Fracture-initiation pressure prediction for transversely isotropic formations

Journal of Petroleum Science and Engineering 176 (2019) 821–835

Contents lists available at ScienceDirect

Journal of Petroleum Science and Engineering journal homepage: www.elsevier.com/locate/petrol

Fracture-initiation pressure prediction for transversely isotropic formations Tianshou Ma

a,b,∗

a,c

a

d

a

a

, Yang Liu , Ping Chen , Bisheng Wu , Jianhong Fu , Zhaoxue Guo

T

a

State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, 610500, China State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, Hubei, 430071, China Department of Mechanical Engineering, The University of Melbourne, Melbourne, VIC 3010, Australia d State Key Laboratory of Hydroscience and Engineering, Department of Hydraulic Engineering, Tsinghua University, Beijing, China b c

ARTICLE INFO

ABSTRACT

Keywords: Fracture-initiation pressure Transversely isotropic formation Failure criterion Tensile strength

Fracture-initiation pressure (FIP) is one of the most important considerations for the exploitation of petroleum and geothermal energy as well as the storage of hydrogen energy, CO2 and hydrocarbon. Wellbore pressures above the FIP maybe lead to formation fracturing, gas leakage and mud lost circulation. The effect of rock anisotropy on FIP is significant, but has not received enough attention. In this paper, the rock mechanical properties were investigated for transverse isotropic rock. Taking into account the anisotropic deformation, modulus, tensile strength and in-situ stress, a FIP prediction model was developed. The FIP models with isotropic situation, anisotropic modulus, anisotropic strength, and fully anisotropic situation were compared. The influence of anisotropic mechanical parameters, horizontal stress ratio (HSR) and pore pressure (PP) on the FIP were also analyzed. The results suggested that the transversely isotropic rock exhibits a distinct anisotropic characteristic. When the anisotropy of rock mechanical parameters was taken into account, the FIP may be higher or lower than that of isotropic model, and the amplitude of variation was about ± 10%. Poisson's ratio anisotropy had a slight impact on the maximum and minimum FIPs. While the more significant the tensile strength anisotropy means the lower FIP. In addition, the influence of anisotropy on the FIP becomes increasingly pronounced with larger HSR and higher PP. The present paper can provide effective guidance to improve the drilling and hydraulic fracturing.

1. Introduction Drilling technology is important for many purposes, including oil and gas development, geothermal energy exploitation, natural gas storage, CO2 geological storage and nuclear waste disposal (Beswick et al., 2014; Fridleifsson and Elders, 2005; Liu et al., 2018; Ma et al., 2016; Zhang and Bachu, 2011). The goal of drilling is to construct highquality underground passages or storage spaces, and fracture-initiation pressure (FIP) is a key parameter for designing a casing program, optimizing drilling mud density, and formulating drilling measures for these spaces (Ma et al., 2017a, 2017b; 2017c). If FIP cannot be predicted accurately, excessive wellbore pressure may break the formation and result in drilling fluids penetrating into rock formations instead of returning to the annulus (Feng et al., 2016). Lost circulation is one of the trickiest challenges encountered in the drilling process (Aadnoy and Looyeh, 2011). Once lost circulation occurs, it not only causes high costs of drilling muds, but also leads to serious downhole accidents due to wellbore instability, blowouts, and pipe-sticking (Cook et al., 2011; Feng and Gray, 2016a, 2016b). These events generally result in huge

∗

economic losses and longer non-production time (NPT), seriously affecting drilling efficiency and operating costs. For geothermal exploitation, especially those from hot dry rock (HDR), hydraulic fracturing could be necessary during the development process (Legarth et al., 2005; Zimmermann et al., 2011; Davies et al., 2013). The fractures created by hydraulic fracturing can connect the injection well and the production well and improve the heat transfer efficiency between the injected fluid and formation and thus the output performance of the geothermal system. As an important aspect of hydraulic fracturing design of geothermal wells, accurate prediction of the FIP can effectively avoid mud loss and induced downhole accidents (Tsang et al., 2005; Li et al., 2006; Celia and Nordbotten, 2009). As for natural gas storage, CO2 geological storage, and nuclear-wastes disposed in the underground formations, the storage pressure should also be ensured not to exceed the FIP. Therefore, accurate prediction of FIP can offer a basic reference for hydraulic fracturing design of oil-gas or geothermal wells, and also provide important guidance and foundation for safe and efficient construction of other engineering projects. Considering the great significance of FIP prediction, many models

Corresponding author. State Key Laboratory of Oil & Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan, 610500, China. E-mail addresses: [email protected], [email protected] (T. Ma).

https://doi.org/10.1016/j.petrol.2019.01.090 Received 7 September 2018; Received in revised form 8 January 2019; Accepted 27 January 2019 Available online 05 February 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

Table 1 Major achievements in FIP prediction. No.

Authors (Year)

Major achievements

Adaptability

1 2 3

Hubbert and Willis (1957) Matthews and Kelly (1967) Haimson and Fairhurst (1967) Eaton (1969) Anderson et al. (1973) Daines (1982) Aadnoy (1988)

Introduced the concept of minimum injection pressure and gave the corresponding expression. Introduced the effective stress coefficient to calculate the FIP of sedimentary formations. Proposed the FIP model of impermeable rock based on Kirsch equation.

Isotropic formation Isotropic formation Isotropic formation

Modified the minimum injection pressure model by using Poisson's ratio of various rock. Presented an empirical model for Gulf Coast sands by considering the variation of formation properties. Superposed tectonic stress coefficient onto Eaton's model to calculate FIP. Proposed a wellbore stability model for an inclined wellbore in transversely isotropic formation with anisotropic modulus and anisotropic shear strength. Derived a closed-form solution for FIP based on fracture mechanics method. Introduced the poroelastic coefficient and Biot coefficient to predict FIP in permeable formation.

Isotropic formation Isotropic formation Isotropic formation Anisotropic formation

Presented a general “design code” to calculate the effects of anisotropy on FIP and provided a systematic approach for optimizing drilling and completion operation. Considered the impact of the poroelasticity, chemical and thermal effects to calculate FIP. Proposed a FIP prediction model for vertical wells in naturally fractured formations. Investigated the FIP, the resulting trace angle and angular position of fractures around the wellbore wall for arbitrarily inclined wellbore. Considered the wellbore breakdown in thermo-poro-elastic media under non-hydrostatic far-field stresses. Proposed the FIP model based on weight function method and fracture mechanics method. Proposed a new FIP model through analyzing leak-off test (LOT) data for offshore drilling. Developed an improved FIP model with a criterion of ATS.

Anisotropic formation

4 5 6 7 8 9 10

Rummel (1987) Detournay and Cheng (1988) Ong and Roegiers (1995)

11 12 13

Chen et al. (2003) Jin et al. (2005) Huang et al. (2012)

14 15 16 17

Wu et al. (2012) Jin et al. (2013) Zhang and Yin (2017) Ma et al. (2017a)

have been proposed to improve the prediction accuracy. The pioneering work on this topic was conducted by Hubbert and Willis (1957). Thereafter, many scholars carry out the relevant research work and have achieved some results (Haimson and Fairhurst, 1967; Matthews and Kelly, 1967; Eaton, 1969; Anderson et al., 1973; Pilkington, 1978; Daines, 1982; Huang, 1984; Rummel, 1987; Constant and Bourgoyne, 1988; Detournay and Cheng, 1988; Chen et al., 2003; Wessling et al., 2009; Oriji and Ogbonna, 2012; Wu et al., 2012; Jin et al., 2013; Guo et al., 2015; Zhang and Yin, 2017; Ma et al., 2017a; Zhang et al., 2018). The primary work and achievements are summarized in Table 1. Although the prediction accuracy of the proposed models has improved with time, and can meet the basic engineering demands to some extent, there are still discrepancies between the predicted and real values for some cases. The main reason is that most of these models were established based on simplifying assumptions, namely the mechanical properties of the rock formations were the same in all directions (isotropic), thus neglecting the distinct anisotropic characteristics of most sedimentary formations. Some scholars realized the necessity of consideration of anisotropy. For example, Aadnoy and Chenevert (1987) and Aadnoy (1988) took the lead by proposing an anisotropic model of wellbore stability based on the theory of anisotropic body. Later, Ong and Roegiers (1993, 1995), Gupta and Zaman (1999), Prioul et al. (2011), Serajian and Ghassemi (2011), and Zhu et al. (2014) adopted a similar approach to further study the hydraulic fracture initiation from deviated wellbore in a transversely isotropic formation. Here the transversely isotropic formation is one with physical properties that are symmetric about an axis that is normal to a plane of isotropy. Generally, each layer has approximately the same properties in-plane but different properties through-the-thickness. This assumption has been accepted and applied widely in many fields. The primary differences between the isotropic and anisotropic formations involve differences in wellbore stress distribution and anisotropic mechanical parameters. Our previous studies indicated that the FIP was obviously influenced by the anisotropic tensile strength (ATS) (Ma et al., 2017a, 2017b), and thus we derived the FIP prediction models for both vertical and inclined wells. Do et al. (2017) also developed an analytical FIP model by involving both effects of hydraulic and mechanical anisotropy. Although the existing models provide accurate prediction of the FIP for some relatively simple cases, they need to be improved to consider the coupling effects of anisotropic modulus, anisotropic Poisson's ratio, and ATS. For this purpose, this paper presents a fully anisotropic model to calculate the FIP for transversely isotropic formations and describes a

Isotropic formation Isotropic formation

Isotropic formation Isotropic formation Isotropic formation Isotropic Isotropic Isotropic Isotropic

formation formation formation formation

systematic analysis of the factors affecting FIP. 2. Anisotropic characteristics of transversely isotropic rock formations 2.1. The anisotropic elastic properties The elastic parameters of anisotropic rocks, such as Young's modulus and Poisson's ratio, affect tremendously both the fracturing and collapse failures of boreholes (Ma et al., 2018; Gui et al., 2018). The deformation of anisotropic elastic rock is also quite different from the isotropic rock. To understand the elastic anisotropy of transversely isotropic formations, layered shale rock samples (collected from the lower Silurian Longmaxi formation in Southern Sichuan Basin of China) with different coring angles were selected for triaxial compression tests, as shown in Fig. 1. Fig. 2 shows the measured stress-strain curves, and the statistical results of the measurement of some key parameters that are presented in Table 2. It can be easily observed that the compressive strength under various coring angles differs from each another. The perpendicular elastic modulus (E3), the parallel elastic modulus (E1), the perpendicular Poisson's ratio (v31), and the parallel Poisson's ratio (v12) for various layered rocks from the literature were collected. Fig. 3 shows the cross plots of the elastic modulus E1 vs. E3 and the Poisson's ratios v12 vs. v31, where the numbers are derived from static or dynamic measurements for different layered rocks (Sayers, 2013). It

Fig. 1. Schematic diagram of the layered shale rock coring. 822

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

Fig. 2. Stress-strain curves for different coring angles. Table 2 The statistical results of elastic parameters under various coring angles. Angle ψ (°)

Diameter (mm)

Length (mm)

0 25.52 48.96 15 25.27 49.65 30 25.56 49.08 45 25.54 49.52 60 25.53 49.60 75 25.44 49.89 90 25.52 49.53 Maximum Minimum Anisotropy ratio (Max./Min.)

Poisson's ratio

0.24 0.10 0.19 0.17 0.22 0.31 0.26 0.31 0.10 3.01

Young's modulus (GPa)

Unconfined compressive strength (UCS) (MPa)

16.48 18.87 19.03 21.12 21.83 22.16 23.49 23.49 16.48 1.43

216.7 218.5 182.5 191.8 96.10 100.3 153.3 218.5 96.1 2.27

Fig. 4. Plot of perpendicular tensile strength Tm vs. parallel tensile strength Tb for various rocks (Data from Ma et al., 2018).

bedding plane with a screwdriver but are stronger across the bedding plane. Fig. 4 shows some typical laboratory results of ATS for sedimentary rocks in the vertical direction (perpendicular to bedding plane) and the horizontal direction (parallel to bedding plane), where the number relates to (1) shale, (2) sandstone, (3) slate, (4) gneiss, (5) schist, (6) coal, (7) marl, the data were collected from Ma et al. (2018). The anisotropic index is defined as the ratio of the perpendicular tensile strength (Tm) to the parallel tensile strength (Tb), i.e. k = Tm/Tb. It can be found that the anisotropic index of various rocks mainly focuses on a range of 1–3 with a few exceeding 4, and the perpendicular tensile strength is always larger than the parallel tensile strength. This illustrates most of sedimentary rocks exhibit significant ATS.

can be seen that there are distinct differences between E1 and E3, and most of the anisotropy index (nE = E1/E3) range from 1 to 3. Similarly, the parallel Poisson's ratio (v12) and perpendicular Poisson's ratio (v31) are also different, and most of the anisotropy index (nv = v12/v31) range from 0.4 to 3. This shows that elastic anisotropy of layered rocks is quite significant.

3. Stress distribution model on the wellbore 3.1. Basic assumptions

2.2. The anisotropic tensile strength (ATS)

For rocks with significant anisotropic characteristics, the effects of anisotropy should be considered so that one can make an accurate FIP prediction. A few assumptions are used to make the present model

The tensile strengths of sedimentary rocks are usually directiondependent. For example, anisotropic rocks can be broken along the

Fig. 3. Plot of (a) E3 versus E1 and (b) ν31 versus ν12 for layered rocks (Reproduced from Sayers, 2013). 823

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

Fig. 5. Transformation between different coordinate systems.

tractable (Gupta and Zaman, 1999): (1) the rock formation is assumed to be continuous, homogeneous and transversely isotropic, (2) the elastic deformation of the rock is small, (3) the stress-strain around the wellbore satisfies the generalized plane strain condition, and (4) seepage, heat transfer and chemical reaction are neglected.

tensors for the TIPCS system; [A] represents the flexibility matrix of the rock medium for the TIPCS; E, v and G represent the elastic modulus, Poisson's ratio and shear modulus with respect to the perpendicular bedding plane, respectively; E', v' and G' represent the elastic modulus, Poisson's ratio, and shear modulus with respect to the parallel bedding plane, respectively. The effective compliance matrix for the BCS is the basis of solving for stress distribution. Based on the coordinate transformation illustrated in Fig. 5, the stress-strain relationship and flexibility matrix for the BCS are expressed as,

3.2. Coordinate transformation For the convenience of obtaining the stress distribution around wellbore circumference, coordinate transformations among different coordinate systems are required and there are five different coordinate systems (Fig. 5). The first is the global coordinate system (GCS) (x,y,z), where z is in the vertical direction, and x and y are in horizontal directions. It can also be called the geodetic coordinate system (N,E,Z). The second is the local coordinate system of in-situ stress (ISCS) (xs,ys,zs), where zs defines the direction of vertical stress (σv), xs and ys are in horizontal directions, and xs coincides with the direction of maximum horizontal stress (σH). To express the relationships between the ISCS and GCS, the azimuth angle (βs) is specified as the angle between the maximum horizontal stress (σH) and the x axis (or N axis). The third system is the local coordinate system of the borehole (BCS) (xb,yb,zb), where zb represents the axis of the borehole, and xb and yb form the local normal cross-section. The BCS is coincident with the ISCS. The fourth system is the cylindrical coordinate system of the borehole (BCCS) (rb,θb,zb), where zb still defines the axis of the borehole, and rb and θb are in the cross-section of the borehole. The fifth system is the local coordinate system of the transverse isotropic plane (TIPCS) (xw,yw,zw), where zw is normal to transverse isotropic plane, and xw and yw form the transverse isotropic plane. The transverse isotropic plane with a dip angle (αw) and a dip direction (βw) is defined to express the relationship between the TIPCS and GCS.

v E v E

[A] =

v E

0

1 E v E

0

v E v E

0

0

0

0

0

0

0

0

1 G

0

0 0 1 G

0

0

0

0

1 G

0

0

0

0

0

= (a31

x

+ a32

y

+ a34

yz

+ a35

xz

+ a36

xy )/a33

(5)

3.4. Closed-form solution of stress distribution The stress distribution around the wellbore is affected by in-situ stress and wellbore pressure. Gupta and Zaman (1999) divided the stress distribution around the wellbore into three components, i.e. induced by (1) the far field in-situ stress, (2) borehole drilling, and (3) wellbore pressure. Therefore, the closed-form solution for stress distribution around the wellbore can be derived and expressed as follows (The details are provided in Appendix A),

0

1 E

(4)

where aij (i,j = 1, to 6) represents the components of the effective flexibility matrix [AT]; The variables σx, σy, σz, τxy, τyz, and τzx represent the stress components.

(1)

0

[AT ] = [M ]T [A][M ]

z

Under the local coordinate system of formation (TIPCS), the constitutive equation of the transversely isotropic rock can be expressed as,

1 E

(3)

where { } xyz and { } xyz represent the stress tensor and strain tensor, respectively, for the BCS; [AT] represents the flexibility matrix of the rock medium for the BCS; [Mσ] represents the stress transformation matrix between the GCS and BCS. According to the hypothesis of generalized plane strain (εz = 0), the stress component (σz) can be expressed as (Zhu et al., 2014),

3.3. Constitutive model

{ } xw yw zw = [A]{ } xw yw zw

{ } xyz = [AT ]{ } xyz

= = z = yz = zx = xy = x

y

where

(2)

where { } xw yw zw and { } xw yw zw respectively denote the stress and strain 824

+ + z,0 + yz,0 + yz,0 + yz,0 + x ,0 y,0

+ x ,b + y, b z, h + z , b yz, h + yz, b zx , h + zx , b xy, h + xy, b x ,h

y, h

(6)

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

coordinate transform relationship, as shown in Fig. 5, the stress components on the wellbore can be transformed into the BCCS and expressed as (Aadnoy and Chenevert, 1987),

= h = H z ,0 = v xy,0 = 0 yz ,0 = 0 xz ,0 = 0 x ,0 y,0

x,h y, h

= b1 = b2

= yz , h = zx , h = xy, h = z,h

x,b y, b

r

+ c1 h + c2

(a31 b3 b4 b5

x,h

H

+ a32

y, h

+ a34

yz , h

+ a35

xz , h

+ a36

xy

sin 2

z

H

h + c3 h + c4 h + c5

xy, h )/ a33

rz

=

yz

sin

+

xz

y

sin 2

+

xy

cos 2

cos

(10)

H H

4. Modeling FIP in transversely isotropic formations

(8)

H

4.1. Failure criterion of ATS

= d1 pm = d2 pm

= (a31 yz , b = d3 pm zx , b = d 4 pm xy, b = d5 pm z,b

= x sin2 + y cos2 = z 0.5 x sin 2 + 0.5 r = z = yz cos xz sin

(7) h

= pm

x ,b

+ a32

y, b

+ a34

yz, b

+ a35

xz , b

+ a36

Ma et al. (2017a, 2017b) compared various failure criteria of ATS and found that the Novae-Zaninetti criterion has the smallest relative error. More importantly, the ATS predicted by the Novae-Zaninetti criterion are relatively conservative with comparison to the experimental result, which is important for safe drilling. Thus, the NovaeZaninetti criterion can be utilized to predict the FIP of transversely isotropic formations and is expressed as (Nova and Zaninetti, 1990),

xy, b )/ a33

(9)

where σx,0, σy,0, σz,0, τxy,0, τyz,0 and τxz,0 are the components of stresses caused by far-field stresses; σx,h, σy,h, σz,h, τxy,h, τyz,h, and τxz,h are the components of the stress tensor caused by drilling the borehole; a is the radius of the borehole; θ is the angle of circumference; σx,b, σy,b, σz,b, τxy,b, τyz,b, and τxz,b are the components of the stress tensor induced by the wellbore pressure, pm. After obtaining the stress distributions for the BCS, they need to be transformed into the BCCS system. Therefore, according to the

T ( b) =

Tb sin2

Tm Tb + Tm cos2

b

b

(11)

where T (βb) is the ATS at a given angle βb; βb is the angle between the tensile stress and bedding normal; Tb is the tensile strength of bedding planes; Tm is the tensile strength of the rock matrix.

Fig. 6. Comparison of the FIP for different models: (a) isotropy, (b) anisotropic modulus, (c) anisotropic strength, and (d) full anisotropy. 825

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

4.2. Anisotropic prediction model of FIP

the critical wellbore pressure that causes wellbore breakout. Therefore, the FIP is,

Wellbore breakout occurs when the compression stress on the borehole wall exceeds the rock strength, and the wellbore tensile failure mainly depend on the formation's ATS T (βb). If the pore pressure (PP) is taken into account, the failure criterion of ATS can be formulated as, 3

pp + T ( b) = 0

pf = min{pmc ( )} where pf is the FIP; θ is the circumferential angle. 4.3. Determination of the angle (βb)

(12)

where σ3 represents the minor principal stress; α represents the Biot coefficient; pp represents pore pressure. For vertical wells, wellbore fracturing happens once the tangential stress on the borehole wall exceeds the ATS of the formation. Therefore, substituting Eqs. (10) and (11) into Eq. (12), the anisotropic prediction model of FIP can be obtained,

f (pm , ) = A

h

+B

H

+ Cpm

pp + T (

b

)= 0

It should be noted that the angle (βb) between the tensile stress and bedding normal should be predetermined before solving for the FIP. Fig. 5 shows that in the GCS (N,E,Z) system, the azimuth angle of the maximum horizontal principal stress is given by βs, the bedding plane is characterized by dip angle αw and azimuth angle βw. Thus, the angle (βb) can be determined by using geometrical relationship and be expressed as,

(13)

where

b

A = sin2 + b1 sin2 + b2 cos2 b5 sin 2 B = cos2 + c1 sin2 + c2 cos2 c5 sin 2 C = d1 sin2 + d2 cos2 d5 sin 2

pp

T(

b

) C

A

h

B

H

= cos

n N

1

n

= cos

N

1

e1 g1 + e2 g2 + e3 g3 e12 + e22 + e32

g12 + g22 + g32

(17)

where

(14)

e1 = sin e2 = sin e3 = cos

Eq. (13) shows that FIP is related to the circumferential angle (θ). For a given circumferential angle (θ), the critical wellbore pressure causing wellbore breakdown can be determined by,

pmc ( ) =

(16)

g1 = cos g2 = sin g3 = 0

(15)

Once the critical wellbore pressure at the given circumferential angle (θ) is reached, the minimum wellbore pressure can be treated as

cos sin

w w

w w

(18)

w

s s

sin sin

+ sin cos

s s

cos cos (19)

where βw represents the bedding planes' azimuth; αw represents the

Fig. 7. Influence of the anisotropic elastic modulus: (a) nE = 1.0, (b) nE = 2.0, (c) nE = 3.0, and (d) nE = 4.0. 826

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

bedding planes' dip; βs represents the azimuth of the maximum horizontal stress (MAXHS).

amplitudes are 0.1%, 0.35%, 0.97%, 1.81%, 2.57%, 3.15%, and 7.01%, compared with the FIP predicted by the isotropic model. However, when βw = 90°, the calculated FIPs increase slightly at αw = 0°. Thereafter, the FIPs first decrease and then increase slightly with increasing dip angle. The predicted FIPs are respectively 69.08 MPa, 68.59 MPa, 67.83 MPa, 66.96 MPa, 66.24 MPa, and 67.51 MPa. The associated reduction ratio is about 0.14%, 0.85%, 1.95%, 3.21%, 4.25%, and 2.41% of the isotropic result. When αw = 90°, the calculated FIP takes on an abnormal variation trend with the azimuth angle βw transformed from 0° to 90°, i.e., first declining, then increasing, and finally declining again. Analogously, if only consider the anisotropic strength, the calculated FIPs show a decreased tendency with the increase of the bedding dip angle except for αw = 0°, as illustrated in Fig. 6(c). This is because the anisotropic tensile strength is closely related to the bedding occurrence and it decreases with the increase of bedding dip angle and azimuth angle. Particularly, the FIP keeps a fixed value of 69.25 MPa when αw = 0°, which is about 0.1% larger than that in the isotropic case. However, when βw = 90°, the FIP decreases with the bedding dip angle increasing from 15° to 90°. The calculated FIP is respectively 68.85 MPa, 67.98 MPa, 67.13 MPa, 66.53 MPa, 66.19 MPa, and 66.08 MPa, and the magnitude of the decline is ∼0.40 MPa, ∼1.27 MPa, ∼2.12 MPa, ∼2.72 MPa, ∼3.07 MPa, and ∼3.18 MPa. Comparing with the isotropic results, the corresponding reduction ratio is about 0.48%, 1.78%, 2.96%, 3.83%, 4.32%, and 4.48%, respectively. Further, if the above anisotropic characteristics are fully covered, the predicted FIPs appear a similar variation behaviors which are above-mentioned. The predicted FIP can be decreased more with the increase of the bedding dip angle, as shown in Fig. 6(d). Similarly, when βw = 0°, the calculated FIPs are consistent with findings presented

5. Results and discussions To investigate the variation of FIP in transversely isotropic formations with different factors, the basic parameters of formation were assumed to be: TVD = 2500 m, a = 0.108 m, σv = 61.2 MPa, σH = 55.5 MPa (N0°E), σh = 47.6 MPa, pp = 30.5 MPa, α = 0.8, E = 23.5 GPa, E' = 16.48 GPa, v = 0.22, v' = 0.22, Tb = 3.17 MPa, and Tm = 6.35 MPa. 5.1. Results and analysis Fig. 6 shows the calculation results for isotropic, anisotropic modulus, anisotropic strength, and fully anisotropic conditions. As exhibited in Fig. 6(a), the FIP stays constant (69.18 MPa) and is independent on the bedding occurrence for the isotropic case. However, after considering formation anisotropy, the effect of bedding occurrence on the predicted FIP is considerable and cannot be neglected. Comparing with the result of isotropic model, involving the anisotropic modulus can increase or lower the predicted FIP depending on the formation occurrence, as illustrated in Fig. 6(b). This anomaly is mainly due to the variation of stress distribution around wellbore circumference caused by the anisotropic modulus. Specifically, when the bedding azimuth angle is in the direction of the minimum horizontal stress (MINHS), i.e., βw = 0°, the FIP increases with the dip angle. When the dip angle increases from 0° to 90° by a step of 15°, the predicted FIPs are 69.25 MPa, 69.42 MPa, 69.85 MPa, 70.43 MPa, 70.96 MPa, 71.36 MPa, and 74.03 MPa, respectively. The respective growth

Fig. 8. Influence of the anisotropic Poisson's ratio: (a) nv = 1.0, (b) nv = 0.75, (c) nv = 0.50, and (d) nv = 0.25. 827

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

previously from the anisotropic modulus model and not explained here. However, when βw = 0°, the FIP increases slightly at αw = 0°, and thereafter, it first decreases and then increases with increased dip angle αw. The predicted FIPs are respectively 68.68 MPa, 67.32 MPa, 65.71 MPa, 64.24 MPa, 63.19 MPa, and 64.34 MPa. Accordingly, the reduction ratio is about 0.72%, 2.69%, 5.02%, 7.14%, 8.66%, and 7.00% compared with the isotropic result. When αw > 0°, the FIP decreases with an increase in the azimuth angle from 15° to 90°. The corresponding reduction magnitude is ∼0.73 MPa, ∼2.54 MPa, ∼4.71 MPa, ∼6.73 MPa, ∼8.17 MPa, and 9.70 MPa for αw = 15°, 30°, 45°, 60°, 75°, and 90°. Through comparisons of four typical cases, it is clearly found that the FIP does not always decrease with increasing dip angle in the case of anisotropy. As long as the anisotropy of elastic modulus exists, the FIP tends to first decrease and then increase. The anisotropic tensile strength does not affect the variation trend of the FIP, but the predictions can be further declined for the fully coupled model. Hence, it is necessary to consider the fully anisotropic features during FIP prediction, especially for some fracture-pressure-sensitive drilling process.

≠ 1.0, as shown in Fig. 7(b)-(d), the FIP is closely related to the bedding occurrence, and the overall variation rules have similar features. When βw = 0°, the predicted FIPs are always the highest. The MAXFPs are 74.03 MPa, 74.46 MPa, and 72.30 MPa for nE = 2.0, 3.0, and 4.0, respectively. The corresponding growth rates are about 7.01%, 7.63%, and 4.51%. When βw = 90°, the predicted FIP is at the minimum value. The MINFPs are 63.18 MPa, 61.02 MPa, and 59.29 MPa for nE = 2.0, 3.0, and 4.0, respectively. The corresponding reduction ratios are about 8.67%, 11.80%, and 14.30%. For αw = 90°, the variation in FIP changes from an initial gradual reduction to an increase and then a decrease. Especially for the case with strong anisotropy of elastic modulus, the stronger the anisotropy of elastic modulus is, the more likely it will affect the FIP. Thus, the elastic modulus anisotropy should be taken into account in the prediction of FIP for a transversely isotropic formation. 5.2.2. Influence of the anisotropic Poisson's ratio Fig. 8 shows the predicted FIP for four Poisson's ratio anisotropy coefficients (PRAC), i.e. nv = 1.00, 0.75, 0.50, and 0.25, when nE = 2.0 and k = 2.0. When βw = 0°, the FIP increases with the bedding dip angle, but the MAXFP decreases gradually with increasing PRAC. The MAXFPs are respectively 74.34 MPa, 74.03 MPa, 73.75 MPa, and 73.46 MPa for nv = 1.0, 0.75, 0.50 and 0.25, and the corresponding growing rates are about 7.46%, 7.01%, 6.61%, and 6.19%. When the bedding azimuth angle βw changes from 0° to 90°, the FIP increases at first and follows by a decrease. This changing trend would be more apparent as the bedding dip angle increases. When βw = 90°, the FIP initially decreases and then increases with increasing the bedding dip angle, but the MAXFP decreases gradually with increasing the PRAC. The MAXFPs are about 63.31 MPa, 63.19 MPa, 63.07 MPa, and 62.95 MPa for nv = 1.0, 0.75, 0.50, and 0.25, respectively, and the

5.2. Factors influencing the FIP 5.2.1. Influence of anisotropic elastic modulus (AEM) Fig. 7 shows the predicted FIPs for four elastic modulus anisotropy situations (nE = 1.0, 2.0, 3.0, and 4.0), where nv = 0.75 and k = 2.0. In the case of nE = 1.0, as illustrated in Fig. 7(a), the FIP achieved a maximum value at βw = 0°. The maximum FIP (MAXFP) is 69.32 MPa, which is about 0.2% larger than that obtained for the isotropic case. When βw = 90°, there exists a minimum FIP (MINFP) of 66.44 MPa, about 3.96% smaller than that obtained for the isotropic case. When nE

Fig. 9. Influence of the ATS: (a) k = 1.0, (b) k = 2.0, (c) k = 3.0, and (d) k = 4.0. 828

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

corresponding growth rates are about 8.49%, 8.66%, 8.83%, and 8.84%. This suggests that the Poisson's ratio anisotropy has slight impact on the FIP.

βw = 0°), the FIP is always the highest. While when the bedding azimuth angle is in the direction of the MAXHS (i.e., βw = 90°), the FIP is usually lowest. When the dip angle αw = 90°, the lowest FIP occurs at an around 10° angle between the bedding azimuth and the MAXHS. When the HSR n > 1, the FIP decreases with increasing the difference in horizontal stresses. Generally speaking, larger horizontal stress difference results in a lower FIP. When βw = 90°, the lowest FIPs are 60.97 MPa, 47.92 MPa, and 34.88 MPa for n = 1.2, 1.4 and 1.6, respectively. The corresponding reduction ratios are about 11.87%, 30.73%, and 49.58%. Obviously, the FIP is very sensitive to the HSR. Moreover, the HSR can lead to FIP drop with a reduction ratio of 50% or much higher. When βw = 0°, the greatest FIPs under various HSRs (n = 1.0, 1.2, 1.4, and 1.6) are 77.15 MPa, 73.05 MPa, 67.26 MPa, and 61.46 MPa, respectively. This means that the FIP can be at its maximum under isotropic horizontal stress situations. The FIP could be decreased with the increase of the HSR, the reductions are ∼4.1 MPa, ∼9.89 MPa, and ∼15.69 MPa for n = 1.2, 1.4, and 1.6, respectively. The corresponding reduction ratios are about 5.31%, 12.82%, and 20.34%, respectively. When HSR n = 1.0, the FIPs under arbitrary bedding occurrence conditions are always greater than that obtained by isotropic model, where the maximum and minimum values are 77.15 MPa and 72.25 MPa, respectively, and the corresponding increasing ratios are 11.52% and 4.44%, respectively. To sum up, the HSR has the greatest impact on FIP. Compared with an isotropic formation, the FIP initially increases and decreases later with increasing the HSR when βw = 0°, and the MAXFP increases by 11.52%. When βw = 90°, the FIP reduces sharply with increasing the HSR, and the maximum reduction ratio is about 50%. Thus, the influence of HSR on FIP cannot be ignored.

5.2.3. Effect of the ATS Fig. 9 displays the effect of the anisotropy index of tensile strength (k = 1.0, 2.0, 3.0, and 4.0) on the FIP, where nE = 2.0 and nv = 0.75. When the bedding azimuth angle is along the MINHS (i.e., βw = 0°), the FIP hardly changes with varying the anisotropy index k, and the MAXFP is 74.03 MPa. When the bedding azimuth angle is along the MAXHS (i.e., βw = 90°), the FIP decreases with the increased ATS. The MINFPs for the above four situations are 66.24 MPa, 63.19 MPa, 62.12 MPa, and 61.58 MPa, respectively, and the corresponding reduction ratios are 4.25%, 8.66%, 10.21%, and 10.96%. For a given bedding dip angle, when the bedding azimuth angle changes from 0° to 90°, the reduction of the FIP has a positive correlation with the ATS. It can be concluded that the extent of the impact of ATS on the FIP grows gradually during the azimuth angle deflection. In short, the FIP does not change with the anisotropy index k in the MINHS direction, but in the MAXHS direction, larger ATS means greater variation of the FIP. 5.2.4. Influence of the horizontal stress ratio (HSR) The HSR, defined by n = σH/σh, is one of the most important considerations for FIP prediction. To study the combined effect of horizontal in-situ stresses and anisotropy parameters, different HSRs (n = 1.0, 1.2, 1.4, and 1.6) are adopted to determine the FIP and the results are shown in Fig. 10. It can be found that the variation of the FIP with bedding occurrence has similar trends for different HSRs. When the bedding azimuth angle has the same direction as the MINHS (i.e.,

Fig. 10. Influence of the horizontal stress ratio: (a) n = 1.0, (b) n = 1.2, (c) n = 1.4, and (d) n = 1.6.

829

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

Fig. 11. Influence of the PP: (a) pp = 30.0 MPa, (b) pp = 35.0 MPa, (c) pp = 40.0 MPa, and (d) pp = 45.0 MPa.

5.2.5. Influence of the PP The PP also plays a vital role in predicting the FIP. To compare its impact on the FIP, the PP are set to be 30.0 MPa, 35.0 MPa, 40.0 MPa, and 45.0 MPa and the results are shown in Fig. 11. It can be seen that when the bedding azimuth angle βw = 0°, the FIP always maintains at the highest level and the predicted FIP using the anisotropic model is larger than that from the isotropic model. The MAXFPs are respectively 74.46 MPa, 70.23 MPa, 66.00 MPa, and 61.77 MPa for pp = 30.0 MPa, 35.0 MPa, 40.0 MPa, and 45.0 MPa. The corresponding variations of FIP are respectively 7.63%, 1.52%, 4.60%, and 10.71%. However, when the bedding azimuth angle βw = 90°, the FIP is lowest and the FIP decreases with increasing PP. The MINFPs are found to be 63.59 MPa, 59.60 MPa, 55.61 MPa, and 51.62 MPa for these four cases, and the corresponding reduction ratios are about 8.08%, 13.85%, 19.62%, and 25.38%. Overall, higher PP will result in lower FIP, which is bad for safe drilling.

MAXFP. The influence degree of each factors on the MAXFP from high to low is HSR, PP, AEM, PRAC and ATS. In contrast, it is found that the effects of anisotropy on the MINFP are more significant than on the MAXFP. As displayed in Fig. 12(b), the predicted MINFP decreases with the variation of anisotropy parameters and the sensitivity degree from large to small is HSR, PP, AEM, ATS and PRAC. Obviously, if the MINFP is selected to optimize drilling mud density, it can greatly reduce the safety risk of drilling operation, but may have an adverse effect on hydraulic fracturing due to the irrational construction parameters. To sum up, for a given PP, it is not hard to see that the HSR and AEM must be put into consideration in the prediction of FIP. Although the ATS can be ignored in predicting MAXFP, it still plays an important role in MINFP. Compared with other factors, the PRAC has a minimal influence on the FIP and therefore it can be neglected.

5.3. Discussion

(1) The characteristics of the anisotropic elastic modulus, Poisson's ratio, and tensile strength of transversely isotropic rocks are investigated. The results indicate that there are distinct differences between the elastic modulus in parallel (E1) and perpendicular (E3) along the bedding plane, and most of the anisotropy ratios (nE = E1/E3) ranged from 1 to 3. The tensile strength in lateral direction (Tm) is always larger than that in the longitudinal direction (Tb). All the results demonstrate prominent anisotropic characteristics of transversely isotropic rocks. (2) The FIP for four typical conditions (isotropy, anisotropic modulus, anisotropic strength, and fully anisotropy) are analyzed. For condition with anisotropic modulus, when the bedding azimuth angle

6. Conclusions

The above results demonstrate that the anisotropic characteristics of rock formation, including AEM, PRAC, ATS, HSR and PP, have different level of effects on the FIP. To clarify the influence degree of each affecting factor, the MAXFP and MINFP calculated for various anisotropy parameters are adopted to make the comprehensive evaluation and comparative analysis. The comparison results are shown in Fig. 12. The error bar marked in the histogram denotes the changing rate of FIP by comparing with the isotropic result. From Fig. 12(a), we can see that the predicted MAXFP increases to some extent after fully considering the effects of anisotropy. Specifically, the HSR has greatest impact on the MAXFP, while the PRAC and ATS have little or no effect on the 830

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

anisotropic strength, when the bedding azimuth angle is in the direction of the MINHS, the FIP remains unchanged with the increase of bedding dip angle. When the bedding azimuth angle changes from along the MINHS to along the MAXHS, the FIP decreased gradually when the dip angle is greater than 0°. For fully anisotropic situation, when the bedding azimuth angle is in the direction of the MINHS, the FIP increases with the bedding dip angle. Meanwhile, as long as an anisotropic modulus exists, the FIP does not always decrease with increasing the dip angle, but tends to decrease initially and then increase. (3) The influence of the anisotropy relating to the elastic modulus, Poisson's ratio and tensile strength on FIP is also investigated. The results imply that the stronger the anisotropy of elastic modulus is, the more likely it will affect the FIP. Poisson's ratio anisotropy also has a little effect on the FIP. When the bedding azimuth angle changes from along the MINHS to the MAXHS, the impact of the anisotropy index of tensile strength on the FIP increases gradually. The FIP does not change with the anisotropy index of tensile strength in the MINHS direction, but in the direction of MAXHS, the variation of FIP becomes greater with increasing the tensile strength anisotropy, and the maximum reduction ratio is 10.96% of the isotropic result. (4) Deep analyses are made to understand the impacts of horizontal stress and PP on the FIP. Compared with isotropic formation, the FIP initially increases and then decreases with increasing the HSRs in the direction of MINHS, and the maximum magnitude reaches 11.52%. In the direction of the MAXHS, the FIP decreases sharply with increasing the HSR, and the maximum reduction ratio is about 50%. Thus, the influence of the horizontal stress should not be neglected when predicting the FIP. Meanwhile, the PP also has a striking impact on the FIP, and higher PP will result in a lower FIP, which is not beneficial for safe drilling. Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant Nos. 41874216 and 51604230); the Scientific Research Foundation of International Cooperation and Exchanges of Sichuan Province (Grant No. 2017HH0061); the Program of Introducing Talents of Discipline to Chinese Universities (111 Plan) (Grant No. D18016); the China Postdoctoral Science Foundation (Grant Nos. 2017T100592 and 2016M600626); and the Young Elite Scientists Sponsorship Program by CAST (2017QNRC001).

Fig. 12. The influence degree of each affecting factor on the FIP. (a) MAXFP; (b) MINFP.

is in the direction of the MINHS, the FIP increases as the bedding dip angle increased. When the bedding azimuth angle is along the MAXHS, the FIP is slightly higher at the dip angle of 0°, after which it decreases, followed by a slow increase. For the condition with Appendix A.1. Governing equation

By ignoring the gravity, the stress distribution under the BCS should satisfy the following equations (Lekhnitskij, 1981), x

+

x xy

x xz

x

+

xy

y y

y

+

xz

+

yz

yz

+

z z z

+

y

z

=0 =0 =0

(A1)

The strain-displacement relationship under generalized plane strain condition is expressed as, x xz

= =

u , x w , x

y

= yz

=

v , z y w , y

=0 xy

=

u y

+

v x

(A2)

where εx, εy, εz, γxy, γyz, and γxz are the strain component; u, v, and w are the displacement component. According to the generalized plane strain hypothesis, the strain components are independent in z. Therefore, the strain compatibility equation can be described by (Gupta and Zaman, 1999),

831

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al. 2

x

y2

2

+

y

zx

yz

y

x

2

=

x2

xy

x y

=0

(A3)

Appendix A.2. Construction of analytic function According to work by Lekhnitskij (1981) and Amadei (1983) for anisotropic plane problems, two stress functions, i.e. F (x, y) and Ψ(x, y), which satisfy the equilibrium equation automatically, are introduced. After considering the plane strain hypothesis, the stress components can be represented by stress functions, in the forms of (Gupta and Zaman, 1999), x

=

2F(x , y )

y2

2F(x , y)

=

y

(x , y ) , x

=

yz

,

zx

x2 (x , y ) , xy y

=

2F(x , y )

=

(A4)

x y

When substituting Eqs. (A4) and (3) into Eq. (A1), we obtain,

L4 F + L3 L3 F + L 2

=0 =0

(A5)

where

L2 =

2 3

L3 = L4 =

ij

4

45 x y

+(

24 x 3 22 x 4

2

2

44 x 2

25

+

+

3

46) x 2 y

4

+2

2

55 y 2

2

26 x 3 y

( 4

16 x y3

+

14

+ (2

3

56 ) x y2 12

+

3

+

15 y3 4

66) x 2 y2

+

4

(A6)

11 y 4

ai3 aj3

= aij

(A7)

a33

Here L2, L3, L4 are the second, third, and fourth order differential operators; βij is the reduced compliance coefficient. Lekhnitskij (1981) found the general solutions of Eq. (A5),

F = 2 Re[F1 (z1) + F2 (z2 ) + F3 (z 3)] = 2 Re[ 1 F 1 (z1) + 2 F 2 (z2 ) + F 3 (z 3)/ 3]

(A8)

where Re is the real component of a complex number; Fk(zk) is a function of complex variable zk = x+μky, k = 1,2,3; μk is the roots of the characteristic equation that corresponds to the strain compatibility equation; λk is a coefficient related to characteristic roots. The characteristic equation corresponding to Eq. (A5) can be solved by the following equation (Amadei, 1983), (A9)

I32 (µ ) = 0

I4 (µ) I2 (µ ) where

I2 (µ ) = I3 (µ ) = I4 (µ ) =

2

22 24

26 µ

+( 2

44

25

+ (2 +

45 µ +

12

2 66) µ

+

46) µ 2 55 µ

(

14

+

3 16 µ 2 )µ +

2 56

+ 15

11 µ µ3

4

(A10)

Lekhnitskij (1981) demonstrated the form of characteristic roots for Eq. (A9), i.e., three of the roots (μ1, μ2, μ3) conjugated with the others (μ∗1, μ∗2, μ∗3). Thus, the coefficient λk in Eq. (A8) can be defined as, 1

=

I3 (µ1) , I2 (µ1)

2

=

I3 (µ 2 ) , I2 (µ 2 )

3

=

I3 (µ3) (A11)

I4 (µ 3 )

To obtain the solutions of the stress distribution around the wellbore, Lekhnitskij (1981) defined three analytical functions 1 (z1)

k (zk),

= F 1 (z1)

2 ( z 2 ) = F 2 (z 2 ) 3 (z 3 )

= F 3 (z 3)/

(A12)

3

where k (zk) denotes arbitrary analytic function; F k (z k ) is the first-order derivative of complex function Fk(zk). Combining Eq. (A8) and (A12), the following expressions can be obtained,

F / x = 2 Re[

1 (z1)

F / y = 2 Re[µ1 = 2 Re[

1

+

2 (z 2 )

1 (z1) + µ 2

1 (z1) +

2

+

3 3 (z 3)]

2 (z 2 ) +

2 (z 2 ) +

3 µ3 3 (z 3)]

3 (z 3)]

(A13)

Hence, when substituting Eq. (A13) into Eq. (A4), the stress components can be expressed in terms of the new analytic function (Aadnoy, 1988; Gupta and Zaman, 1999),

832

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

x

= 2 Re[µ12

y

= 2 Re[

1 (z1)

+ µ22

1 (z1) +

2 Re[

2 (z 2 )

2 (z 2 ) +

yz

=

1 (z1)

zx

= 2 Re[ 1 µ1

1 (z1) +

xy

=

1 (z1) + µ 2

1

2 Re[µ1

+

3

2 µ2

3 (z 3)]

3 (z 3)]

2 (z 2 )

2

2 3 µ3

+ +

3 (z 3)]

2 (z 2 ) + µ 3

2 (z 2 ) +

3 (z 3)]

3 µ3

3 (z 3)]

(A14)

Eq. (A14) can be considered as the governing equation for the stress distribution around the wellbore. Obviously, the original problem has been converted to finding the three analytic functions, which must be in conjunction with the corresponding boundary conditions. Appendix A.3. Closed-form solution of stress distribution Gupta and Zaman (1999) divided the stress distribution around the wellbore into three parts: (1) one induced by the far-field in-situ stress, (2) one induced by drilling the borehole, and (3) one induced by the wellbore pressure. (1) Components of the far field in-situ stress tensor {σ0}xyz Before drilling the borehole, the stress components equals to the in-situ stresses,

= h = H z ,0 = v xy,0 = 0 yz ,0 = 0 xz ,0 = 0 x ,0 y,0

(A15)

where σH, σh, and σv are the maximum horizontal stress (MAXHS), minimum horizontal stress (MINHS), and vertical stress, respectively,; σx,0, σy,0, σz,0, τxy,0, τyz,0, and τxz,0 are the components of the far-field stresses. (2) Components of the stress tensor induced by drilling the borehole {σh}xyz After drilling the borehole, if the internal pressure is ignored, the stresses on the borehole wall can be considered as zero. Thus, for any point (a, θ) located on the wellbore wall, its boundary conditions can be described as, x xy

cos + cos +

zx

cos

xy y

+

sin sin

yz

= =

sin

h H

cos sin

=0

(A16)

After substituting Eq. (A14) into Eq. (A16), the analytic function can be solved. Then, substituting the obtained analytic function into Eq. (14) leads to the corresponding stress components {σh}xyz (Pei, 2008), x,h y, h

= b1 = b2

= yz , h = zx , h = xy, h = z,h

+ c1 + c2

h h

(a31 b3 b4 b5

H H

+ a32

x,h

+ c3 h + c4 h + c5 h

y, h

+ a34

yz , h

+ a35

xz , h

+ a36

xy, h )/ a33

H H

(A17)

H

where

b1 =

Re[i 1 µ12 (

b2 =

Re[i 1 (

b3 = Re[i b4 =

b5 = Re[i 1 µ1 (

Re[ 1 µ12 (µ 2

c2 =

Re[ 1 (µ 2 1 1 (µ 2

+i

1 3)

µ3

2 3)

µ3

2 3)

µ3

2 3)

+

3 3( 1

+ i 3(

+

2 2 ( 1 3 µ3

2 3)

+

+

1 3 µ3 2 µ2 2 (

2 µ2 (

1 3 µ3

3( 1

µ1 ) + µ1 ) +

2 )]

1 2 )]

2 3 µ 3 3 ( µ1 2

3 3 ( µ1 2

1 3 µ3

2 )]

2 )]

+ i 3 µ3 (

µ1 ) +

µ1) +

3( 1 2 )]

1

+ i 3 µ3

2 2 µ 2 ( 1 3 µ3

2(

2 3)

1 3)

1 3)

+

µ3

+ i 3 µ32

3 ( µ1 2

µ1) + 3 µ3

µ 2 1)]

µ 2 1)] µ 2 1)] 3 µ 3 ( µ1

3 (µ1 2

2

µ 2 1)]

µ 2 1)]

(A18)

1

= k

= (µ 2

1 3)

2 2 (1

1) + i 2 µ 2 (1

µ3

1 3)

1) + i 2 µ 2 2 (1

2 3

Re[ 1 µ1 1 (µ 2

c5 = Re[ 1 µ1 (µ 2 k

1) + i

2 3

c1 =

c4 =

1) + i 2 (1

2 3

1 1( 2 3

Re[i 1 µ1 1 (

c3 = Re[

1) + i 2 µ 22 (1

2 3

(z k / a ) 2

µ1) +

1

2 3 ( µ1

µk2

µ3 ) +

(A19) 1 3 (µ 3

(A20)

µ2 )

833

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al.

k

(z k / a ) 2

zk /a +

=

1

z k = a (cos

µk2

1

(A21)

iµk

(A22)

+ µk sin )

Here σx,h, σy,h, σz,h, τxy,h, τyz,h, and τxz,h are the components of the stress tensor caused by drilling the borehole; a is the radius of the borehole; θ is the angle of circumference, (°). (3) Components of the stress tensor induced by wellbore pressure {σb}xyz If the internal pressure is taken into account, the stresses on the borehole wall equals to the wellbore pressure pm. Thus, for any point (a,θ) located on the wellbore wall, the boundary conditions can be described as,

cos + cos +

x xy zx

cos

xy y

+

sin sin

yz

= pm cos = pm sin

sin

=0

(A23)

Similarly, the stress components { h} xyz under this condition can be obtained (Pei, 2008), x,b y, b

= d1 pm = d2 pm

= (a31 = d3 pm zx , b = d 4 pm xy, b = d5 pm z,b

x ,b

+ a32

+ a34

y, b

yz, b

+ a35

xz , b

+ a36

xy, b )/ a33

yz , b

(A24)

with,

d1 = Re d2 = Re d3 =

1 (µ 2

+

µ3

3 3 ( µ1 2 1 µ1 (µ 2

Re

d 4 = Re d5 =

2 µ3 2 3 + i 2 3 1 µ1 (µ 2 + 3 µ32 3 (µ1 2 µ 2 1 + i 1

+ 3 µ3

µ3

1 1 µ1 (µ 2

µ3 2

2 3

µ2

1

+i

1

2 3

+i

2 3

1

+i

1

i

1 3)

µ1 + i

i

1 3)

2 µ 2 ( 1 3 µ3

µ1 + i

i

1 3)

i 2)

1

i) +

2 3

1

µ2

2 ( 1 3 µ3

i) +

2 3

+i

+i

µ1 + i

i 2)

1

+i

µ2

2 2 µ2 ( 1 3 µ3

i 2)

i) +

2 3

+i

1

2 3

µ3 2

1 1 (µ 2

+ 3 ( µ1

+i

µ2

3 ( µ1 2

+ 3 µ 3 (µ 1

Re

2 3

i) +

2 1 µ 2 ( 1 3 µ3

µ1 + i

i

1 3)

i 2) i) +

2 1 ( 1 3 µ3

µ1 + i

i

1 3)

i 2)

(A25)

where σx,b, σy,b, σz,b, τxy,b, τyz,b, and τxz,b are the components of the stress tensor induced by wellbore pressure. Finally, the stress distribution around the wellbore in transversely isotropic formation can be obtained by superposing the above-mentioned three stress components,

= = z = yz = zx = xy = x

y

+ + z,0 + yz,0 + yz,0 + yz,0 + x ,0 y,0

+ x ,b + y, b z, h + z , b yz, h + yz, b zx , h + zx , b xy, h + xy, b x ,h

y, h

(A26)

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.petrol.2019.01.090.

from well logs. J. Petrol. Technol. 25 (11), 1259–1268. Beswick, A.J., Gibb, F.G., Travis, K.P., 2014. Deep borehole disposal of nuclear waste: engineering challenges. Proc. Inst. Civ. Eng.-Energy 167 (2), 47–66. Celia, M.A., Nordbotten, J.M., 2009. Practical modelling approaches for geological storage of carbon dioxide. Gr. Water 47 (5), 627–638. Chen, G., Chenevert, M.E., Sharma, M.M., Yu, M., 2003. A study of wellbore stability in shales including poroelastic, chemical, and thermal effects. J. Petrol. Sci. Eng. 38 (3–4), 167–176. Constant, D.W., Bourgoyne, A.T., 1988. Fracture-gradient prediction for offshore wells. SPE Drill. Eng. 3 (2), 136–140. Cook, J., Growcock, F., Guo, Q., Hodder, M., van Oort, E., 2011. Stabilizing the wellbore to prevent lost circulation. Oilfield Rev. 23 (4), 26–35.

References Aadnoy, B.S., 1988. Modelling of the stability of highly inclined boreholes in anisotropic rock formations. SPE Drill. Eng. 3 (3), 259–268. Aadnoy, B.S., Chenevert, M.E., 1987. Stability of highly inclined boreholes. SPE Drill. Eng. 2 (4), 364–374. Aadnoy, B., Looyeh, R., 2011. Petroleum Rock Mechanics: Drilling Operations and Well Design. Gulf Professional Publishing. Amadei, B., 1983. Rock Anisotropy and the Theory of Stress Measurements. Springer Verlag, Berlin. Anderson, R.A., Ingram, D.S., Zanier, A.M., 1973. Determining fracture pressure gradients

834

Journal of Petroleum Science and Engineering 176 (2019) 821–835

T. Ma et al. Daines, S.R., 1982. Prediction of fracture pressures for wildcat wells. J. Petrol. Technol. 34 (4), 863–872. Davies, R., Foulger, G., Bindley, A., Styles, P., 2013. Induced seismicity and hydraulic fracturing for the recovery of hydrocarbons. Mar. Petrol. Geol. 45, 171–185. Detournay, E., Cheng, A.D., 1988. Poroelastic response of a borehole in a non-hydrostatic stress field. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 25 (3), 171–182. Do, D.P., Tran, N.H., Hoxha, D., Dang, H.L., 2017. Assessment of the influence of hydraulic and mechanical anisotropy on the fracture initiation pressure in permeable rocks using a complex potential approach. Int. J. Rock Mech. Min. Sci. 100, 108–123. Eaton, B.A., 1969. Fracture gradient prediction and its application in oilfield operations. J. Petrol. Technol. 21 (10), 1–353. Feng, Y., Gray, K.E., 2016a. A fracture-mechanics-based model for wellbore strengthening applications. J. Nat. Gas Sci. Eng. 29, 392–400. Feng, Y., Gray, K.E., 2016b. A parametric study for wellbore strengthening. J. Nat. Gas Sci. Eng. 30, 350–363. Feng, Y., Jones, J.F., Gray, K.E., 2016. A review on fracture-initiation and-propagation pressures for lost circulation and wellbore strengthening. SPE Drill. Complet. 31 (2), 134–144. Fridleifsson, G.O., Elders, W.A., 2005. The Iceland Deep Drilling Project: a search for deep unconventional geothermal resources. Geothermics 34 (3), 269–285. Gui, J.C., Ma, T.S., Chen, P., Yuan, H.Y., Guo, Z.X., 2018. Anisotropic damage to hard brittle shale with stress and hydration coupling. Energies 11 (4), 926. Gupta, D., Zaman, M., 1999. Stability of boreholes in a geologic medium including the effects of anisotropy. Appl. Math. Mech. 20 (8), 837–866. Guo, J., He, S., Deng, Y., Zhao, Z., 2015. New stress and initiation model of hydraulic fracturing based on nonlinear constitutive equation. J. Nat. Gas Sci. Eng. 27, 666–675. Haimson, B., Fairhurst, C., 1967. Initiation and extension of hydraulic fractures in rocks. SPE J. 7 (03), 310–318. Huang, R., 1984. A model for predicting formation fracture pressure. J. Univ. Pet. (China) 9 (4), 335–346. Huang, J., Griffiths, D.V., Wong, S.W., 2012. Initiation pressure, location and orientation of hydraulic fracture. Int. J. Rock Mech. Min. Sci. 49, 59–67. Hubbert, M.K., Willis, D.G., 1957. Mechanics of hydraulic fracturing. AIME Petroleum Transactions 210, 153–168. Jin, X., Shah, S.N., Roegiers, J.C., Hou, B., 2013. Breakdown pressure determination-a fracture mechanics approach. In: SPE Annual Technical Conference and Exhibition, 30 September-2 October, 2013, New Orleans, Louisiana, USA. (SPE 166434). Jin, Y., Zhang, X., Chen, M., 2005. Initiation pressure models for hydraulic fracturing of vertical wells in naturally fractured formation. Acta Pet. Sin. 26 (6), 113–114. Lekhnitskij, S.G., 1981. Theory of the Elasticity of Anisotropic Bodies. Mir Publishers, Moscow. Legarth, B., Huenges, E., Zimmermann, G., 2005. Hydraulic fracturing in a sedimentary geothermal reservoir: results and implications. Int. J. Rock Mech. Min. Sci. 42 (7–8), 1028–1041. Li, Z., Dong, M., Li, S., Huang, S., 2006. CO2 sequestration in depleted oil and gas reservoirs —caprock characterization and storage capacity. Energy Convers. Manag. 47 (11–12), 1372–1382. Liu, Y., Ma, T.S., Chen, P., Yang, C., 2018. Method and apparatus for monitoring of downhole dynamic drag and torque of drill-string in horizontal wells. J. Petrol. Sci. Eng. 164, 320–332. Matthews, W.R., Kelly, J., 1967. How to predict formation pressure and fracture gradient. Oil Gas J. 65 (8), 92–106. Ma, T.S., Chen, P., Zhao, J., 2016. Overview on vertical and directional drilling technologies for the exploration and exploitation of deep petroleum resources. Geomech. Geophys. Geo-Energy Geo-Resourc. 2 (4), 365–395. Ma, T.S., Wu, B.S., Fu, J.H., Zhang, Q.B., Chen, P., 2017a. Fracture pressure prediction for

layered formations with anisotropic rock strengths. J. Nat. Gas Sci. Eng. 38, 485–503. Ma, T.S., Zhang, Q.B., Chen, P., Yang, C.H., Zhao, J., 2017b. Fracture pressure model for inclined wells in layered formations with anisotropic rock strengths. J. Petrol. Sci. Eng. 149, 393–408. Ma, T.S., Liu, Y., Chen, P., Wu, B.S., 2017c. Fracture-initiation pressure analysis of horizontal well in anisotropic formations. In: 51st US Rock Mechanics/Geomechanics Symposium, 25-28 June, 2017, San Francisco, California, USA. (ARMA-2017-0031). Ma, T.S., Peng, N., Zhu, Z., Zhang, Q.B., Yang, C.H., Zhao, J., 2018. Brazilian tensile strength of anisotropic rocks: review and new insights. Energies 11 (2), 304. Nova, R., Zaninetti, A., 1990. An investigation into the tensile behaviour of a schistose rock. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 27 (4), 231–242. Ong, S.H., Roegiers, J.C., 1993. Influence of anisotropies in borehole stability. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 30 (7), 1069–1075. Ong, S.H., Roegiers, J.C., 1995. Fracture initiation from inclined wellbores in anisotropic formations. In: International Meeting on Petroleum Engineering, 14-17 November, 1995, Beijing, China. (SPE 29993). Oriji, A., Ogbonna, J., 2012. A new fracture gradient prediction technique that shows good results in Gulf of Guinea wells. In: Abu Dhabi International Petroleum Conference and Exhibition, 11-14 November, 2012, Abu Dhabi, UAE. (SPE 161209). Pei, J., 2008. Strength of Transversely Isotropic Rocks. PhD Dissertation. Massachusetts Institute of Technology, USA. Pilkington, P.E., 1978. Fracture gradient estimates in Tertiary basins. Petrol. Eng. Int. 8 (5), 138–148. Prioul, R., Karpfinger, F., Deenadayalu, C., Suarez-Rivera, R., 2011. Improving fracture initiation predictions on arbitrarily oriented wells in anisotropic shales. In: Canadian Unconventional Resources Conference, 15-17 November, 2011, Calgary, Alberta, Canada. (SPE 147462). Rummel, F., 1987. Fracture mechanics approach to hydraulic fracturing stress measurements. In: Atkinson, B.K. (Ed.), Fracture Mechanics of Rock, pp. 217–239. Sayers, C.M., 2013. The effect of anisotropy on the Young's moduli and Poisson's ratios of shales. Geophys. Prospect. 61 (2), 416–426. Serajian, V., Ghassemi, A., 2011. Hydraulic fracture initiation from a wellbore in transversely isotropic rock. In: Proceedings of the 45th US Rock Mechanics/Geomechanics Symposium, 26-29 June, San Francisco, California. (ARMA-11-201). Tsang, C.F., Bernier, F., Davies, C., 2005. Geohydromechanical processes in the Excavation Damaged Zone in crystalline rock, rock salt, and indurated and plastic clays—in the context of radioactive waste disposal. Int. J. Rock Mech. Min. Sci. 42 (1), 109–125. Wessling, S., Pei, J., Bartetzko, A., Dahl, T., Wendt, B.L., Marti, S.K., Stevens, J.C., 2009. Calibrating fracture gradients-An example demonstrating possibilities and limitations. In: International Petroleum Technology Conference, 7-9 December, 2009, Doha, Qatar. (IPTC 13831). Wu, B., Zhang, X., Jeffrey, R.G., Wu, B., 2012. A semi-analytic analysis of a wellbore in a non-isothermal low-permeable porous medium under non-hydrostatic stresses. Int. J. Solid Struct. 49 (13), 1472–1484. Zhang, J.C., Yin, S.X., 2017. Fracture gradient prediction: an overview and an improved method. Petrol. Sci. 14 (4), 720–730. Zhang, Y., Zhang, J., Yuan, B., Yin, S., 2018. In-situ stresses controlling hydraulic fracture propagation and fracture breakdown pressure. J. Petrol. Sci. Eng. 164, 164–173. Zhang, M., Bachu, S., 2011. Review of integrity of existing wells in relation to CO2 geological storage: what do we know? Int. J. Greenhouse Gas Control 5 (4), 826–840. Zhu, H.Y., Guo, J.C., Zhao, X., Lu, Q., Luo, B., Feng, Y.C., 2014. Hydraulic fracture initiation pressure of anisotropic shale gas reservoirs. Geomech. Eng. 7 (4), 403–430. Zimmermann, G., Blöcher, G., Reinicke, A., Brandt, W., 2011. Rock specific hydraulic fracturing and matrix acidizing to enhance a geothermal system—concepts and field results. Tectonophysics 503 (1–2), 146–154.

835