Variation rules of fracture initiation pressure and fracture starting point of hydraulic fracture in radial well

Journal of Petroleum Science and Engineering 140 (2016) 41–56

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Variation rules of fracture initiation pressure and fracture starting point of hydraulic fracture in radial well D.G. Gong, Z.Q. Qu, T.K. Guo n, Y. Tian, K.H. Tian Department of Petroleum Engineering, China University of Petroleum (East China), No. 66 , Changjiang West Road, Huangdao District, Qingdao 266580, China

art ic l e i nf o

a b s t r a c t

Article history: Received 25 June 2015 Received in revised form 20 October 2015 Accepted 7 January 2016 Available online 8 January 2016

Radial well technology in combination with hydraulic fracturing technology has gained encouraging achievements as a new method of increasing production in oil fields. Compared with conventional perforation fracturing, radial well fracturing has an obvious advantage in breaking through polluted borehole areas, and it is superior to horizontal well fracturing because of the shorter construction period, less consumption of fracturing fluid and lower damage to the reservoir. Currently, in China, the study on the position of fracture starting point and fracture propagation of radial well is still at the preliminary stage and the fracture initiation pressure and position of fracture starting point remain unclear. Consequently, it's difficult to design radial well completion parameters (length of radial well, diameter of radial well and fracturing truck units) precisely and implement the technology more efficiently. Based on fluidsolid coupling effect and the maximum tensile-stress criterion, this paper follows the concept of dynamic analysis and analyzes the influence rule of the length, diameter and azimuth of radial well, horizontal insitu stress and natural fracture on fracture initiation pressure and fracture starting point under the stress of strike-slip fault by using ABAQUS to simulate and study local stress accumulation situation caused by drilling vertical well section, radial well section and fracturing section through finite element method. The result shows that initiation pressure and distance between well and fracture starting point increases as the length, diameter and azimuth of radial well section rise. Azimuth is most influenced, followed by length, and lastly diameter of radial well section. When the horizontal in-situ stress ratio (sH:sh) is decreased from 1.9 to 1.1, if the azimuth is 0°, the initiation pressure increases by 41.35%; if the azimuth is 90°, the initiation pressure declines 0.8%. However, the positions of these two fracture starting point remain unchanged. The permeability increases by four orders of magnitudes and the fracture initiation pressure goes up 34.5%, with no influence on the fracture starting point. When there existed natural fractures in the reservoir, the intersection between fracture section and radial well section firstly shows fracturing. Besides, the fracture initiation pressure and the hydrostatic fluid column pressure in vertical wellbore are equivalent. Reduced length and diameter of radial well section as well as reservoir permeability and properly increased azimuth of radial well are conducive to fracture at the toe end of the radial well section. On the contrary, fracture at shaft linings of vertical well section is easily occurred. The research result can be used to predict the direction of fracture propagation to some extent and is favorable for designing parameters of radial well completion and fracturing operation. & 2016 Elsevier B.V. All rights reserved.

Keywords: Fracture initiation pressure Fracture starting point Hydraulic fracturing Radial well Fluid–solid coupling Finite element analysis

1. Introduction Radial well is a horizontal well whose radius of curvature is much smaller than that of conventional well. The boreholes are usually produced by hydraulic jetting or drilling, showing the length of 30–100 m and the diameter of 25–50 mm (Wade et al.,1992; Wu, 1994; Li et al., 2000). The technique combined radial

n

Corresponding author. E-mail addresses: [email protected] (D.G. Gong), [email protected] (T.K. Guo). http://dx.doi.org/10.1016/j.petrol.2016.01.006 0920-4105/& 2016 Elsevier B.V. All rights reserved.

well and hydraulic fracturing as an emerging stimulation treatment has been used in Jiangsu, Shengli and Liaohe oil fields. This technique has the following advantages: (1) the borehole of radial well performs the function of guiding the direction of fractures and increasing the penetration of fractures; (2) through the branches of fracture, a larger producing area of reservoir can be connected. Compared with conventional fracturing, the fracturing effect is better and the effective period is longer; (3) the radial branches reduce the fracture initiation pressure as well as the operation pressure; (4) the height of fracture propagation is effectively controlled so as to prevent the water layers from

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connecting. The thin interbeds can be chosen for fracturing. The technique has a shortened operation time and causes less damage to oil layers from the foreign liquid. Currently, research on the fracture initiation pressure in oriented perforation is large in amount, and the following three methods are commonly used: experimental method, numerical analysis method and numerical simulation method. Ketterij, Huang and Lhomme used experimental method on the influence of perforation parameters of highly deviated well on fracture initiation pressure and fracture starting point. Stress calculations were performed to obtain the pattern of influence of perforation angle on fracture initiation pressure and fracture propagation (van de Ketterij and de Pater, 1997). True triaxial apparatus was used to impose triaxial stress on the rock samples. By changing such parameters as depth and diameter of perforation and the included angle α between the axis of perforation and the maximum horizontal stress, the variation rules of fracture initiation pressure under different perforation parameters was analyzed (Huang and Li, 2007). The rule of fracture initiation was studied by using horizontal fluid to impose pressure on open-hole section through experimental method. It showed that the fracture initiation had a direct connection with the seepage flow mechanism and the microscopic structure of the rock mass (Lhomme et al., 2002). Luo and Zhu employed numerical analysis method to solve the problem of hydraulic fracture initiation in oriented perforation. Based on the stress state on the intersection between the borehole and perforation tunnel in the deviated well with casing perforation, Luo Tianyu proposed the method to calculate the tangential stress near the hole edge. Moreover, the calculations of stress field at the intersection between the two holes, fracture initiation pressure and fracture starting point in the presence of micro-circular plane were investigated. The method also was discussed for the calculation of fracture initiation pressure at the intersection of the two holes in annular space with tight cementation (Luo et al., 2007). The analytical model of stress was derived around the cased wells by Zhu Haiyan. Combining with Hossain's model and the maximum tensile-stress criterion, the model of predicting the fracture initiation pressure and fracture initiation angle on the perforation tunnel during the oriented casing perforation was established. Later, the effective stress theory was utilized to establish the calculation model of fracture initiation pressure around the wellbore in shale gas reservoir. The influence of stratum dip direction and in-situ stress azimuth on the fracture initiation pressure also was analyzed (Zhu et al., 2013, 2014). Numerical simulations based on computer have been carried out for hydraulic fracture initiation in oriented perforation. The influence of perforation on the fracture initiation pressure and fracture morphology in hydraulic fracturing was discussed. The fracture propagation models were established under different positions of perforation and different far-field principal stress. The impact of perforation parameters on fracture initiation pressure and fracture morphology was analyzed (Zhuang et al., 2008). The numerical model of fluid-solid coupling was constructed by using finite element method. The fracture initiation and propagation near the wellbore were discussed (Salehi and Nygaard, 2015). A 3D finite element method was applied in predicting the fracture initiation pressure under helix distribution perforation. The 3D numerical model of wellbore and the formation was built under cased hole completion (with the existence of cement mantle and casing) (Biao et al., 2011). These scholars produced many research results of perforation parameters and fracture initiation pressure, but few of them paid attention to the research of such radial well completion parameters as length, diameter and fracture initiation pressure. As for

the initiation position of radial well studies, no result has been found. In China, the study on fracture initiation and propagation in the fracturing of radial well is still at the preliminary stage. This technique is more frequently applied in the development of coalbed methane. The integration of operation of short-radius radial horizontal well and hydraulic fracturing was proposed for the increase of gas production in Hunchun Basin that has lowrank coal and good permeability and favorable gas-bearing conditions (Xian et al., 2010). Michael Patrick Megorden suggested the use of radial well in coalbed to improve and guide the geometry of fractures, and a certain effect was achieved (Megorden et al., 2013). At present, the fracture initiation pressure and position are still unclear; as a result, such radial well completion parameters as length and diameter of radial well and fracturing truck unit cannot be effectively designed. Jiangsu Oil Field has taken the lead in carrying out radial well fracturing test in China and achieved satisfactory results, which proves the feasibility of radial well fracturing. Jiangsu Oil Field is located in Gaoyou sinking area in the south of Northern Jiangsu Basin where exist many strike-slip faults, such as Shigang, Chajian and Wubu faults. The radial well parameters and rock parameters applied in this study are all from X zone in Jiangsu Oil Field and the target system belongs to strikeslip faults. In this paper, the author firstly introduces the theoretical basis and finite element discretization method of establishing fluidstructure interaction numerical model by ABAQUS, revealing the stress–percolation interaction mechanical mechanism of Pore Fluid-Stress module in ABAQUS software, which helps to better understand the applicable range and assumption conditions of the model. As the target system pertained to strike-slip fault area, strike-slip fault's stress mechanism was selected as the ground stress mechanism and 3D numerical modeling study was carried out concerning the radial well fracture initiation pressure and fracture starting point. Elasto-plastic model is adopted as the material property and given that the rock around the shaft lining is subject to the triaxial pressure and stress incurred by well sections removal. Drucker–Prager yield criterion is applied to judge the yield stress. In the process of hydraulic fracturing, the maximum tensile–stress criterion is adopted to judge whether fracture occurs to rock under tensile stress. The numerical simulation process is, in order, divided into the following three steps: simulation of the process of removing vertical well section, simulation of the process of removing radial section and simulation of fracturing process. Because the stress concentration phenomenon caused by removing rocks from drilling the vertical well section with initial stress and radial well section is taken into consideration (Hubbert and Willis, 1956; Valko et al., 1995; Aadnoy, 1988), the numerical model established is able to better simulate the stress changing situation around vertical well section and radial well section. Real reservoir rock parameters were taken as the criterion of the model, and the single factor influence on fracture initiation pressure and position was studied by using different parameters like the azimuth, diameter and length of radial well, horizontal in-situ stress ratio, permeability and natural fracture position through analysis on the numerical simulation result. Meanwhile, we also made in-depth analysis about “the region susceptible to fracture initiation” at the vertical well section and the toe ends of the radial well in case of different azimuth, diameters and lengths of radial well, horizontal in-situ stress ratios, permeability and natural fracture positions, and found how “the region susceptible to fracture initiation” regularly distributes. The research results are conducive to designing the radial well completion parameters and fracturing operation parameters by predicting the propagation direction of fractures to some extent.

D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

2. ABAQUS finite element discretization method and fluid– solid coupling controlling equation set

⎛ ∇p ⎞ −nT kk r ⎜ o − g⎟ = qo ⎝ ρo ⎠

Well drilling and completion can lead to local changes of effective stress of porous medium matrix and changes the permeability and porosity of formation. During the pumping of fracturing fluid, the fluid in the pores of rock will have an impact on the strength and deformation of rock. As a result, it is necessary to consider the coupling between the stress field of rock and the seepage field (Jia et al., 2009; Wang et al., 2004). Based on the above consideration, “Pore Fluid-Stress” module in ABAQUS software is utilized to study the process of removing open-hole vertical well section and radial well section from the formation and the completion process of hydraulic fracturing, as well as to investigate how the stress changes in key areas; in this way, the influence law of ground stress field and well completion parameters on fracture initiation pressure and position of radial well are determined. Brief introduction is given to the internal mechanical mechanism of “Pore Fluid-Stress” module and Fluid–Solid coupling controlling equation set. 2.1. ABAQUS finite element discretization method By inducting shape functions, the ABAQUS platform takes shape functions as the interpolation functions in the unit. Displacement of any point in the unit is expressed by nodal displacement as the weighting functions in weighted residual method to deal with the external load and make the distributed force equivalent to the single force and moment on the nodes (Yue and Li, 2006; Chen et al., 2009). Definition of shape function

⎧ u = Nu u¯ ⎪ ⎨ ε = Bu¯ ⎪ p = N p¯ p o ⎩ o

(1) T

⎡

∫V δϵT Dep ddtϵ dV + ∫V δ ϵT Dep ⎢⎣ m

(s w + po ϵ) dpo ⎤ ⎥ dV 3Ks dt ⎦

dp

−

∫V δ ϵT m (sw + po ξ ) dto dV

=

∫V δ uT ddtf dV + ∫S δ uT ddtt dS dp¯ du¯ df +C o = dt dt dt

2.2. Fluid solid coupling control equations Combining (3) and (4), the stress-seepage coupling Eq. (5) is obtained. Using ABAQUS finite element solver to solve the equation, and the rules of distribution of stress, strain, displacement, porosity, permeability and saturation of the region of interest are obtained.

⎧ df ⎫ ⎡ K C ⎤ d ⎧ u¯ ⎫ ⎡ 0 0⎤ ⎧ u¯ ⎫ ⎪ ⎪ ⎨ ⎬ ⎨ ⎬ +⎢ = ⎨ dt ⎬ ⎥ ⎢⎣ ⎥ E G ⎦ dt ⎩ p¯o ⎭ ⎣ 0 F ⎦ ⎩ p¯o ⎭ ⎪ ⌢ ⎪ ⎩ f ⎭

(5)

where

⎡

⎛

mT Dep ⎞ ⎤ ⎟ B⎥ dV 3Ks ⎠ ⎥⎦

E=

∫V NpT ⎢⎢⎣ so ⎜⎝ mT −

F=

∫V (∇Np )T kkr ∇Np dV

G=

∫V NTp ⎨ so ⎢⎢ ⎜⎜ 1 K−s n

⌢ f =

⎧ ⎪

⎡⎛

⎪

⎣⎝

⎩

−

⎫ ⎪ mT Dep m ⎞ ⎤⎥ ⎟⎟ (so + po ξ ) + ξn + n so ⎬ Np dV 2 ⎪ ⎥ K (3Ks ) ⎠ ⎦ o⎭

∫S NTp qob dS − ∫V (∇Np )T kkr gdv

⎧ ⎫ dp mT Dep m 1−n s ⎨ ξn + n o + so [ ⎬ o =0 +⎪ − ](so + po ξ ) ⎪ 2 3 K K 3 K ( ) o s ⎩ ⎭ dt S ⎪

Formula (9) is derived in Appendix A. Formula (10) is derived in Appendix B. Formula (11) is derived in Appendix C.

p¯o is pore pressure for cell nodes; a, b is an arbitrary function; A¯ is the control equation; B¯ is the continuity equation for the boundary; qob is flow of boundary; Np is the form function.

3. Numerical Simulation of hydraulic fracture initiation in radial well

(3)

3.1. Introduction of the model

⎛ ⎡ ⎛ ∇p ⎞⎤ mT Dep ⎞ dε ⎟⎟ So ⎜⎜ mT − − ∇T ⎢ k 0k r ⎜ o − g⎟ ⎥ 3KS ⎠ dt ⎝ ρo ⎠ ⎥⎦ ⎣⎢ ⎝

3

(4)

(9)

Using Galerkin's method, formula (10) and (11) are taken as A¯ and B¯ in formula (2), respectively. Introducing formula (1) into formula (2) and supposing a = − b , it is simplified into formula (4)2,3:

2

⌢ dp¯ du¯ + Fp¯o + G o = f dt dt

(2)

Introducing (1) into (9), the finite element formulation of solid phase is obtained1:

1

(11)

where u¯ is displacement for element nodes;

¯ ¯ =0 + ∫ b BdS ∫V aT AdV S

K

E

43

⎪

(10)

3.1.1. Assumptions (1) The reservoir is made of isotropic homogeneous material with oil reservoir saturation of 1. (2) Assuming the reservoir matrix as elasto-plastic material, its mechanical behavior obeys the elastic damage theory. It is believed that the rock satisfies the maximum tensile-stress criterion. When the maximum tensile stress acting on the rock exceeds its tensile strength, it will undergo tensile damage and fracture. (3) Before fracturing, the fluid in the radial well is almost in the static state. The frictional resistance of fracturing fluid in radial well and that of perforation tunnel are not considered in the calculations.

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Fig. 1. Stereogram of numerical model. (a) Part and geometry of vertical well and radial well, (b) schematic diagram of geometry of the model, (c) CPE4P element division. Table 1 Basic parameters of Gao X Radial Well in Jiangsu Oilfield. Parameter

Value

Parameter

Value

Well No. Position of perforation permeability Oil reservoir saturation Initial pore pressure Initial porosity Maximum horizontal principal stress(sH) Minimum horizontal principal stress(sh) Vertical stress(sv)

Gao X well 1950 m 0.1
10  3 μm2 1 18 Mpa 0.05 52.9 Mpa 36.6 Mpa 47.7 Mpa

Tensile strength of the rock Constitutive model of the rock Plastic yield criterion Cohesive element damage evolution criterion Shear dilatancy of the rock Outer diameter of vertical well Diameter of vertical well Elastic modulus of the reservoir Poisson's ratio

3.0 Mpa Elastic plastic model Drucker–Prager Elastic damage evolution criterion 54.0 139.7 mm 30 mm 17 GPa 0.2

3.1.2. Basic model parameters Take the Gao X radial well in Jiangsu Oilfield as an example, and a numerical model of 10 m
10 m
60 m is established. The maximum of grid number of the model exceeds 100 thousand. See Fig. 1(a)–(c). The model parameters were sourced from the parameters of Gao X Radial Well in Jiangsu Oilfield. The basic parameters are listed in Table 1. Considering the state of triaxial stress on shaft lining and the situation removing vertical well section and radial well section having an compressive stress on the formation, elasto-plastic model is taken as the analysis of constitutive model. Drucker–Prager yield criterion is applied to judge whether the rock near borehole zones reaches the plastic state after being subject to shear stress in the

well sections removal process. In this study, followed the thought of dynamic analysis, studied local stress superposition caused by drilling vertical well section, perforating radial well section and fracturing, combining the maximum tensile–stress damage criterion, we analyzed the fracture initiation pressure and fracture starting point during the fracturing in radial well. The initial stress field was balanced by using the keyword *Geostatic (Dai et al., 2012). The removal of vertical and radial well sections was simulated by stiffness discount using the keywords *Model Change, remove and command *field (Wang and Chen, 2006). The injection of fracturing fluid and the changes of pressure were simulated by using the keywords *Dsload and *Amplitude (Zhang et al., 2012; Abell and Choi, 2012; Brooker and Ronalds, 2001).

Fig. 2. Schematic diagram of the simulated removal of well sections and fracturing. (a) Balancing of initial stress field, (b) simulation of the removal of vertical section, (c) simulation of the removal of radial section, (d) simulation of the injection of fracturing fluid.

D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

22 20 18 16

70

14 12

60

10 8

50

6 4

40

fracture starting point/m

initiation pressure/MPa

24

initiation pressure the distance between fracture starting point and vertical well

2 0

30

azimuth of radial well

2

0

6

maximum stress at the wall of vertical well maximum stress at the toe of radial well

4

-2

2

-4

0

-6

-2

-8

-4

-10

-6

maximum stress at the toe of radial well /MPa

90

maximum stress at the wall of vertical well /MPa

(b)

(a)

80

45

azimuth of radial well

Fig. 3. Analysis of the influence of azimuth of radial well on the fracture initiation in radial well. (a) Rule of variation of fracture initiation pressure and fracture starting point with the azimuth of radial well, (b) rule of variation of maximum stress at the wall of vertical well and at the toe of radial well with the azimuth of radial well.

The analysis consisted of the following four steps: (1) (2) (3) (4)

Balancing of initial stress field (Fig. 2(a)) Simulation of the removal of vertical section (Fig. 2(b)) Simulation of the removal of radial section (Fig. 2(c)) Simulation of the injection of fracturing fluid (see Fig. 2(d))

3.2. Calculation results and analysis 3.2.1. Influence of azimuth of radial well on the fracture initiation pressure and fracture starting point Using the length of 50 m as the benchmark, the variation rules of fracture initiation pressure and fracture starting point was analyzed when the azimuths (included angle between the perforation direction of radial well and the direction of maximum horizontal principal stress) are 0°, 15°, 45°, 75° and 90°, respectively. Under ABAQUS software, the initial in-situ stress is converted into three normal stresses sxx, syy and szz under the coordinate system of the well bore and into three tangential stresses τxy, τyz and τzx. Thus the redistribution of in-situ stress field is achieved. The normal stresses and tangential stresses value under different azimuths are shown in Table 2. (1) Analysis of fracture initiation pressure and fracture starting point Numerical simulation showed that the plastic deformation of the rock does not occur in the process of removing the vertical and radial well sections. The fracture initiation pressure increased as the azimuth of radial well rose (Fig. 3(a)).The azimuths of radial well went up from 0 to 90, and the corresponding fracture initiation pressure jumped by 40.84%.When the azimuth was equal to or smaller than 45°, the fracture starting point was maintained on the inner surface of radial well 0.499 m away from the vertical well section (Figs. 4(c) and 5(c)). When the azimuth was larger Table 2 Normal stress and shear stress values near the shaft under different azimuth of radial well. Azimuth of radial well (deg.)

szz (MPa)

sxx (MPa)

syy (MPa)

τxy (MPa)

0 15 45 75 90

47.7 47.7 47.7 47.7 47.7

52.9 51.8 44.7 37.6 36.6

36.6 37.6 44.7 51.8 52.9

0  4.1 8.2  4.1 0

than 45°, the fracture starting point changed. The maximum tensile stresses exceeding 3 MPa first occurred on the inner surface of the radial well at 14.96 m from the vertical well section (Fig. 6(c)). This distance kept constant and increased by 14.5 m. With the increase of azimuths of radial well, the nephogram of stress distribution near the perforation tunnel changed. When the azimuth of vertical well was smaller than or equal to 45°, the maximum tensile stress occurred on the upper and lower ends of the perforation tunnel (Fig. 7(a) and (b)), which was conducive to forming longitudinal fracture. When the azimuth of radial well was larger than 45°, the maximum tensile stress occurred on the left and right ends of the perforation tunnel (Fig. 7(c)), facilitating in forming transverse fractures. According to the minimum energy principle, the fracture will propagate along the direction of minimum resistance. The stress distribution around the borehole of vertical well section differs with the azimuth of radial well. Different azimuths bring about different stress distributions near the radial well and different angle distributions of plane of minimum resistance, and then the fracture initiation pressure and starting point would change. (2) Analysis of “the region susceptible to fracture initiation” When the tensile strength reaches at a certain position of the radial well, the phenomenon of stress concentration will also occur at the shaft lining of vertical well section (Fig. 8(a)) and at the toe of radial well section (Fig. 8(b)). The maximum stress at the shaft lining of vertical well section and at the toe of radial well section indicates whether these two regions are susceptible to fracture initiation. In ABAQUS, the compressive stress is negative, tensile stress positive. Calculation showed that as the azimuth of radial well increased, the compressive stress acting on shaft lining of vertical well section (sum of in-situ stress, pore pressure and pressure of fracturing fluid) climbed gradually. The maximum stress occurred at the azimuth of 45°. When the azimuth was above 45°, the compressive stress acting on the shaft lining of the vertical well decreased. Fig. 3(b) shows the variation rules of maximum stress on the shaft lining of vertical well section. With the increase of azimuth, the compressive stress acting on the toe of radial well rose as well, reaching the maximum under the azimuth of 45°. When the azimuth was larger than 45°, tensile stress acting on the toe of radial well rose sharply. After reaching the peak, the tensile stress gradually decreased. According to the regions with large tensile stress, the maximum tensile stress was 2.50 MPa at the toe of radial well under

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D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

Fig. 4. The cloud image at the azimuth of 0°. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

the azimuth of 75°; therefore, it was “the region susceptible to fracture initiation”. 3.2.2. Influence of diameter of radial well on fracture initiation pressure and fracture starting point (1) Analysis of fracture initiation pressure and fracture starting point Using the azimuth of 90° and the radial well length of 30 m as

benchmark, the variation rules of fracture initiation pressure and fracture starting point under the inner diameter of perforation tunnel on the radial well of 10 mm, 20 mm, 30 mm, 40 mm and 50 mm were analyzed, respectively. Numerical simulation showed that under the inner diameter of 10 mm, the fracture initiation pressure was 54.6 MPa and the fracture initiation occurred 8.46 m away from the vertical well section (Fig. 9 (c)). Under the inner diameter of 50 mm diameter, the fracture

Fig. 5. The cloud image at the azimuth of 45°. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

47

Fig. 6. The cloud image at the azimuth of 0°. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

initiation pressure increased by 3.66%. With the increase of inner diameter, the fracture initiation pressure went up, and the distance from the fracture starting point to the vertical well rose as well. When the inner diameter was above 40 mm, the fracture initiation occurred 11.45 m away from the vertical well section and roughly stayed unchanged. Compared to the diameter of 10 mm, the distance from the vertical well section increased by 2.49 m (Fig. 10(c)). Thus a proper inner diameter of radial well section can help control the fracture starting point. However, the distance from the fracture starting point to the vertical well section does not increase linearly with the increase of the inner diameter (Fig. 11(a)). As the diameter of radial well increased, there was an ascent of the contact area between the fracturing fluid and the rock, resulting in serious filtration. This would further put on the required displacement and the fracture initiation pressure. (2) Analysis of “the region susceptible to fracture initiation” With the increase of the inner diameter of radial well section,

the compressive stress acting on the area of shaft lining of vertical well gradually decreased, and the tensile stress at the toe of radial well section reduced as well (Fig. 11(b)). As the diameters of radial well rose from 10 mm to 30 mm, the maximum stresses across the stress-concentration region at the toe of the radial well section were 2.42 MPa, 2.26 MPa and 2.10 MPa, respectively. In these cases, the regions were “regions susceptible to fracture initiation”. However, the stress acting on the shaft lining of the vertical well did not satisfy the requirements, therefore the shaft lining of the vertical well section was not “the region susceptible to fracture initiation”. 3.2.3. Influence of the length of radial well on fracture initiation pressure and fracture starting point Using the azimuth of radial well of 90° and the inner diameter of 30 mm as benchmark, the variation rules of fracture initiation pressure and fracture starting point was analyzed when the length

Fig. 7. The stress distribution image of the transverse section of the radial well at 0°, 45° and 90°. (a) Stress distribution in 0° azimuth, (b) stress distribution in 45° azimuth, (c) stress distribution in 90° azimuth.

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D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

Fig. 8. Comparison of stress distribution over the wall of vertical well and at the toe of radial well at the azimuth of 90°. (a) Stress-concentration region over the wall of vertical well at the azimuth of 90°, (b) stress-concentration region at the toe of radial well at the azimuth of 90°.

of radial well section was 10 m, 20 m, 30 m, 40 m and 50 m, respectively. (1) Analysis of fracture initiation pressure and starting point As shown from the numerical simulation, the fracture initiation pressure reached the minimum of 53.4 MPa when the length of radial well section was 10 m. At this moment, the fracture starting point was 0.49 m away from the vertical well section (Fig. 12(c)). As the length of radial well section increased, the fracture initiation pressure went up to a certain extent. The fracture initiation pressure was 57.5 MPa when the length of radial well section was 50 m, and the fracture starting point was 14.96 m from the vertical well section (Fig. 13(c)). The initiation pressure rose by 7.68% and the distance by 14.5 m. According to by the numerical simulation results, increasing the length of the radial well section leads to fracture initiations in the region far away from the vertical well section (Fig. 14(a)). As the radial well length increased, there was an ascent of the contact area between the fracturing fluid and the rock, resulting in higher filtration. With greater discharge that is required, the fracture initiation pressure increased correspondingly.

(2) Analysis of “the region susceptible to fracture initiation” With the increase of length of radial well, the compressive stress acting on the shaft lining of vertical well section gradually decreased, so did the tensile stress acting on the toe of radial well (Fig. 14(b)). As the analysis of the regions with large tensile stress indicated, when the length of radial well section increased from 10 m to 40 m, the maximum stresses across the stress-concentration region at the toe of radial well were 2.37 MPa, 2.15 MPa, 2.10 MPa and 2.01 MPa, respectively; therefore they were “the regions susceptible to fracture initiation” at the toe of radial well. However, the stress acting on the shaft lining of vertical well section did not satisfy the requirement, and the shaft lining of vertical well section was not “the region susceptible to fracture initiation”. 3.2.4. Influence of horizontal in-situ stress ratio on fracture The inner diameter of radial well section of 30 mm and the length of radial well section 50 m were taken as reference. The overburden pressure of 47.7 MPa and the maximum horizontal in-

Fig. 9. Schematic diagram of the distribution of fracture starting point and stress in 10 mm diameter. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

49

Fig. 10. Schematic diagram of the distribution of fracture starting point and stress in 50 mm diameter. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

14

initiation pressure/MPa

62

initiation pressure the distance between fracture starting point and vertical well

60

13

12

58 11 56 10

54

9

52 50

10

20

30

40

50

fracture starting point/m

64

8

diameter of radial well/mm

-5.0

3.0

maximum stress at the wall of vertical well maximum stress at the toe of radial well

-5.5

2.5 -6.0

-6.5

2.0

-7.0 1.5 -7.5

-8.0

10

20

30

40

50

1.0

maximum stress at the toe of radial well /MPa

(b) maximum stress at the wall of vertical well /MPa

(a)

diameter of radial well/mm

Fig. 11. Analysis of the influence of diameter on fracture initiation in radial well. (a) Rule of variation of fracture initiation pressure and fracture starting point with the diameter of radial well, (b) rule of variation of maximum stress over the wall of vertical well and at the toe of radial well with the diameter of radial well.

situ stress of 52.9 MPa kept constant. The minimum horizontal insitu stress was changed properly, and the fracture initiation pressure and starting point were solved under the horizontal in-situ stress ratios (sH:sh) of 1.1, 1.3, 1.5, 1.7 and 1.9, respectively. (1) Analysis of fracture initiation pressure and starting point ① Azimuth of 0° As the minimum horizontal principal stress increased, the fracture initiation pressure climbed gradually. When the horizontal in-situ stress ratio was 1.9, the fracture initiation pressure was 41.6 MPa; when the horizontal in-situ stress ratio was 1.1, the fracture initiation pressure was 58.8 MPa and increased by 41.35%. The fracture starting point did not change, and the distance from the vertical well section was 0.49 m. See Fig. 15(a). ② Azimuth of 90° The calculation showed a different variation rules compared with the situation where the azimuth was 0°. With the increase of

the minimum horizontal principal stress, the fracture initiation pressure decreased to a certain extent. When the horizontal in-situ stress ratio was 1.9, the fracture initiation pressure was 59.01 MPa; when the horizontal in-situ stress ratio was 1.1, the fracture initiation pressure was 58.54 MPa and decreased by 0.80%. The fracture starting point did not change, and the distance from the vertical well section was maintained at 14.95 m (Fig. 15(b)). (2) Analysis of “the region susceptible to fracture initiation” ① Azimuth of 0° As the minimum horizontal principal stress increased the compressive stress acting on the shaft lining of vertical well section went up gradually, so did the tensile stress at the toe of radial well (Fig. 16 (a)). When the horizontal in-situ stress ratio was 1.5, the compressive stress at the toe of radial well reached the maximum of 4.5 MPa. As the minimum horizontal principal stress rose, the compressive stress at the toe of radial well decreased, while the tensile stress increased.

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Fig. 12. Schematic diagram of the distribution of fracture starting point and stress in 10 m length. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

When the horizontal in-situ stress ratio was 1.1, the tensile stress reached the maximum of 2.21 MPa. In this case, the toe of radial well was “the region susceptible to fracture initiation”. However, the stress acting on the shaft lining of vertical well did not satisfy the requirement, and the shaft lining of vertical well was not “the region susceptible to fracture initiation”.

② Azimuth of 90° Along with the increase of minimum horizontal principal stress, the compressive stress acting on the shaft lining of vertical well section increased gradually, so did the tensile stress at the toe of radial well (Fig. 16(b)). When the horizontal in-situ stress ratio was 1.1, the compressive stress reached the maximum of 15.5 MPa,

Fig. 13. Schematic diagram of the distribution of fracture starting point and stress in 50 m length. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

60

20

16

58 12 56 8

54

4

52 50

fracture starting point/m

initiation pressure/MPa

24

initiation pressure the distance between fracture starting point and vertical well

10

20

30

40

50

0

radial well length/m

2.8

-2

maximum stress at the wall of vertical well maximum stress at the toe of radial well

-4

2.6

2.4

-6

2.2

-8

2.0 -10 1.8 -12 1.6 -14

10

20

30

40

maximum stress at the toe of radial well /MPa

64

maximum stress at the wall of vertical well /MPa

(b)

(a)

62

51

50

radial well length/m

Fig. 14. Analysis of the influence of radial well length on fracture initiation. (a) Rule of variation of fracture initiation pressure and fracture starting point with the length of radial well, (b) rule of variation of maximum stress on the wall of vertical well and at the toe of radial well with radial well length.

and the tensile stress reached the maximum of 2.52 MPa. Calculation showed that when the horizontal in-situ stresses ratio changed from 1.7 to 1.1, the toe of radial well was “the region susceptible to fracture initiation”. However, the stress acting on the shaft lining of vertical well section did not satisfy the requirement, and therefore the shaft lining of vertical well was not “the region susceptible to fracture initiation”. 3.2.5. Influence of permeability on fracture initiation pressure and starting point Taking inner diameter of 30 mm, length of 50 m and azimuth of 0° of radial well as reference, the permeability of numerical models was set respectively as 0.01
10  3 μm2, 0.1
10  3 μm2, 1
10  3 μm2, 10
10  3 μm2 and 100
10  3 μm2 to study the effect of permeability on fracture initiation pressure and starting point. (1) Analysis of fracture initiation pressure and starting point The result of numerical stimulation showed that the fracture initiation pressure increased correspondingly as the permeability increased. When the permeability was 0.01
10  3 μm2, the fracture initiation pressure was 37.1 MPa (Fig. 17(c)). With the increase of permeability, the fracture initiation pressure increased obviously; when the permeability was 100
10  3 μm2, the fracture initiation pressure was 49.9 MPa, increased by 34.5% (Fig. 18(c)). Meanwhile, it was found in the research that when the permeability changed from 10
10  3 μm2 to 100
10  3 μm2, the

fracture initiation pressure changed in a small extent, increased by 3.33%. There was a certain range of the influence of permeability on fracture initiation pressure from the analyzation and when the permeability increased to a certain degree, the fracture initiation pressure tended to be a constant. In addition, it was found in the research that changing permeability would not cause change of fracture initiation positions. Compared fracture initiation points with different permeability, it was found that fracture initiation position did not change and the fracture starting point was 0.49 m from the shaft lining of vertical well section, as shown in (Fig. 19 (a)). With the increase of permeability, the permeable channels of rock increased, which is conducive to fluid flow and is less likely to cause pressure-out phenomena. Therefore, the fracture initiation pressure increased. (2)Analysis of “the region susceptible to fracture initiation” With the increase of permeability, the compressive stress acting on the shaft lining of vertical well section and toe end of radial well section decreased gradually (Fig. 19(b). Analyzed on region with relatively larger compressive stress, when the permeability increased from 0.01
10  3 μm2 to 100
10  3 μm2, the maximum point in the region of stress around shaft lining of vertical well section and toe end of radial well is 1.34 MPa at the permeability of 100
10  3 μm2. In this case, there was not “the region susceptible to fracture initiation”, indicating that the change of permeability has little impact on forming “the region susceptible to fracture initiation”.

Fig. 15. Influence of horizontal in-situ stress ratio on fracture initiation in radial well. (a) Rule of variation of fracture initiation pressure and starting point with horizontal insitu stress ratio at the azimuth of 0°, (b) rule of variation of fracture initiation pressure and starting point with horizontal in-situ stress ratio at the azimuth of 90°.

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D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

Fig. 16. Influence of horizontal in-situ stress ration on the stress over the wall of vertical well and at the toe of radial well. (a) Rule of variation of maximum stress over the wall of vertical well and at the toe of radial well along with the changes of horizontal in-situ stress ratio at the azimuth of 0°, (b) rule of variation of maximum stress over the wall of vertical well and at the toe of radial well along with the changes of horizontal in-situ stress ratio at the azimuth of 90°.

3.2.6. Influence of natural fractures on fracture initiation pressure and starting point The inner diameter of radial well of 30 mm, the length of radial well of 50 m and the azimuth of 90° were taken as reference. “Cohesive” module in ABAQUS was added to the positions at 10 m, 20 m, 30 m, 40 m and 50 m from the vertical well section, respectively, in order to simulate natural fractures. Parameter SDEG was used to determine whether the natural fractures began to fracture. When SDEG 40.5, the fracture was considered to start fracturing; when SDEG 41, the fractures had completely fractured (Alfano et al., 2007; Yao et al., 2015). Then the fracture initiation pressure and starting point in radial well were predicted in the presence of natural fractures. The cohesive element material parameters were shown in Table 3.

As shown by numerical simulation, the natural fractures at all five positions had split open under the pressure of 19.5 MPa (1950 m hydrostatic liquid column pressure). Take the fracture at 40 m as an example (Fig. 20 (a)). After the radial well section removed, the maximum SDEG of “Cohesive” module was 0.3121, which was not up to the standard of SDEG 4 0.5. After the vertical wellbore was filled with fracturing fluid, SDEG ¼0.5096 (Fig. 20 (e)), indicating that the natural fractures had begun to fracture. At this moment, the maximum stresses acting on the inner wall of vertical well section and the radial well section were  15.5 MPa and  16.4 MPa, respectively, far lower than the fracture initiation pressures. It is obvious that the natural fractures, if any, are the places where fractures firstly occur.

Fig. 17. Schematic diagram of the distribution of fracture starting point and stress in 0.01
10  3 μm2 permeability. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

53

Fig. 18. Schematic diagram of the distribution of fracture starting point and stress in 100
10  3 μm2 permeability. (a) Simulation of the removal of vertical section, (b) simulation of the removal of radial section, (c) the cloud image of the maximum principal stress at the initial stage of fracture initiation, (d) the cloud image of pore pressure at the initial stage of fracture initiation.

4. Conclusions ABAQUS software for finite element analysis was used to analyze the phenomenon of stress concentration caused by the drilling and removal of rock mass in vertical well section and radial well section subject to initial stress with the consideration of fluid–solid coupling. The 3D fracture initiation model of radial well was established. Under the mechanism of stress in strike-slip fault, the influence of in-situ stress, azimuth, length and diameter of radial well and natural fractures on fracture initiation pressure and fracture starting point was analyzed. The following conclusions are reached: 1. Under the assumption condition of the numerical stimulation, as the length, diameter and azimuth of radial well increase, fracture initiation pressure and distance from fracture initiation

Table 3 The material parameters of Cohesive element. Parameters

Numerical value Parameters

Numerical value

Nominal stress normalonly mode Nominal stress first direction Nominal stress second direction Top/bottom coefficient Damage evolution type

3 MPa

Elastic E/Knn

3
104 MPa

3 MPa

Elastic G1/Kss

3
104 MPa

3 MPa

Elastic G2/Ktt

3
104 MPa

6
10  7 m3/s Displacement

Mode mix ratio Energy Degradation Maximum

position to vertical well section also increase, vice versa; when the length increases from 10 mm to 50 m, the fracture initiation pressure increases by 7.68% and the distance from the fracture

Fig. 19. Analysis of the influence of permeability on fracture initiation in radial well. (a) Rule of variation of fracture initiation pressure and fracture starting point with the permeability of rock, (b) rule of variation of maximum stress over the wall of vertical well and at the toe of radial well along with the changes of permeability.

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D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

Fig. 20. Schematic diagram of the distribution of fracture starting point and stress with natural fracture. (a) A fractures at 40 m, (b) simulation of the removal of vertical section, (c) simulation of the removal of radial section, (d) SDEG calculation results of cohesive element after radial well segment removal, (e) SDEG calculation results of cohesive elements at the early stage of fracture initiation, (f) the partial enlargement of graph (e).

initiation position to vertical well section increases by 14.5 m; when the diameters increase from 10 mm to 50 mm, the fracture initiation pressure increases by 3.66% and the distance from the fracture initiation position to vertical well section inches up by 2.49 m; when the radial well azimuths soar from 0° to 90°, the fracture initiation pressure jumpes by 40.84% and the distance from the fracture initiation position to vertical well section increases by 14.5 m. Fracture initiation occurs in the region far away from the vertical well section at the cost of putting on fracture initiation pressure with relatively increasing length, diameter and azimuth of radial well. 2. The influence of horizontal in-situ stress ratio on fracture initiation pressure and fracture starting point is related to azimuth of radial well: when the azimuth of radial well is 90° and the horizontal in-situ stress ratios (sH:sh) decrease from 1.9 to 1.1, the fracture initiation pressure reduces by 0.8%; when the azimuth of radial well is 0° and the horizontal in-situ stress ratios (sH:sh) decrease from 1.9 to 1.1, the fracture initiation pressure increases to 41.35%. Therefore, the change of horizontal

in-situ stress has no impact on the fracturing point. 3. The increase of permeability leads to obvious increase of the fracture initiation pressure. When the permeability increases from 0.01
10  3 μm2 to 100
10  3 μm2, the fracture initiation pressure by 34.5%. However there is no influence on the fracture initiation position. It is superior for the formation with low permeability to use radial well fracture with lower fracture initiation pressure. 4. With the presence of natural fractures, the fractures take first the priority in fracturing at the intersection between the natural fractures and the radial well section. Besides, the fracture initiation pressure and the hydrostatic fluid column pressure in vertical wellbore are equivalent. 5. Decreasing the length, diameter of radial well and permeability, increasing the azimuth of radial well may lead to fracture initiation at the toe of radial well. Otherwise, the fracture initiation would occur on the shaft lining of vertical well section. 6. When the azimuth is 0° and the horizontal in-situ stress ratio ( sH:sh) is larger than 1.5, increasing the horizontal in-situ stress

D.G. Gong et al. / Journal of Petroleum Science and Engineering 140 (2016) 41–56

may lead to fracture initiation at the toe of radial well. When the horizontal in-situ stress ratio (sH:sh) was smaller than 1.5, the reduction of horizontal in-situ stress may lead to fracture initiation at the toe of radial well. When the azimuth is 90°, the reduction of horizontal in-situ stress may lead to fracture initiation at the toe of radial well. The reduction of horizontal insitu stress ratio may lead to fracture initiation at the shaft lining of vertical well section. However, it is independent from the azimuth of radial well.

55

Dep is the elastic-plastic matrix; t is time; m ¼ [1, 1, 1, 0, 0, 0]T sw is water saturation; ks is the Compression modulus of solid particles; ds ξ = dpo is the parameters of the relationship between capillary o pressure and saturation.

Appendix B. Establishment of continuity equation Acknowledgments The authors would like to acknowledge the financial support of the National Natural Science Foundation of China, China (Grant no. 51404288). The authors would like to express their gratitude to “the Fundamental Research Funds for the Central Universities (Grant no. 15CX02012A)” and China University of Petroleum, Qingdao, China (East China) Graduate innovation project funded projects (YCX2014010), for providing continuous financial supports during the course of studies.

Appendix A. Stress equilibrium equation The stress equilibrium equation of porous rock is represented by the principle of virtual work. That is, the virtual work of the rock at a certain time point is equivalent to that done by physical force and planar force on the rock (Zhang and Dai, 2010), i.e.

∫V δ ϵT dσdV − ∫V δ uT dfdV − ∫S δ uT d tdS = 0

(6)

where σ is effective stress of rock; t is the planar force of the rock mass; f is the physical force of the rock mass; δε is virtual displacement and δu is virtual strain; dV is the volume infinitesimal; dS is the area infinitesimal. The Biot's effective stress in porous media is expressed as follows (Mao, 2003):

σ ′ij = σ ij +αδij ⎡⎣ χp o+(1 − χ ) pa ⎤⎦

(7)

where σ′ij is the effective stress of porous media; σij is the total stress of the rock mass; δ ij is Kronecker symbol; α is Biot coefficient; χ is porosity. Assume χ = s and α ¼1. Introducing the pore fluid pressure and the pore gas pressure, the above formula is simplified into

σ ′ij = σ ij +δij ⎡⎣ so p o+(1 − so ) pa ⎤⎦ = σij + δij p¯

(s w + po ϵ) dpo ⎤ ⎥ dV 3Ks dt ⎦

dp

−

∫V δ ϵT m (sw + po ξ ) dto dV

=

∫V δ uT ddtf dV + ∫S δ uT ddtt dS

⎧ ⎫ ⎡1 − n ⎪ ⎪ dp mT Dep m ⎤ s ⎥ (so + po ξ ) ⎬ o = 0 ⎨ ξn + n o + so ⎢ +⎪ − 2 ⎪ Ko (3KS ) ⎥⎦ ⎣⎢ 3Ks ⎩ ⎭ dt

(10)

where k 0 is the product of tensor of initial permeability coefficient and fluid density; kr is specific permeability coefficient; ρo is liquid density; g is the vector form of gravitational acceleration; n is rock porosity; Ko is the modulus of the volume of fluid inside the rock.

Appendix C. Boundary conditions (1) Flow boundary conditions (Fei, 2010)

⎛ ∇p ⎞ −nT kk r ⎜ o − g⎟ = qo ρ ⎝ o ⎠

(11)

where n is unit normal direction for flow boundary; k is permeability coefficient tensor; qo is the total amount of fluid passing the boundary in unit time. (2) Pore pressure boundary condition Pore pressure boundary condition can be expressed by the following formula: po = pob . It is assumed that the pore pressure on the boundary is a fixed value pob .

References

Neglecting the viscosity of fluid in the rock, formula (8) is introduced into formula (6), and derivative is taken with respect to time:

⎡

⎛ ⎡ ⎛ ∇p ⎞⎤ mT Dep ⎞ dε ⎟⎟ So ⎜⎜ mT − − ∇T ⎢ k 0k r ⎜ o − g⎟ ⎥ ⎢ 3KS ⎠ dt ⎝ ρo ⎠ ⎥⎦ ⎣ ⎝

(8)

where so is the saturation of pore fluid; po is the pore fluid pressure; pa is pore gas pressure; p¯ is the average pressure of fluid and gas.

∫V δϵT Dep ddtϵ dV + ∫V δ ϵT Dep ⎢⎣ m

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