+G1+2cosφsinθ0cosθ0H1,σz0FP‐RB1=sin2φA−G1+cos2φC,τr0θ0FP‐RB1=A+G1−cos2φA−G1+sin2φCsinθ0cosθ0−cosφcos2θ0−sin2θ0H1,τθ0z0FP‐RB1=−sinφcosφsinθ0C−A−G1+sinφcosθ0H1,τz0r0FP‐RB1=sinφcosφcosθ0C−A−G1+sinφsinθ0H1.The above some components are shown in the formula.(26)x1=cosφr0cosθ0−z0tanφ,y1=r0sinθ0−L,z1=z0cosφ+sinφr0cosθ0−z0tanφ.


## 2.5. Total Stress Components around the Radial Borehole 0

All stress components generated by in situ stress, internal pressure, and fracturing fluid percolation of all boreholes have been obtained separately and converted into (r0, θ0, z0) coordinate system through the above derivation. According to stress superposition principle, the total stress components around the radial borehole 0 can be obtained in (r0, θ0, z0) coordinate system as follow.
(27)σr0Total=σr0In‐Situ+σr0IP‐MB+σr0IP‐RB0+σr0FP‐RB0+σr0IP‐RB1+σr0FP‐RB1,σθ0Total=σθ0In‐Situ+σθ0IP‐MB+σθ0IP‐RB0+σθ0FP‐RB0+σθ0IP‐RB1+σθ0FP‐RB1,σz0Total=σz0In‐Situ+σz0IP‐MB+σz0IP‐RB0+σz0FP‐RB0+σz0IP‐RB1+σz0FP‐RB1,τr0θ0Total=σr0θ0In‐Situ+σr0θ0IP‐MB+σr0θ0IP‐RB0+σr0θ0FP‐RB0+σr0θ0IP‐RB1+σr0θ0FP‐RB1,τθ0z0Total=σθ0z0In‐Situ+σθ0z0IP‐MB+σθ0z0IP‐RB0+σθ0z0FP‐RB0+σθ0z0IP‐RB1+σθ0z0FP‐RB1,τz0r0Total=σz0r0In‐Situ+σz0r0IP‐MB+σz0r0IP‐RB0+σz0r0FP‐RB0+σz0r0IP‐RB1+σz0r0FP‐RB1.


## 2.6. Fracture Initiation from Radial Borehole 0

There are many fracture criteria used in predicting fracture initiation, such as maximum tensile stress criterion, maximum tensile strain criterion, Tresca criterion, and Von Mises criterion. The maximum tensile stress criterion is chosen as the prediction principle in this paper. The three principal stresses can be calculated by formula (28) according to elasticity mechanics theory.
(28)σ1=σr0Total,σ2=12σθ0Total+σz0Total+σθ0Total−σz0Total2+4τθ0z0Total2,σ3=12σθ0Total+σz0Total−σθ0Total−σz0Total2+4τθ0z0Total2.In terms of the porous medium, the total stress should be substituted by effective stress. For the convention that pressure stress is defined as positive, the maximum tensile stress of research point (r0, θ0, z0) can be written as formula (29).
(29)σtr0,θ0,z0=−σ3r0,θ0,z0−Pp.The condition of fracture initiation is given as:(30)σtr0,θ0,z0>σT,whereσT is the tensile strength of rock, MPa.


## 3. Case Calculation and Sensitivity Analysis

To study fracture initiation from radial borehole 0,r0 takes the value of R0 to calculate the stress on the wall of borehole 0, and a serial of sensitivity analysis of different parameters are conducted, such as intersection angle of radial boreholes, the radius of radial boreholes, depth difference between radial boreholes, Biot coefficient, and the number of radial boreholes. Meanwhile, as illustrated in Figure 3, the wall of radial borehole 0 is flattened into a plane, called radial borehole 0 plane, to visually display the situation of borehole wall.Figure 3
Schematic diagram of flattening the radial borehole 0 wall.The increment of maximum tensile stress on the wall of radial borehole 0, which caused by the presence of radial borehole 1, can be considered as the interference between boreholes and forms the guidance of radial borehole 1 to the fracture from radial borehole 0. Based on the difference in the maximum tensile stress distribution with single radial borehole 0 and the maximum tensile stress distribution with both radial boreholes, the increment of maximum tensile stress caused by radial borehole 1 can be calculated, and the maximum tensile stress increasing zone can be plotted on radial borehole 0 plane. Through the above methodology, the interference between radial boreholes is quantified and analyzed. The basic calculation parameters are shown in Table1.Table 1
Basic calculation parameters.
ParameterValueUnitMaximum horizontal in situ stress,σH48MPaMinimum horizontal in situ stress,σh40MPaVertical in situ stress,σv50MPaPoisson’s ratio of rock,vr0.2—Young’s modulus of rock,Er50GPaYoung’s modulus of the casing,Ec135GPaExterior radius of the main wellbore,Re0.18mInterior radius of the main wellbore,Ri0.165mRadius of radial borehole 0,R00.015mRadius of radial borehole 1,R10.015mInternal pressure,Pw45MPaInitial pore fluid pressure,Pp20MPaPermeability coefficient,δ0.9—Formation porosity,ϕ0.15—Biot coefficient,ε0.55—Azimuth angles of radial borehole 0,β045DegreeIntersection angle of radial boreholes,φ0DegreeDepth difference between radial boreholes,L0.05mTensile strength of rock,σT8MPa


### 3.1. The Effect of Radial Borehole Intersection Angle

In this part, the other parameters are set to basic calculation parameters, and the intersection angle of radial boreholes,φ, is changed to analyze its effect on the interference between radial boreholes and fracture initiation pressure.


#### 3.1.1. The Sensitivity of FIP

As shown in Figure4, as the intersection angle of radial boreholes increasing from 0° to 25°, FIP increases rapidly at first and then stays pretty constant when the intersection angle exceeds 10°. When the intersection angle between two radial boreholes is small, the additional maximum tensile stress increase caused by radial borehole 1 roughly overlaps with the maximum tensile stress concentration region of radial borehole 0 wall, the zone around θ0=90°. Thus, lower FIP can be attained when the intersection angle is below 10°. The additional maximum tensile stress decreases, and the region with additional stress increase deviates from θ0=90° when intersection angle, φ, increases, which conduce the rapid growth of FIP. When the intersection angle of radial boreholes is more than 10°, it can be considered that the expansion of the angle does not affect FIP anymore.Figure 4
FIP with different intersection angles between radial boreholes.


#### 3.1.2. The Sensitivity of Interference

As shown in Figures5 and 6, the inference caused by radial borehole 1 decreases with the growth of intersection angle, φ. When φ=0°, the maximum tensile stress increasing region is distributed along the whole length of radial hole 0, and the highest increase of maximum tensile stress caused by radial hole 1 is the largest. At this time, radial borehole 1 has an excellent guiding strength to the fracture from radial borehole 0, encouraging fracture to extend toward radial boreholes 1 and connect two boreholes. With the further increase of φ from 5° to 20°, the maximum growth of maximum tensile stress gradually decreases, and the maximum tensile stress increasing region also shrinks and shifts. The distance between two boreholes starts to enlarge along radial borehole 0 when φ>0°, and the growth rate of distance positively correlates with intersection angle, φ. Therefore, the region near the root of borehole 0, about s<0.7m, still can be guided well at φ=5°, and nearly, no area can be directed at φ=20°. The highest increase of maximum tensile stress falls to 0 at φ=25°, and it can be considered that radial borehole 1 no longer has interference to the fracture from radial borehole 0.Figure 5
Maximum tensile stress increasing region with different intersection angles.Figure 6
Highest increment of maximum tensile stress with different intersection angles.


### 3.2. The Effect of Radial Borehole Depth Difference

For radial borehole 1 virtually does no influence FIP of radial borehole 0 whenφ>10°, the depth difference is changed to analyze the effect of depth difference in the condition of that the intersection angle, φ, is fixed to 0°, and the other parameters take the value of basic calculation parameters.


#### 3.2.1. The Sensitivity of FIP

As illustrated in Figure7, with the increase of depth difference from 40 mm to 90 mm, FIP increases rapidly firstly (depthdifference<60mm), then ascends slowly, because the increasing distance between boreholes weakens the action of additional stress field caused by radial borehole 1 and heightens FIP.Figure 7
FIP with different depth differences of radial boreholes.


#### 3.2.2. The Sensitivity of Interference

As illustrated in Figures8 and 9, the area of stress increasing region and the increasing growth of maximum tensile stress reduce with the increase of depth difference from 40 mm to 90 mm, representing the declining trend of interference and guidance of radial borehole 1. When the depth difference is small (L<50mm), a higher increment of maximum tensile stress can be attained, and even both top and bottom sides (θ0=90°, 270°) of radial borehole 0 can be affected by addition stress field. As depth difference continues to increase, the stress growth region moderately decreases, and the rising degree of maximum tensile stress gradually decreases. The highest increase of maximum tensile stress on the wall of radial borehole 0 has been reduced to 0.1076 MPa when the depth difference reaches 90 mm. It can be considered that radial borehole 0 is beyond the influencing domain of additional stress field caused by radial borehole 1, and radial borehole 1 has no guidance to the fracture from radial borehole 0 when depth difference exceeds 90 mm.Figure 8
Maximum tensile stress increasing region with different depth differences.Figure 9
Highest increment of maximum tensile stress with different depth differences.


### 3.3. The Effect of Radial Borehole Radius

According to the previous study (Liu et al., 2018), the radius of the radial borehole has virtually no influence on its own FIP in the hydraulic fracture guided by single radial boreholes. However, the situation will be different in the case of multiradial boreholes. Due to the interference between radial boreholes, the radius of a particular radial borehole may affect FIP of the other radial boreholes. Thereby, depth difference is fixed as 90 mm; the other parameters are set to basic calculation parameters, and the radius of radial borehole 1 is changed to analyze its effect on radial borehole 0.


#### 3.3.1. The Sensitivity of FIP

As illustrated in Figure10, fracture initiation pressure decreases at a nearly uniform speed as the radius of radial borehole 1, R1, increasing. The increase of radial borehole 1 radius enhances the interference between boreholes and benefits the fracture initiation from the radial borehole 0.Figure 10
FIP with different radius of radial borehole 1.


#### 3.3.2. The Sensitivity of Interference

As illustrated in Figures11 and 12, both of the maximum tensile stress increasing region and the growth of maximum tensile stress augment in the pace of radius of radial borehole 1 increasing. The case of R1=25mm can be considered as a critical value for that only one side of borehole 0, top side, is affected by radial borehole 1 when R1<25mm, and both sides of radial borehole 0 will be affected by radial borehole 1 if R1≥25mm. To get further study about the influence of radial borehole radius on the magnitude of the region where radial borehole can generate guidance, two groups of calculations are carried out, respectively. In one group, fix the depth difference, L, to 50 mm and calculate the intersection angle when the maximum tensile stress increment decreases to 0 under different radius and define the angle at this time as the critical intersection angle, φc. In the other group, fix the intersection angle, φ, to 0° and calculate the depth difference when the maximum tensile stress increment decreases to 0 under different radius and define the depth difference at this time as the critical depth difference, Lc.Figure 11
Maximum tensile stress increasing region with different radius of radial borehole 1.Figure 12
Highest increment of maximum tensile stress with different radius of radial borehole 1.From Figures13 and 14, both of the critical depth difference and critical intersection angle increase as the radius of radial borehole 1 increasing. The critical depth difference represents the longitudinal size of the region where radial borehole can generate interference and guidance. As the radius increases, the critical depth difference increases at a uniform speed. Every time the radius increases by 1 mm, the critical depth difference increases by 5.2 mm. Meanwhile, the critical intersection angle represents the transverse size of the region where radial borehole can generate interference and guidance. The critical angle increases with the increase of radius, but the increasing speed is decreasing gradually.Figure 13
Critical depth difference with different radius of radial borehole 1.Figure 14
Critical intersection angle with different radius of radial borehole 1.


### 3.4. The Effect of Biot Coefficient

Biot coefficient is one of the basic parameters of formation rock and plays an essential role in hydraulic fracture design. In this part, Biot coefficient is changed, and the other parameters adopt the value of basic calculation parameters to study the effect of Biot coefficient.


#### 3.4.1. The Sensitivity of FIP

As illustrated in Figure15, fracture initiation pressure increases as the Biot coefficient rising. Besides, the growth rate of FIP accelerates slightly with the increase of Biot coefficient. Thus, a low Biot coefficient is helpful to fracture initiation and obtain a low fracture initiation pressure.Figure 15
FIP with different Biot coefficient.


#### 3.4.2. The Sensitivity of Interference

It can be seen from Figures16 and 17 that with the increase of Biot coefficient, the maximum tensile stress increasing region and the highest growth of maximum tensile stress are both shrinking. Besides, there is an excellent linear relationship between the highest increment of maximum tensile stress and Biot coefficient. When the Biot coefficient increases by 0.1, the highest increment of maximum tensile stress decreases by 0.89 MPa. Similar to the analysis of radius of radial borehole 1, the critical depth difference, Lc, and critical intersection angle, θc, are calculated separately with the fixed intersection angle of 0° and the fixed depth difference of 50 mm, respectively, to study the influence of Biot coefficient on the interference range of radial borehole 1.Figure 16
Maximum tensile stress increasing region with different Biot coefficient.Figure 17
Highest increment of maximum tensile stress with different Biot coefficient.As shown in Figures18 and 19, with the increase of Biot coefficient, the critical intersection angle and critical depth difference have a similar decreasing trend, which means that the interference region of radial borehole 1 decreases with the increase of Biot coefficient. This means that the low Biot coefficient is conducive to expand the influence range of additional stress field caused by radial borehole 1, which guides fracture to connect radial boreholes and conduces the form of the fracture network. Also, the declining pace of both critical intersection angle and critical depth difference decreases with the increment of Biot coefficient.Figure 18
Critical depth difference with different Biot coefficient.Figure 19
Critical intersection angle with different Biot coefficient.


### 3.5. The Effect of Radial Borehole Number

In this part, the multi-radial borehole model is compared with the single radial borehole model proposed by Liu [20] to analyze the influence of additional radial boreholes. Meanwhile, the existing model can be extended easily to conditions with arbitrary number of radial boreholes through simple steps. The specific steps are not described in this article. In this part, the other parameters are set to basic calculation parameters, and the number of radial boreholes is changed. All radial boreholes have the same azimuth angles, and the depth difference between adjacent radial boreholes is fixed to 50 mm. The radial borehole in the middle is taken as the research object. According to the change of FIP and highest increment of maximum tensile stress, the influence of radial boreholes number is studied.


#### 3.5.1. The Sensitivity of FIP

It can be seen from Figure20, with the increase of the number of radial boreholes, FIP decreases promptly first, then slows down gradually and finally decreases by 23.7% to 31.3 MPa. The maximum decrease of FIP occurs when the number of radial boreholes changes from 1 to 2. Compared with single radial borehole, FIP of double radial boreholes decreases by 20.2% due to one additional radial borehole. Therefore, in the formation where fracture is hard to initiate, multi-radial boreholes, compared with single borehole, can effectively reduce FIP and induce fracture initiation. However, when number of radial boreholes adds up to 3, continually increasing the number of radial boreholes has little effect on the FIP. At this time, it is no longer a feasible choice to reduce FIP by increasing the number of radial boreholes.Figure 20
FIP with different number of radial boreholes.


#### 3.5.2. The Sensitivity of Interference

As shown in Figures21 and 22, highest increment of maximum tensile stress grows up with the increase of radial borehole number. Comparing double radial boreholes with single radial borehole, the additional radial borehole increases the maximum tensile stress on borehole wall by 6.12 MPa. Meanwhile, due to the same orientation of the radial boreholes, maximum tensile stress increasing region appears near the top and bottom of radial borehole (θ=90° and 180°). However, the increase rate slows down rapidly with the increase of radial borehole number. When the number of radial boreholes exceeds 3, the influence of radial borehole number on highest increment of maximum tensile stress nearly can be neglected.Figure 21
Maximum tensile stress increasing region with different radial borehole number.Figure 22
Highest increment of maximum tensile stress with different radial borehole number.


## 3.1. The Effect of Radial Borehole Intersection Angle

In this part, the other parameters are set to basic calculation parameters, and the intersection angle of radial boreholes,φ, is changed to analyze its effect on the interference between radial boreholes and fracture initiation pressure.


### 3.1.1. The Sensitivity of FIP

As shown in Figure4, as the intersection angle of radial boreholes increasing from 0° to 25°, FIP increases rapidly at first and then stays pretty constant when the intersection angle exceeds 10°. When the intersection angle between two radial boreholes is small, the additional maximum tensile stress increase caused by radial borehole 1 roughly overlaps with the maximum tensile stress concentration region of radial borehole 0 wall, the zone around θ0=90°. Thus, lower FIP can be attained when the intersection angle is below 10°. The additional maximum tensile stress decreases, and the region with additional stress increase deviates from θ0=90° when intersection angle, φ, increases, which conduce the rapid growth of FIP. When the intersection angle of radial boreholes is more than 10°, it can be considered that the expansion of the angle does not affect FIP anymore.Figure 4
FIP with different intersection angles between radial boreholes.


### 3.1.2. The Sensitivity of Interference

As shown in Figures5 and 6, the inference caused by radial borehole 1 decreases with the growth of intersection angle, φ. When φ=0°, the maximum tensile stress increasing region is distributed along the whole length of radial hole 0, and the highest increase of maximum tensile stress caused by radial hole 1 is the largest. At this time, radial borehole 1 has an excellent guiding strength to the fracture from radial borehole 0, encouraging fracture to extend toward radial boreholes 1 and connect two boreholes. With the further increase of φ from 5° to 20°, the maximum growth of maximum tensile stress gradually decreases, and the maximum tensile stress increasing region also shrinks and shifts. The distance between two boreholes starts to enlarge along radial borehole 0 when φ>0°, and the growth rate of distance positively correlates with intersection angle, φ. Therefore, the region near the root of borehole 0, about s<0.7m, still can be guided well at φ=5°, and nearly, no area can be directed at φ=20°. The highest increase of maximum tensile stress falls to 0 at φ=25°, and it can be considered that radial borehole 1 no longer has interference to the fracture from radial borehole 0.Figure 5
Maximum tensile stress increasing region with different intersection angles.Figure 6
Highest increment of maximum tensile stress with different intersection angles.


## 3.1.1. The Sensitivity of FIP

As shown in Figure4, as the intersection angle of radial boreholes increasing from 0° to 25°, FIP increases rapidly at first and then stays pretty constant when the intersection angle exceeds 10°. When the intersection angle between two radial boreholes is small, the additional maximum tensile stress increase caused by radial borehole 1 roughly overlaps with the maximum tensile stress concentration region of radial borehole 0 wall, the zone around θ0=90°. Thus, lower FIP can be attained when the intersection angle is below 10°. The additional maximum tensile stress decreases, and the region with additional stress increase deviates from θ0=90° when intersection angle, φ, increases, which conduce the rapid growth of FIP. When the intersection angle of radial boreholes is more than 10°, it can be considered that the expansion of the angle does not affect FIP anymore.Figure 4
FIP with different intersection angles between radial boreholes.


## 3.1.2. The Sensitivity of Interference

As shown in Figures5 and 6, the inference caused by radial borehole 1 decreases with the growth of intersection angle, φ. When φ=0°, the maximum tensile stress increasing region is distributed along the whole length of radial hole 0, and the highest increase of maximum tensile stress caused by radial hole 1 is the largest. At this time, radial borehole 1 has an excellent guiding strength to the fracture from radial borehole 0, encouraging fracture to extend toward radial boreholes 1 and connect two boreholes. With the further increase of φ from 5° to 20°, the maximum growth of maximum tensile stress gradually decreases, and the maximum tensile stress increasing region also shrinks and shifts. The distance between two boreholes starts to enlarge along radial borehole 0 when φ>0°, and the growth rate of distance positively correlates with intersection angle, φ. Therefore, the region near the root of borehole 0, about s<0.7m, still can be guided well at φ=5°, and nearly, no area can be directed at φ=20°. The highest increase of maximum tensile stress falls to 0 at φ=25°, and it can be considered that radial borehole 1 no longer has interference to the fracture from radial borehole 0.Figure 5
Maximum tensile stress increasing region with different intersection angles.Figure 6
Highest increment of maximum tensile stress with different intersection angles.


## 3.2. The Effect of Radial Borehole Depth Difference

For radial borehole 1 virtually does no influence FIP of radial borehole 0 whenφ>10°, the depth difference is changed to analyze the effect of depth difference in the condition of that the intersection angle, φ, is fixed to 0°, and the other parameters take the value of basic calculation parameters.


### 3.2.1. The Sensitivity of FIP

As illustrated in Figure7, with the increase of depth difference from 40 mm to 90 mm, FIP increases rapidly firstly (depthdifference<60mm), then ascends slowly, because the increasing distance between boreholes weakens the action of additional stress field caused by radial borehole 1 and heightens FIP.Figure 7
FIP with different depth differences of radial boreholes.


### 3.2.2. The Sensitivity of Interference

As illustrated in Figures8 and 9, the area of stress increasing region and the increasing growth of maximum tensile stress reduce with the increase of depth difference from 40 mm to 90 mm, representing the declining trend of interference and guidance of radial borehole 1. When the depth difference is small (L<50mm), a higher increment of maximum tensile stress can be attained, and even both top and bottom sides (θ0=90°, 270°) of radial borehole 0 can be affected by addition stress field. As depth difference continues to increase, the stress growth region moderately decreases, and the rising degree of maximum tensile stress gradually decreases. The highest increase of maximum tensile stress on the wall of radial borehole 0 has been reduced to 0.1076 MPa when the depth difference reaches 90 mm. It can be considered that radial borehole 0 is beyond the influencing domain of additional stress field caused by radial borehole 1, and radial borehole 1 has no guidance to the fracture from radial borehole 0 when depth difference exceeds 90 mm.Figure 8
Maximum tensile stress increasing region with different depth differences.Figure 9
Highest increment of maximum tensile stress with different depth differences.


## 3.2.1. The Sensitivity of FIP

As illustrated in Figure7, with the increase of depth difference from 40 mm to 90 mm, FIP increases rapidly firstly (depthdifference<60mm), then ascends slowly, because the increasing distance between boreholes weakens the action of additional stress field caused by radial borehole 1 and heightens FIP.Figure 7
FIP with different depth differences of radial boreholes.


## 3.2.2. The Sensitivity of Interference

As illustrated in Figures8 and 9, the area of stress increasing region and the increasing growth of maximum tensile stress reduce with the increase of depth difference from 40 mm to 90 mm, representing the declining trend of interference and guidance of radial borehole 1. When the depth difference is small (L<50mm), a higher increment of maximum tensile stress can be attained, and even both top and bottom sides (θ0=90°, 270°) of radial borehole 0 can be affected by addition stress field. As depth difference continues to increase, the stress growth region moderately decreases, and the rising degree of maximum tensile stress gradually decreases. The highest increase of maximum tensile stress on the wall of radial borehole 0 has been reduced to 0.1076 MPa when the depth difference reaches 90 mm. It can be considered that radial borehole 0 is beyond the influencing domain of additional stress field caused by radial borehole 1, and radial borehole 1 has no guidance to the fracture from radial borehole 0 when depth difference exceeds 90 mm.Figure 8
Maximum tensile stress increasing region with different depth differences.Figure 9
Highest increment of maximum tensile stress with different depth differences.


## 3.3. The Effect of Radial Borehole Radius

According to the previous study (Liu et al., 2018), the radius of the radial borehole has virtually no influence on its own FIP in the hydraulic fracture guided by single radial boreholes. However, the situation will be different in the case of multiradial boreholes. Due to the interference between radial boreholes, the radius of a particular radial borehole may affect FIP of the other radial boreholes. Thereby, depth difference is fixed as 90 mm; the other parameters are set to basic calculation parameters, and the radius of radial borehole 1 is changed to analyze its effect on radial borehole 0.


### 3.3.1. The Sensitivity of FIP

As illustrated in Figure10, fracture initiation pressure decreases at a nearly uniform speed as the radius of radial borehole 1, R1, increasing. The increase of radial borehole 1 radius enhances the interference between boreholes and benefits the fracture initiation from the radial borehole 0.Figure 10
FIP with different radius of radial borehole 1.


### 3.3.2. The Sensitivity of Interference

As illustrated in Figures11 and 12, both of the maximum tensile stress increasing region and the growth of maximum tensile stress augment in the pace of radius of radial borehole 1 increasing. The case of R1=25mm can be considered as a critical value for that only one side of borehole 0, top side, is affected by radial borehole 1 when R1<25mm, and both sides of radial borehole 0 will be affected by radial borehole 1 if R1≥25mm. To get further study about the influence of radial borehole radius on the magnitude of the region where radial borehole can generate guidance, two groups of calculations are carried out, respectively. In one group, fix the depth difference, L, to 50 mm and calculate the intersection angle when the maximum tensile stress increment decreases to 0 under different radius and define the angle at this time as the critical intersection angle, φc. In the other group, fix the intersection angle, φ, to 0° and calculate the depth difference when the maximum tensile stress increment decreases to 0 under different radius and define the depth difference at this time as the critical depth difference, Lc.Figure 11
Maximum tensile stress increasing region with different radius of radial borehole 1.Figure 12
Highest increment of maximum tensile stress with different radius of radial borehole 1.From Figures13 and 14, both of the critical depth difference and critical intersection angle increase as the radius of radial borehole 1 increasing. The critical depth difference represents the longitudinal size of the region where radial borehole can generate interference and guidance. As the radius increases, the critical depth difference increases at a uniform speed. Every time the radius increases by 1 mm, the critical depth difference increases by 5.2 mm. Meanwhile, the critical intersection angle represents the transverse size of the region where radial borehole can generate interference and guidance. The critical angle increases with the increase of radius, but the increasing speed is decreasing gradually.Figure 13
Critical depth difference with different radius of radial borehole 1.Figure 14
Critical intersection angle with different radius of radial borehole 1.


## 3.3.1. The Sensitivity of FIP

As illustrated in Figure10, fracture initiation pressure decreases at a nearly uniform speed as the radius of radial borehole 1, R1, increasing. The increase of radial borehole 1 radius enhances the interference between boreholes and benefits the fracture initiation from the radial borehole 0.Figure 10
FIP with different radius of radial borehole 1.


## 3.3.2. The Sensitivity of Interference

As illustrated in Figures11 and 12, both of the maximum tensile stress increasing region and the growth of maximum tensile stress augment in the pace of radius of radial borehole 1 increasing. The case of R1=25mm can be considered as a critical value for that only one side of borehole 0, top side, is affected by radial borehole 1 when R1<25mm, and both sides of radial borehole 0 will be affected by radial borehole 1 if R1≥25mm. To get further study about the influence of radial borehole radius on the magnitude of the region where radial borehole can generate guidance, two groups of calculations are carried out, respectively. In one group, fix the depth difference, L, to 50 mm and calculate the intersection angle when the maximum tensile stress increment decreases to 0 under different radius and define the angle at this time as the critical intersection angle, φc. In the other group, fix the intersection angle, φ, to 0° and calculate the depth difference when the maximum tensile stress increment decreases to 0 under different radius and define the depth difference at this time as the critical depth difference, Lc.Figure 11
Maximum tensile stress increasing region with different radius of radial borehole 1.Figure 12
Highest increment of maximum tensile stress with different radius of radial borehole 1.From Figures13 and 14, both of the critical depth difference and critical intersection angle increase as the radius of radial borehole 1 increasing. The critical depth difference represents the longitudinal size of the region where radial borehole can generate interference and guidance. As the radius increases, the critical depth difference increases at a uniform speed. Every time the radius increases by 1 mm, the critical depth difference increases by 5.2 mm. Meanwhile, the critical intersection angle represents the transverse size of the region where radial borehole can generate interference and guidance. The critical angle increases with the increase of radius, but the increasing speed is decreasing gradually.Figure 13
Critical depth difference with different radius of radial borehole 1.Figure 14
Critical intersection angle with different radius of radial borehole 1.


## 3.4. The Effect of Biot Coefficient

Biot coefficient is one of the basic parameters of formation rock and plays an essential role in hydraulic fracture design. In this part, Biot coefficient is changed, and the other parameters adopt the value of basic calculation parameters to study the effect of Biot coefficient.


### 3.4.1. The Sensitivity of FIP

As illustrated in Figure15, fracture initiation pressure increases as the Biot coefficient rising. Besides, the growth rate of FIP accelerates slightly with the increase of Biot coefficient. Thus, a low Biot coefficient is helpful to fracture initiation and obtain a low fracture initiation pressure.Figure 15
FIP with different Biot coefficient.


### 3.4.2. The Sensitivity of Interference

It can be seen from Figures16 and 17 that with the increase of Biot coefficient, the maximum tensile stress increasing region and the highest growth of maximum tensile stress are both shrinking. Besides, there is an excellent linear relationship between the highest increment of maximum tensile stress and Biot coefficient. When the Biot coefficient increases by 0.1, the highest increment of maximum tensile stress decreases by 0.89 MPa. Similar to the analysis of radius of radial borehole 1, the critical depth difference, Lc, and critical intersection angle, θc, are calculated separately with the fixed intersection angle of 0° and the fixed depth difference of 50 mm, respectively, to study the influence of Biot coefficient on the interference range of radial borehole 1.Figure 16
Maximum tensile stress increasing region with different Biot coefficient.Figure 17
Highest increment of maximum tensile stress with different Biot coefficient.As shown in Figures18 and 19, with the increase of Biot coefficient, the critical intersection angle and critical depth difference have a similar decreasing trend, which means that the interference region of radial borehole 1 decreases with the increase of Biot coefficient. This means that the low Biot coefficient is conducive to expand the influence range of additional stress field caused by radial borehole 1, which guides fracture to connect radial boreholes and conduces the form of the fracture network. Also, the declining pace of both critical intersection angle and critical depth difference decreases with the increment of Biot coefficient.Figure 18
Critical depth difference with different Biot coefficient.Figure 19
Critical intersection angle with different Biot coefficient.


## 3.4.1. The Sensitivity of FIP

As illustrated in Figure15, fracture initiation pressure increases as the Biot coefficient rising. Besides, the growth rate of FIP accelerates slightly with the increase of Biot coefficient. Thus, a low Biot coefficient is helpful to fracture initiation and obtain a low fracture initiation pressure.Figure 15
FIP with different Biot coefficient.


## 3.4.2. The Sensitivity of Interference

It can be seen from Figures16 and 17 that with the increase of Biot coefficient, the maximum tensile stress increasing region and the highest growth of maximum tensile stress are both shrinking. Besides, there is an excellent linear relationship between the highest increment of maximum tensile stress and Biot coefficient. When the Biot coefficient increases by 0.1, the highest increment of maximum tensile stress decreases by 0.89 MPa. Similar to the analysis of radius of radial borehole 1, the critical depth difference, Lc, and critical intersection angle, θc, are calculated separately with the fixed intersection angle of 0° and the fixed depth difference of 50 mm, respectively, to study the influence of Biot coefficient on the interference range of radial borehole 1.Figure 16
Maximum tensile stress increasing region with different Biot coefficient.Figure 17
Highest increment of maximum tensile stress with different Biot coefficient.As shown in Figures18 and 19, with the increase of Biot coefficient, the critical intersection angle and critical depth difference have a similar decreasing trend, which means that the interference region of radial borehole 1 decreases with the increase of Biot coefficient. This means that the low Biot coefficient is conducive to expand the influence range of additional stress field caused by radial borehole 1, which guides fracture to connect radial boreholes and conduces the form of the fracture network. Also, the declining pace of both critical intersection angle and critical depth difference decreases with the increment of Biot coefficient.Figure 18
Critical depth difference with different Biot coefficient.Figure 19
Critical intersection angle with different Biot coefficient.


## 3.5. The Effect of Radial Borehole Number

In this part, the multi-radial borehole model is compared with the single radial borehole model proposed by Liu [20] to analyze the influence of additional radial boreholes. Meanwhile, the existing model can be extended easily to conditions with arbitrary number of radial boreholes through simple steps. The specific steps are not described in this article. In this part, the other parameters are set to basic calculation parameters, and the number of radial boreholes is changed. All radial boreholes have the same azimuth angles, and the depth difference between adjacent radial boreholes is fixed to 50 mm. The radial borehole in the middle is taken as the research object. According to the change of FIP and highest increment of maximum tensile stress, the influence of radial boreholes number is studied.


### 3.5.1. The Sensitivity of FIP

It can be seen from Figure20, with the increase of the number of radial boreholes, FIP decreases promptly first, then slows down gradually and finally decreases by 23.7% to 31.3 MPa. The maximum decrease of FIP occurs when the number of radial boreholes changes from 1 to 2. Compared with single radial borehole, FIP of double radial boreholes decreases by 20.2% due to one additional radial borehole. Therefore, in the formation where fracture is hard to initiate, multi-radial boreholes, compared with single borehole, can effectively reduce FIP and induce fracture initiation. However, when number of radial boreholes adds up to 3, continually increasing the number of radial boreholes has little effect on the FIP. At this time, it is no longer a feasible choice to reduce FIP by increasing the number of radial boreholes.Figure 20
FIP with different number of radial boreholes.


### 3.5.2. The Sensitivity of Interference

As shown in Figures21 and 22, highest increment of maximum tensile stress grows up with the increase of radial borehole number. Comparing double radial boreholes with single radial borehole, the additional radial borehole increases the maximum tensile stress on borehole wall by 6.12 MPa. Meanwhile, due to the same orientation of the radial boreholes, maximum tensile stress increasing region appears near the top and bottom of radial borehole (θ=90° and 180°). However, the increase rate slows down rapidly with the increase of radial borehole number. When the number of radial boreholes exceeds 3, the influence of radial borehole number on highest increment of maximum tensile stress nearly can be neglected.Figure 21
Maximum tensile stress increasing region with different radial borehole number.Figure 22
Highest increment of maximum tensile stress with different radial borehole number.


## 3.5.1. The Sensitivity of FIP

It can be seen from Figure20, with the increase of the number of radial boreholes, FIP decreases promptly first, then slows down gradually and finally decreases by 23.7% to 31.3 MPa. The maximum decrease of FIP occurs when the number of radial boreholes changes from 1 to 2. Compared with single radial borehole, FIP of double radial boreholes decreases by 20.2% due to one additional radial borehole. Therefore, in the formation where fracture is hard to initiate, multi-radial boreholes, compared with single borehole, can effectively reduce FIP and induce fracture initiation. However, when number of radial boreholes adds up to 3, continually increasing the number of radial boreholes has little effect on the FIP. At this time, it is no longer a feasible choice to reduce FIP by increasing the number of radial boreholes.Figure 20
FIP with different number of radial boreholes.


## 3.5.2. The Sensitivity of Interference

As shown in Figures21 and 22, highest increment of maximum tensile stress grows up with the increase of radial borehole number. Comparing double radial boreholes with single radial borehole, the additional radial borehole increases the maximum tensile stress on borehole wall by 6.12 MPa. Meanwhile, due to the same orientation of the radial boreholes, maximum tensile stress increasing region appears near the top and bottom of radial borehole (θ=90° and 180°). However, the increase rate slows down rapidly with the increase of radial borehole number. When the number of radial boreholes exceeds 3, the influence of radial borehole number on highest increment of maximum tensile stress nearly can be neglected.Figure 21
Maximum tensile stress increasing region with different radial borehole number.Figure 22
Highest increment of maximum tensile stress with different radial borehole number.


## 4. Conclusion and Recommendation

In this paper, an analytical model for fracture initiation in hydraulic fracturing guided by multi-radial boreholes is established. The model can be used to calculate the fracture initiation pressure of one particular radial borehole and the additional stress field on its wall caused by the other radial boreholes. Then, a series of sensitivity analysis is performed by examining the effects of different parameters, including the intersection angle between radial boreholes, the depth difference, radius of the radial borehole, Biot coefficient, and the number of radial boreholes. The major conclusions are:(1)
In the hydraulic fracturing guided by multi-radial boreholes, FIP declines with the decrease of intersection angle only when the intersection angle between radial boreholes is small (φ<25°). When the intersection angle is 0°, the fracture initiation pressure decreases to the minimum. Therefore, if it is necessary to obtain low fracture initiation pressure, all radial boreholes should protrude in the same direction as far as possible. Besides, the increase of intersection angle will weaken the mutual interference between boreholes in the range and degree, resulting in only the area near the main wellbore being guided, which is useless to the formation of fracture network between radial boreholes(2)
In the hydraulic fracturing guided by multi-radial boreholes, with the increase of radial depth difference between boreholes, FIP increases rapidly at first and then slowly. Therefore, to obtain low fracture initiation pressure, the depth difference between radial boreholes can be reduced to a certain extent within the allowable range of field conditions and casing strength. At the same time, the increase of the depth difference between radial boreholes will weaken the mutual interference between boreholes, which is not conducive to the formation of the fracture network between radial boreholes(3)
Although the radius of the radial borehole does not affect FIP when single radial borehole exists alone, the increase of radius will reduce FIP in the presence of multiradial boreholes. Therefore, the radius of radial boreholes can be appropriately increased in the formation with high fracture initiation pressure. At the same time, the increase of the radius of radial borehole can significantly expand the interference range of radial borehole, enhance the ability to guide fracture, and is conducive to the formation of fracture network between radial boreholes(4)
In the hydraulic fracturing guided by multi-radial boreholes, low Biot coefficient is beneficial to fracture initiation and obtain low FIP. At the same time, low Biot coefficient can enhance the interference between radial boreholes, expand the influence range of additional stress field, and conduce the fracture network connecting radial boreholes(5)
Compared with single radial borehole, multi-radial boreholes can reduce FIP significantly. Due to one additional radial borehole, FIP of double radial boreholes is reduced by 20.2%. If the number of radial boreholes continues to increase, FIP can be reduced by 23.7% at most. Thus, in the formation with a high FIP, increasing the number of radial boreholes is a good choice to obtain a lower FIP, which can reduce the requirements of operation equipment and enhance the operation safety. However, the declining rate of FIP decreases as the number of radial boreholes increasing. When the number of radial boreholes exceeds 3, FIP basically remains steady. At this time, it will no longer be a good option to reduce FIP by continually increasing the number of boreholes for the low reduction of FIP and the expense of additional radial drilling(6)
In the design of hydraulic fracturing guided by multi-radial boreholes, the analytical model can be used to calculate the FIP of each radial borehole and analyze the interference between radial boreholes to optimize the layout and parameters of radial boreholes, to obtain lower fracture initiation pressure and better formation of the fracture network between radial boreholes



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*Source: 6657788-2021-02-11.xml*