Wellbore stability analysis in transverse isotropic shales with anisotropic failure criteria

Journal of Petroleum Science and Engineering 176 (2019) 982–993

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Wellbore stability analysis in transverse isotropic shales with anisotropic failure criteria

T

Yuwei Lia,b,c,d, Ruud Weijermarsb,∗ a

Department of Petroleum Engineering, Northeast Petroleum University, Daqing, 163318, China Harold Vance Department of Petroleum Engineering, Texas A&M University, College Station, TX, 77843, USA c Institute of Unconventional Oil & Gas, Northeast Petroleum University, Daqing, 163318, China d PetroChina Dagang Oilfield Company, Tianjin, 300280, China b

ARTICLE INFO

ABSTRACT

Keywords: Transverse isotropy Borehole stability Shale Breakdown pressure Collapse pressure

The stress concentrations in the wall of a horizontal well in transversely isotropic shale are quantified and the stable mud weight window is established using a modified Hoek-Brown failure criterion. The upper margin of the safe drilling window is assumed due to the tensile breakdown pressure, which is computed with an anisotropic failure criterion considering the tensile strength and degree of anisotropy of the shale rock. Shear failure occurs at the lower boundary of the safe drilling window and is computed observing the rock's compressive strength anisotropy. The results show that when the degree of tectonic in-situ stress anisotropy increases, the safe pressure window between the tensile breakdown and shear collapse pressures decreases. We suggest that the narrowing of the safe drilling window as compared to isotropic cases may explain the increased occurrence of wellbore instability when drilling in shale. When the anisotropy of the elastic moduli increases further, both the breakdown pressure and collapse pressure decrease. Consequently, the range of recommended mud weights in the window for stable wellbores quickly narrows. Sensitivity analyses of the Poisson's ratio anisotropy effect on the breakdown pressure and collapse pressure indicate that the influence of Poisson's ratio anisotropy on borehole stability is very limited. The opposite holds for the Young's moduli, which are key factors contributing to narrowing of the safe drilling window when these moduli become more anisotropic.

1. Introduction

reasons being that the prediction of borehole tensile breakdown and compressive collapse pressures are not accurate enough to manage drilling fluid pressure effectively. In addition, shale is chemically active (Ekbote and Abousleiman, 2006) and drilling fluid seepage in shale formations will greatly affect the borehole stability, especially when drilling with water-based muds. Shale exhibits a high degree of mechanical (elastic) anisotropy, notably higher than in sandstone. If the influence of anisotropic mechanical characteristics on the stress distribution and rock strength around the wellbore can be modelled more accurately, the actual drilling of wells in shale will encounter fewer wellbore stability problems. Due to the difference of mechanical parameters in different directions (Fig. 1a&b), the stress concentrations around boreholes in anisotropic shales attain higher values than in isotropic formations (Li et al., 2015), which is one factor why shale wells are more prone to borehole instability. At present, the models used to calculate the stress around boreholes in transversely isotropic formations include single-porosity and multi-porosity poroelastic models. Multi-porosity poroelastic models are more complicated than

The drilling of stable wells in unconventional shale plays is technically challenging, because shales commonly are well-bedded and possess pervasive anisotropy with mechanical properties displaying transverse isotropy. Rather than approximating well behavior using an over-simplified isotropic model, the pre-drill and real-time stability analysis of shale wells need to take into account the transversely isotropic elastic properties. An appropriate wellbore stability model for shale needs to quantify the resulting stress concentrations near the well and evaluate failure conditions, especially for horizontal wells in the shale target zone. Improved borehole stability control is a key factor for the effective development of shale oil and gas fields. Not only will drilling time be shortened, but the required hydraulic fracture treatment would be less prone to borehole collapses. Borehole instability problems still occur during the drilling and completion of shale wells (Chen et al., 2003; Akhtarmanesh et al., 2013; Zhang, 2013; You et al., 2014), one of the

∗

Corresponding author. E-mail address: [email protected] (R. Weijermars).

https://doi.org/10.1016/j.petrol.2019.01.092 Received 26 April 2018; Received in revised form 17 January 2019; Accepted 28 January 2019 Available online 29 January 2019 0920-4105/ © 2019 Elsevier B.V. All rights reserved.

Journal of Petroleum Science and Engineering 176 (2019) 982–993

Y. Li and R. Weijermars

(2017b) proposed a numerical model to investigate the impact of fluid pressure on the breakdown. They came to the conclusion that shale deformation, permeability anisotropy and fluid properties have significant impacts on the tensile breakdown pressure. The studies cited above advanced the spectrum of concurrent theoretical methods developed for borehole stability analysis in shale formations. However, when calculating the stress distribution around the borehole in anisotropic rocks, the impact on the effective stress needs to be considered together with the changes in the breakdown and collapse pressures due to anisotropic strength characteristics. In the present study, a novel borehole stability analysis model for transverse isotropic shale formations is established by considering the impact of transverse isotropic properties of shale on the stress concentrations together with the changes in tensile breakdown and shear collapse criteria. The model presented below for wellbore stability is new because the failure criteria not only consider the mechanical characteristics of shale when calculating the stress distribution, but also consider the strength anisotropy when judging wellbore rock failure.

Fig. 1. Rock mechanical parameters in formations, without (a) and with (b) layered anisotropy.

models using single-porosity poroelastic model. Not only the fluid pressure in the primary pores, but also the fluid pressure in the other pore types such as fractures, will affect the stress distribution around boreholes. Abousleiman and Cui (1998) present closed-form solutions for the pore pressures and stress fields for the inclined borehole and the cylinder using the single-porosity poroelastic model. Their research showed that the anisotropic material coefficients play an important role in calculating the in-plane stress fields. Many previous studies have considered the key factor of anisotropy in the analysis of borehole stability (Lekhnitskii, 1963; Amadei, 1983; Karpfinger et al., 2011; Aadnoy, 1987; Wang, 1999; Zhao et al., 2013; Lu and Chen, 2013; Dong et al., 2015). For example, Jin et al. (2011), Liu et al. (2002) and Yuan et al. (2012) studied the borehole collapse pressure and breakdown pressure quantitatively by combining the traditional linear elastic borehole stress distribution model with weak plane strength theory, and analyzed the relationship between the borehole stability and a weak surface under different slope angles. Yan et al. (2013) found that shale gas reservoirs are mechanically anisotropic and borehole instability therefore is prone to occur; a borehole stability model in shale was created based on transverse isotropic theory and single plane of weakness theory. Liu et al. (2014) set up a calculation model for a horizontal open-hole well in a rock comprising multiple-weak-planes to analyze the factors affecting the collapse volume, which was used in a field application. Liu and Yu (2016) analyzed the collapse pressure of Bonan shale (Jiyang Depression, China) with the consideration of tri-axial stresses, and they established a more accurate prognosis model for the prediction of collapse pressure based on the weak plane strength theory. Considering the occurrence of each weak plane for variable borehole azimuth, borehole stress conditions and in-situ stress, Chen et al. (2015) established a collapse pressure prediction model for horizontal wells in shale formation with multiple weak planes. Liu et al. (2015) presented an in-situ stress model of three-dimensional fluid-solid coupling using percolation mechanics, and included the breakdown pressure according to the maximum tensile stress theory. Li et al. (2015) established a 3D numerical model to improve borehole breakdown predictions of horizontal borehole in laminated shales by incorporating the anisotropic elastic deformation and hydro-mechanical coupling effects. Ma et al. (2016) stated that mechanical-chemical coupling is the most important factor of borehole stability for horizontal drilling in shale gas reservoirs. Their effective stress tensor around a borehole is based on a pore pressure propagation model coupled with a stress distribution model using failure criteria for shale with mechanical weakness planes. In order to predict the breakdown pressure more accurately, Shan et al. (2017) proposed a staged finite element model approach to obtain the stress distribution around a horizontal borehole. Zhang et al. (2017a) modelled the stress field around a borehole in anisotropic shale and analyzed the effects of the elastic parameters' anisotropic ratios on the well circumferential stresses. Zhang et al.

2. Stress distribution around anisotropic boreholes 2.1. Model properties When drilling in conventional reservoirs such as sandstone, we often assume that the rock responds as an isotropic medium (Fig. 1a). For such isotropic cases, the elastic engineering constants are equal in all directions: (1)

Ex = E y = Ez = E xy

=

G=

yz

=

xz

(2)

=

E 2(1 + )

(3)

In contrast, for transverse isotropic shale formations (Fig. 1b), the elastic engineering constants are no longer isotropic, but become tensor quantities:

Ex = E y = Eh; xz

=

Gh =

=

v;

Eh 2(1 +

h)

yz

(4)

Ez = E v xy

;

=

(5)

h

Gxy = G h;

Gv =

Ex Ez Ex + Ez + 2

xz

Ez

;

Gyz = Gxz = Gv

(6) where Ex, Ey, Ez, Eh and Ev are the Young's moduli in different directions of the formation for a conveniently oriented coordinate system (Fig. 1b); υxy, υyz, υxz, υh and υv are Poisson's ratios; Gxy, Gyz, Gxz, Gh and Gv are the corresponding shear moduli. There are significant differences between the isotropic mechanical and the transverse isotropic mechanical properties [cf. Eqs. (1)–(6)]. Depending on whether the assumed model is an isotropic (Fig. 1a) or transverse isotropic one (Fig. 1b), the stress response to penetration by a borehole will be different for the same drilling parameters. Our aim is to apply a model to calculate the stress in transverse isotropic shale formations when perforated by a cylindrical drill hole (Fig. 1b), and establish the safe drilling window using anisotropic failure criteria. 2.2. Coordinate transformations Shale formations are mainly developed using horizontal and extended reach wells, and there is a certain angle between the direction of the borehole axis and the geographic coordinate system or the stress coordinate system. Coordinate system transformations are required to transform the far-field in-situ stresses from a geographic coordinate system to a borehole coordinate system (Jaeger et al., 2009). First, the far-field in-situ stress distribution is transformed from the principal stress coordinate system to the geographic coordinate system 983

Journal of Petroleum Science and Engineering 176 (2019) 982–993

Y. Li and R. Weijermars

Fig. 2. Coordinate system transformation (a) The principal stress coordinate system transform to the geographic coordinate system (b) The geographic coordinate system transform to the borehole coordinate system.

(Fig. 2a). Assume σp is the far-field stress tensor under the principal stress coordinate system; σg is the far-field stress distribution tensor under the geographic coordinate system; αpg is the azimuth of maximum horizontal principal stress; βpg is the angle between the direction of σv and Zg-axis. The transformation of the stress tensor σp from the principal stress coordinate system, to a stress tensor σg referring to the geographic coordinate system is given by (Lu et al., 2015): g

= R Tpg ×

p

=

p

0

H

0 0

h

1 Ex

a11 a21 a31 a41 a51 a61

(7)

× R pg cos

0 0 ; R pg =

0

presence and the pressure on the wellbore. Assume aij represents the anisotropic compliance matrix:

pg

pg

pg

sin

cos

v

cos

sin

sin

pg

pg

cos

cos

pg

sin

pg

pg

sin

pg

sin

pg

a12 a22 a32 a42 a52 a62

a13 a23 a33 a43 a53 a63

a14 a24 a34 a44 a54 a64

a15 a25 a35 a45 a55 a65

xy

a16 a26 a36 a46 = a56 a66

pg

cos

xz

Ez yz

1 Ey

Ey

0

xy

Ey

yz

Ez

Ez

0

0

0

0

0

1 Ez

0

0

0

0

0

Ez

xz

0

0

0

0

1 G yz

0

0

0

0

1 G xz

0

0

0

0

0

0

1 G xy

pg

(8)

(11)

Next, the far-field in-situ stress tensor σg under the geographic coordinate system, is transformed to the stress tensor σb under the borehole coordinate system (Fig. 2b). The stress transformation relationship are as follows:

The stresses at the margin of the borehole with internal pressure pw and outer boundary stresses σb due to an in-situ stress field can be computed from (Amadei, 1983):

xx , b b

= R bg ×

g

cos R bg = cos

×

bg

RTbg

cos

sin

bg

bg

sin

=

bg

xy, b

xz, b

yx , b

yy, b

yz , b

zx , b

zy, b

zz , b

sin

bg

cos

cos bg

sin

bg

bg

sin

bg

0

bg

sin

(9)

bg

cos

bg

x

=

xx , b

+ +

y

=

yy, b

xy

=

xy , b

xz

=

xz , b

yz

=

yz , b

z

(10)

=

+

x,i

=

xx , b

+ 2Re [µ12

y, i

=

yy, b

+ 2Re [

=

xy , b

2Re [µ1

xz , i

=

xz , b

+ 2Re [ 1 µ1

yz , i

=

yz , b

1 (a a33 31 x , i

zz , b

1 (z1)

xy , i

2Re [

+ a32

y, i

1

+ µ22

1 (z1)

+

2 (z 2 )

1 (z1)

+ a34

+ µ2

1 (z1)

1 (z1)

2 (z 2 )

+

+

yz, i

+

3

2 (z 2 ) 2 µ2

+

2 (z 2 )

+ a35

xz, i

3 (z 3)]

3 (z 3)]

+

2 (z 2 )

2

2 3 µ3 3 µ3

3 (z 3)]

+ µ3

+

3 (z 3)]

+ a36

xy, i )

3 (z 3)]

(12)

where σb is the far-field stress distribution tensor under the borehole coordinate system; σxx,b, σyy,b, σzz,b, τxy,b, τxz,b, τyz,b are the relevant farfield stress components under the borehole coordinate system; αbg is the azimuth of borehole axis; βpg is the deviation angle of the borehole axis.

where σxx,b, σyy,b, σzz,b, τxy,b, τxz,b, τyz,b are the far-field stress tensor elements under the borehole coordinate system; Re is the real component of a complex number; Φ’i are derivatives of the stress analytical functions; zi, μi and λi are imaginary numbers. In Eq. (12), Φ’i can be calculated from Eqs. (13–15):

2.3. Stress distribution algorithms When the coordinate system transformation is completed, we can directly apply the far-field in-situ stress to calculate the stress distribution around the wellbore in the borehole coordinate system. The calculation method above is based on the theoretical works of Lekhnitskii (1981), Amadei (1984), Serajian and Ghassemi (2011) and Lu et al. (2015). The far-field stress tensor components, σij,b, will modify near the borehole when the drill penetrates the rock. The induced stress distribution around the borehole can be calculated by taking into account the anisotropic perturbations due to the hole's

1 (z1)

1

= 2

+(

z1 2 a

( )

1

xy , b

i

xx , b

1

µ12

+ ipw )(

(i 2 3

xy , b

yy, b

1) + (

+ pw )(µ 2

yz , b

i

2 3 µ3 )

xz , b ) 3 (µ3

µ 2)] (13)

984

Journal of Petroleum Science and Engineering 176 (2019) 982–993

Y. Li and R. Weijermars

2 (z 2 )

1

= 2

+(

z2 2 a

( )

2

i

xy , b

xx , b

(i

µ22

1

xy , b

+ ipw )(1

yy, b

2 3)

+(

+ pw )( i

yz , b

1 3 µ3

µ1 )

xz , b ) 3 (µ1

1

= 2

+(

z3 2 a

( )

3

i

xy , b

xx , b

(i

µ32

1

+ ipw )(

xy , b 2)

1

yy, b

+(

yz , b

+ pw )( 2 µ1 i

xz , b )(µ 2

i

2 3 ( µ1

= ei ,

i = 1,2,3

µ3) +

1 3 (µ 3

µ1)]

= z =

z

11

2

45 µ +

2

µ3

µ4

16

44

+ (2

12

+

66

) µ2

2

26 µ

+

22

=0

(20)

In the above expressions, βij can be calculated by Eq. (21): ij

a i3

= aij

aj3 a33

,

i , j = 1,2,4,5,6

(21)

1

=

l3 (µ 2) ; l2 (µ 2 )

=

2

3

=

(22)

1

And the functions l3(μi) and l2(μi) are as follows:

l2 (µi ) =

2 55 µi

l3 (µi ) =

3 15 µi

2 (

45 µi + 14

+

2 56 ) µi

+(

25

+

46) µi

24

(24)

The stress distribution around the borehole given by Eq. (12) can be solved according to the above computational schedule. 2.4. Pore pressure Because shale is a porous medium, the effective stresses σ′j that impacts on the borehole stability can be determined from the total stress of Eq. (12) by subtracting the pore pressure Pp as follows (Shao, 1998; Tan et al., 2015): x

=

x

(1

M11 + M12 + M13 ) pp 3K s

y

=

y

(1

2M13 + M33 ) pp 3K s

z

=

z

(1

M11 + M12 + M13 ) pp 3K s

Ex (Ez M12 = (1 + h )(Ez M13 = M23 =

Ez

h

+ Ex Ez h

Ex Ez Ez h

2 v)

2Ex

v

2Ex v2

=

3

(

x)

y

+ cos 2

xy

(30)

3

+

3

m

ci

0.5

+s

(31)

ci

3

=

3

+

m

ci

3

0.5

+s

(32)

ci

Because of the tensile strength σt = σ3, Eq. (32) can be converted to:

4

(25)

2 v)

m

t

s=0

(33)

ci

Considering that both m and s are positive for the rock material constants, σt and σci are opposite, and the root is obtained from Eq. (33) as follows:

(26)

t 2 v)

0.5

t ci

Here Mij is stiffness matrix, and it can be calculated by Eqs. 26–29:

Ex (Ez + Ex v2 ) M11 = (1 + h )(Ez Ez h 2Ex

rz = 0 1 sin 2 2

where σci is the uniaxial compressive strength of the rock; m (m > 0) and s (0 ≤ s ≤ 1) are material constants. The values of m and s depend on the properties of the rock mass. For example, a rock mass subjected to strong disturbance, will have m = 0.001, hard and complete rock mass, m = 25. For a completely broken rock mass, s = 0, and completely in tact rock mass, s = 1 (Liu et al., 2013). Rock tensile strength can be established by conducting a Brazilian Test. According to Li et al. (2016), the shear stress is 0 in the disc center, and the minimum principal stress σ3 and maximum principal stress σ1 has the relationship: σ1 = −3σ3. By combining Eq. (31), the criterion for empirical tensile failure can be expressed as:

(23)

44

xy

An initial empirical tensile failure criteria was proposed by Hoek and Brown (1980) as follows:

l3 (µ3) l2 (µ3)

sin 2

3.1. Tensile failure

In Eq. (12), λi can be calculated by Eq. (22):

l3 (µ1) ; l2 (µ1)

y

z

Significant leak-off in the matrix will occur when the pressure of the drilling mud column in the borehole is greater than the tensile strength of borehole rock. Both the tensile and compressive strengths of transverse isotropic shale are also anisotropic material properties, which means that both the tensile and compressive strengths of the borehole rock are different in different directions of the borehole radial.

(19)

=0

+ cos2

3. Anisotropic borehole stability model

Here zi is a pure imaginary number; μ1 is the positive root of Eq. (19), and μ2 and μ3 are the positive roots of Eq. (20): 2 55 µ

x

The results of Eq. (30) are used in the remainder of our study to analyze the effect of anisotropy on the tensile and shear failure of the wellbore, when drilling in the transverse isotropic shale formation.

(18)

i = 1,2,3

=

r

(17)

+ µi sin ),

(29)

2 v

2E x

pp

sin2

=

As shown in Fig. 2b, under the borehole coordinate system, zi can be expressed as:

z i = a (cos

= pw

r

(16)

µ2)

h) h

The circumferential and radial stresses around the borehole when drilling in the shale formation can be converted from Cartesian to polar coordinates (Jaeger et al., 2009):

1 µ2)

where a is the radius of the borehole; pw is fluid pressure in the well. The parameters Δ, ηi and zi can be calculated using Eqs. 16–18:

µ1 +

Ez

2.5. Conversion from Cartesian to polar coordinates

(15)

= µ2

Ez

where Pp is the pore pressure and Ks is the bulk modulus of the solid phase.

µ3)] (14)

3 (z 3)

Ez2 (1

M33 =

(27)

=

ci

m 32

m2 s + 1024 16

0.5

(34)

The empirical formula for the variation of the compressive strength of different inclination angle of the bedding for intact rocks was established by Jaeger (1960) and Donath (1961):

(28) 985

Journal of Petroleum Science and Engineering 176 (2019) 982–993

Y. Li and R. Weijermars ci

=A

B [ cos 2(

(35)

)]

m

where A and B are constants, their unit is MPa; A and B are determined by the mechanical properties of the rocks. They can be calculated by equation (35) using the experimental results of the uniaxial compressive strength test. βm is the inclination angle of the bedding when the sample uniaxial compressive strength is minimum; β is the inclination angle of the bedding when uniaxial failure test is conducted, β = θ when at the borehole wall. Equation (35) is an empirical correction, and the authors suggest that this equation should be used when drilling through a shale formation with bedding development. Moreover, the formation depth should not be too deep such that the influence of thermal stress can be ignored. To investigate tension failure along the inclination angle of the bedding, we substitute Eq. (35) into Eq. (34) and the corresponding mechanical model is established. This yields an equation for the tensile strength of layered shale: t

= {A

B [cos 2(

)]}

m

m 32

m2 s + 1024 16

Fig. 3. Variations in circumferential stress around the wellbore computed using Eq. (30) after conversion to field units with input data from Table 1 and compared with reference curve of Serajian and Ghassemi (2011, their Fig. 4).

0.5

(36)

Eq. (36) reflects the characteristics of anisotropic tensile strength of shale, and the fracture initiates where and when the effective circumferential stress magnitude equals the rock's tensile strength: σ′θ = σt. The circumferential tangential stress required for tensile failure of the wellbore can be expressed as:

=

{A

B [cos 2(

)]}

m

m2 s + 1024 16

m 32

results (Fig. 3), which confirms that our calculations are accurate. Next, the model of anisotropic borehole stresses can be used to evaluate the wellbore stability of anisotropic shales. We will calculate the borehole tensile breakdown pressure and shear collapse pressure in a transverse isotropic shale formation by applying our model in a synthetic case study.

0.5

(37)

4.2. Synthetic case study

The tensile breakdown pressure can be calculated by using Eq. (37), which indicates that the breakdown pressure will vary with the direction of the borehole radial.

We assume that the horizontal well extends along the direction of the horizontal maximum principal stress (Fig. 4), and the in-situ stress coordinate system is consistent with the geographical coordinate system. Under this assumption, the borehole coordinates Xb, Yb and Zb are in the same direction as native, far-field stresses σv, σh and σH, used in Eq. (8), respectively. The rock mechanical parameters and in-situ stress data used for our synthetic case study calculations are shown in Table 2. The circumferential stress variation with the angle around the borehole was calculated using Eq. (30) with Table 2 inputs (Fig. 5a). The circumferential stress varies periodically with the tangential angle, the maximum tensile stress of −1.35 MPa occurs at 90° +π, and the maximum compressive stress of 36.23 MPa occurs at 0° +π. The results indicate that the borehole rock is prone to tensile rupture in the vertical direction (90° +π). The synthetic example assumes Andersonian, normal fault stress conditions. Normal faulting Andersonian stress regimes in rocks with isotropic elastic properties will also form vertical cracks when the borehole pressure exceeds the critical strength (Thomas and Weijermars, 2018). However, in transverse isotropic shale the stress concentrations around the wellbore are higher as follows from the comparison in Fig. 5b. The impact of the modulus anisotropy expressed by the ratio Ev/Eh was also evaluated with several discrete values (0.2, 0.6, 1, 2 and 6), using Pw = 50.48 MPa. The results of Fig. 5b show that with transverse isotropy in the horizontal plane as in Fig. 4, which occurs when 0 < Ev/Eh < 1, the tangential stress maximum occupies a broader region with a lower maximum stress as compared to the isotropic case Ev/Eh = 1. Reversely, when the transverse isotropy is parallel to the vertical direction (rotated 90° from that shown in Fig. 4), which occurs when 1 < Ev/Eh < ∞, the tangential stress peaks at higher values as comared to the isotropic case (Fig. 5b).

3.2. Compressive shear failure For compressive strength of a layered shale, the empirical HoekBrown failure criterion was initially proposed by Hoek and Brown (1980) for intact rocks, and theoretical calculation methods of compressive strength were revised and consummated by various authors (Ramamurthy et al., 1988; Saroglou and Tsiambaos, 2008; Ismael et al., 2014). The collapse of the borehole occurs when the drilling pressure is too small to maintain the stress balance, and shear failure is initiated under the condition of compressive stress. According to the shear failure criterion of Eq. (31), and because of the stress equivalences σ′θ = σ1 and σ′r = σ3, one can substitute Eq. (35) into Eq. (31), and we get: 0.5

=

r

+ {A

B [cos 2(

m

)]} m

A

B [cos 2(

m

)]

+1 (38)

Combining Eqs. (38) and (30), the shear collapse pressure when drilling in a transverse isotropic shale formation can be determined. 4. Model application 4.1. Stress model validation The circumferential stress of the borehole must be calculated correctly in order to determine the tensile breakdown pressure and the shear collapse pressure for wellbore stability analysis. Therefore, we first verify the calculation result of circumferential stress obtained by the stress model of Section 2. The circumferential stress profile computed with our model (Fig. 3) is compared to that of Serajian and Ghassemi (2011), using the same input parameters for a nearly isotropic test case used in their study (Table 1). The circumferential stresses calculated by our model are consistent with the reference

4.3. Tensile strength variation The anisotropy of the tensile strength of the wellbore is calculated using Eq. (36). The tensile strength of the borehole wall rock is not a constant value, but varies with the tangential position around the well 986

Journal of Petroleum Science and Engineering 176 (2019) 982–993

Y. Li and R. Weijermars

Table 1 Input parameters for the verification model (after Serajian and Ghassemi, 2011). No. 1 2 3 4 5 6 7 8 9 10

Parameters maximum horizontal principal stress minimum horizontal principal stress vertical stress, MPa; pore pressure Young's modulus in orientation of Xg-axis Young's modulus in orientation of Yg-axis Young's modulus in orientation of Zg-axis Poisson's ration in planes of XgYg Poisson's ration in planes of YgZg Poisson's ration in planes of XgZg

Symbol

Unit

σH σh σv pp Ex Ey Ez υxy υyz υxz

Value

Applied in Eqs 6

psi psi psi psi psi psi psi / / /

0.0085 × 10 0.008 × 106 0.01 × 106 0.0061 × 106 14.00001 × 106 14.00001 × 106 1.4 × 106 0.2 0.19999 0.19999

(7), (8) (7), (8) (7), (8) (25) (11) (11) (11) (11) (11) (11)

Note: The results of the stress calculation are verified, so we only need the parameters in Table 1. Other parameters can be set to 0.

compaction caused by the minimum horizontal principal stress. When the well circumference angle is 0°, tensile strength of borehole rock is caused by the compression in the direction of vertical, maximum principal stress. The vertical stress is much larger than the minimum horizontal principal stress, and the deviatoric stress is compressional whereas the horizontal principal deviatoric stress is tensional. The tensile strength of the anisotropic shale is smallest in the direction of the minimum horizontal principal stress. When the circumferential angle of the borehole is 90°, the tensile strength is at a minimum. Leakoff will occur when the effective circumferential stress magnitude is equal to or greater than the tensile strength (Fig. 6b). Finally, according to the data in Table 2, the tensile breakdown pressure and shear collapse pressure are calculated to be 50.48 MPa and 30.37 MPa, respectively. 4.4. Sensitivity analysis: impact of in-situ stress anisotropy coefficient on wellbore stability In order to analyze the influence of stress anisotropy on borehole stability, we define the stress anisotropy coefficient:

K =

v

(39)

h

Fig. 4. Model of the case study borehole with assumed boundary conditions.

In the sensitivity analysis, we keep σh constant, increase σv continuously, and increase Kσ from 1.0 to 1.4 with incremental increases of 0.1. The collapse pressure and breakdown pressure vary with the stress anisotropy coefficient as shown in Fig. 7. The effect of stress anisotropy on rock collapse pressure and breakdown pressure was analyzed by using Eqs. (30), (37) and (38). The collapse pressure and the breakdown

(Fig. 6a). The tensile strength changes cyclically, reaching a minimum value of 1.35 MPa at 90°, and maximum of 19.71 MPa at 0°. The variation is relatively easy to explain: When the well circumference angle is 90°, the tensile failure of the borehole needs to overcome the Table 2 The parameters for the using of model calculation. No

Parameters

Symbol

Unit

Value

Applied equations

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

maximum horizontal principal stress minimum horizontal principal stress vertical stress, MPa; pore pressure Young's modulus in orientation of Xg-axis Young's modulus in orientation of Yg-axis Young's modulus in orientation of Zg-axis Poisson's ration in planes of XgYg Poisson's ration in planes of YgZg Poisson's ration in planes of XgZg radius of the borehole bulk modulus of solid phase material constant material constant material constant material constant inclination angle of the bedding when the sample uniaxial compressive strength is minimum

σH σh σv pp Ex Ey Ez υxy υyz υxz a Ks A B m s βm

MPa MPa MPa MPa GPa GPa GPa / / / m GPa MPa MPa MPa MPa °

38.4 36.2 45.2 18.1 25.1 25.1 19.3 0.19 0.17 0.17 0.1 69 72.7 63.4 0.79 0.45 45

(7), (8) (7), (8) (7), (8) (25) (11) (11) (11) (11) (11) (11) (18) (25) (36), (37), (38) (36), (37), (38) (36), (37) (36), (37) (36), (37), (38)

Note: The parameters No. 1–11 according to Ma and Chen (2014), Chen (2017), Serajian and Ghassemi (2011); No. 12 according to Tan et al. (2015); No.13–14 according to Liu et al. (2013), and No. 15–17 according to Shi et al. (2016). 987

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Fig. 5. The tangential stress variation for (a) Table 2 data with Ev/Eh = 0.77 (b) Table 2 data with sensitivity to Ev/Eh. Fig. 6. Tensile strength and leak-off point of borehole rock (a) Tensile strength of borehole rock varies with the circumferential angle (b) Leak-off occurs when effective circumferential stress magnitude is equal to the tensile strength.

pressure both decrease when the stress anisotropy coefficient increases (Fig. 7). The tensile breakdown pressure decreases from 61.71 to 43.65 MPa, and the collapse pressure for shear failure decreases from 35.96 to 27.07 MPa. Consequently, the safe drilling fluid density window gradually narrows when the stress anisotropy increases. Our conclusion is that the anisotropy of in-situ stress has a significant influence on the stability of the wellbore, which is why any local increase in the anisotropy of stress generally prompts drilling engineers to pay closer attention to wellbore stability analysis and control. The reason for the drastic reduction in the breakdown pressure is mainly the change in the circumferential stress around the wellbore (using Eq. (37)). Figs. 8 and 9 show that the circumferential wellbore stress σθ reaches the shear breakdown pressure at different in-hole pressures when the stress anisotropy coefficient Kσ changes. For example, when the stress anisotropy coefficient increases, the borehole circumferential stress anisotropy will increase accordingly. When the stress anisotropy coefficient Kσ increases from 1.0 to 1.4, the maximum value of borehole circumferential stress increases from 0.11 to 58.35 MPa: An increase of nearly 530 times. Increase of the circumferential stress due to elastic anisotropy makes the wellbore more prone to tensile failure and thus promotes instability of the borehole.

Fig. 7. Influence of the stress anisotropy coefficient [Eq. (39)] on the safe drilling margins of the wellbore stability window. 988

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Fig. 10. Sensitivity of the circumferential stress anisotropy intensity [Eq. (40)] (black line, left scale) to the stress anisotropy coefficient [Eq. (39)] (horizontal scale). Breakdown pressure (red line, right scale) varies with stress anisotropy coefficient [Eq. (39)]. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 8. The influence of the stress anisotropy coefficient [Eq. (39)] on the circumferential stress magnitude.

4.5. Sensitivity analysis: impact of circumferential stress anisotropy coefficient on wellbore stability In an alternative approach, we analyze the effect of the in-situ stress anisotropy on the borehole circumferential stress using the

Fig. 9. Cloud image of circumferential stress distribution for three different stress anisotropy coefficients (a) 1.0, (b) 1.2 and (c) 1.4. 989

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Fig. 11. Cloud image of the difference between circumferential stress and radial stress for three different stress anisotropy coefficients (a) 1.0, (b) 1.1 and (c) 1.3.

circumferential stress anisotropy intensity normalized by the circumferential stress anisotropy for the isotropic elastic continuum case: K

=

K max 1.0 max

K min 1.0 min

,

K = 1.0, 1.1, ...1.4.

Next, we define the intensity ratio of circumferential stress and radial stress difference as: K

(40)

=

( (

K r )max 1.0 r )max

,

K = 1.0, 1.1, ...1.4.

(41)

The variation of the intensity ratio (of the circumferential stress and radial stress difference) is calculated (according to Eq. (41)), and plotted in Fig. 12. When the stress anisotropy increases, the circumferential stress and radial stress difference increases linearly. For example, when the stress anisotropy coefficient Kσ increases from 1.0 to 1.4, ΔΔKσ increases from 1.0 to 8.72 (Fig. 12, black line, left scale). The corresponding borehole collapse pressure decreases linearly (Fig. 12, red line, right scale). The results of Fig. 12 show that due to the enhanced anisotropy of formation stress, the shear collapse pressure decreases. This means that with the increase of the anisotropy of the formation stress, the mud weight pressure used to ensure borehole does not occur collapse is reduced. But the anisotropy of formation stress reduces the difference between the breakdown pressure (Fig. 7) and collapse pressure, and the controllable range of drilling fluid density in safety drilling becomes smaller, which requires more precise control of borehole stability to ensure safe drilling. Higher anisotropy of in-situ stress is less favorable to wellbore stability due to a reduction in safe drilling margins. The results show that due to the enhanced anisotropy of formation stress, a smaller hydrostatic pressure of drilling fluid can maintain the stress balance of the rock wall. Such stress balance of the borehole is

The variation of circumferential stress anisotropy according to Eq. (40) is included in Fig. 10 (left-hand scale). An increase in the anisotropy of the in-situ stress, translates to the circumferential stress anisotropy intensity of the borehole increasing linearly. The anisotropy coefficient of in-situ stress Kσ increases from 1.0 to 1.4, and the circumferential stress anisotropy ΔKσ increases from 1.0 to 48.31. The corresponding breakdown pressure decreases linearly, which again shows that and increase of the anisotropy of the formation will increase the circumferential stress anisotropy. The stress concentration on the borehole will be more pronounced for increased stress anisotropy. As a direct consequence, the wellbore will be more prone to tensile failure. Fig. 11 visualizes the influence of in-situ stress anisotropy on the collapse pressure of borehole. When the anisotropy of in-situ stress increases, the difference between the circumferential stress and the radial stress increases, so that shear failure can occur under the lower mud weight pressure, and the collapse pressure decreases with the increase of anisotropy of in-situ stress. The main reason of the wellbore collapse is that the pressure of the drilling fluid cannot maintain the stress balance of the borehole rock, and does not preclude the circumferential stress and radial stress difference from reaching the critical condition of shear failure. 990

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Fig. 12. Sensitivity of the difference between the circumferential stress and radial stress [Eq. (41)] (black line, left scale) to the stress anisotropy coefficient [Eq. (39)] (horizontal scale). Collapse pressure (red line, right scale) varies with stress anisotropy coefficient [Eq. (39)]. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 13. Influence of the elastic modulus anisotropy coefficient [Eq. (42)] on the safe drilling margins of the wellbore stability window.

aimed for to make the borehole remains stable and easy to control by density of the fluid used in the drilling process However, the anisotropy of shale makes the difference between breakdown pressure and collapse pressure smaller, and the controllable range of drilling fluid density for safe drilling becomes smaller, which reduces borehole stability. Therefore, the stronger the anisotropy of in-situ stress is, the more unfavorable to the control of borehole stability. 5. Sensitivity analysis: impact of anisotropic elastic engineering modulus on wellbore stability The differences in wellbore stability between transverse isotropic shale and isotropic rock are mainly reflected in the elastic moduli and Poisson's ratio. For the transverse isotropic shale these mechanical parameters are different in horizontal and vertical directions. In order to better understand and explain the influence of the special mechanical properties of transverse isotropic shale on borehole stability, the respective effects of the elastic moduli and Poisson's ratio on collapse pressure and breakdown pressure of borehole are separately evaluated below.

Fig. 14. The influence of the Poisson's ratio anisotropy coefficient [Eq. (43)] on the safe drilling margins of the wellbore stability window.

K =

We define the elastic modulus anisotropy coefficient:

Ev Eh

(43)

We keep υh constant and change υv and use Kυ with stepwise decreases from 1.0 to 0.6 to analyze the influence of the change of Poisson's ratio in different directions on the borehole stability. When the anisotropy of Poisson's ratio of transverse isotropic shale increases (Fig. 14), the collapse and breakdown pressures only show a slight decrease, which impact is not very significant. Although the difference between breakdown pressure and the collapse pressure of the safety drilling range is slightly decreased, the decrease is not significant which implies that the rock Poisson's ratio anisotropy has very limited influence on the borehole stability.

5.1. Elastic moduli anisotropy and wellbore stability

KE =

v h

(42)

For our sensitivity analysis we keep Eh constant and change Ev, and make KE decrease from 1.0 to 0.6 (at increments of 0.1) to analyze the influence of the change of elastic modulus in different directions on the borehole stability. When the elastic modulus anisotropy of transverse isotropic shale increases (Fig. 13.), then both the breakdown and collapse pressures will decrease. The breakdown pressure is more affected by the anisotropy of elastic modulus. The difference between the breakdown and collapse pressures decreases when the elastic modulus are more anisotropic and the safe drilling window narrows.

6. Discussion 6.1. Principal insights In this paper, the wellbore stability model for transverse isotropic shale is advanced by considering the various mechanics parameters in the vertical direction and the horizontal direction as well as the anisotropy of tensile strength and shear failure strength in different directions of the borehole. The influence of the in-situ stress conditions and the anisotropic characteristics of rock mechanics parameters on the collapse pressure and breakdown pressure of borehole in drilling process is calculated and analyzed. The transverse isotropic mechanical

5.2. Poisson's ratio anisotropy and wellbore stability Similarly, we define the Poisson's ratio anisotropy coefficient: 991

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properties have a significant effect on the borehole stability of shale wells. When the anisotropy of the elastic moduli increases, the impact on the borehole stability reduces the safe drilling window. Increased anisotropy of the in-situ stress also has a significant impact on the stability. The larger the stress anisotropy, the more difficult will be the preservation of wellbore stability, because the safe drilling window is substantially narrower for strongly anisotropic in-situ stresses.

neglected as in previous studies (Wang and Shen, 2017; Leea and Pietruszczakb, 2017). However, the intermediate principal stress may impact the rock failure process (Mohammad and Pinnaduwa, 2016). A broad range of failure criteria was compared under isotropic stress assumptions (Wang and Weijermars, 2019). The effect of anisotropic failure criteria is first advanced in the present study. 7. Conclusions

6.2. Practical application and model inputs

This paper considers the geo-mechanical impact of a layered shale as a transverse isotropic elastic medium, and accounts for the associated anisotropy of its tensile strength and shear failure strength. A wellbore stability model that accounts for both the enhanced stress anisotropy and change in failure criteria due to the elastic anisotropy shale provides an important supplement to, and improvement over, isotropic WBA methods. The following systematic insights were obtained by applying the model to a synthetic case study, expanded with a sensitivity analysis to various key parameters:

The present study analyzes borehole stability under the assumption of transversely isotropic elasticity. Shale formations are known to be transverse isotropic. Measurements of the elastic stiffnesses on shale have confirmed these are commonly transverse anisotropic (Laubie and Ulm, 2014). Rather than measuring the stiffnesses under laboratory conditions, sonic dipole logging tools can measure the five non-vanishing stiffness tensor components (C11, C13, C, C44, and C55) directly in the borehole (e.g. see Brooks et al., 2015; Aderibigbe et al., 2016). The methodology to obtain such parameters for the sonic dipole tool have been detailed in Herwanger and Koutsabeloulis (2011). The stiffnesses can be translated using standard conversion expressions to obtain the elastic engineering constants (in our model Ex, Ey, Ez, Eh and Ev; υxy, υyz, υxz, υh and υv). Further inputs required for computing the magnitude and orientation of the resulting anisotropic stress concentrations near the borehole are the local tectonic stress tensor components (as in Table 1). Separately, a number of intrinsic rock parameters are required for quantifying the anisotropic strength, based on which the safe drilling window for a certain depth can be calculated. The acquisition of key failure strength parameters (Ks, A, B, m, s and βm) is currently only possible via laboratory tests (as per references cited in Table 2). The limited availability of such input parameters remains an obstacle for wider practical application of our model until such data become available for the specific rock formation studied. Our study highlight the necessity to secure such experimental data of anisotropic failure as a key step for more accurate wellbore stability model analyses when drilling shale formations.

(1) The strength of horizontal shale is intrinsically anisotropic, which is why the anisotropy of tensile and compressive strengths (on breakdown pressure and collapse pressure, respectively) should be considered in the wellbore stability analysis. (2) Separately, the in-situ stress anisotropy significantly influences the stability of any borehole. When the in-situ stress anisotropy increases, the difference between the circumferential stress and radial stress also increases. The breakdown pressure and collapse pressure will then decrease, and the safe drilling window decreases gradually, which requires close attention in wellbore stability analysis to ensure safe drilling conditions. (3) With increasing anisotropy in the elastic moduli, both the breakdown pressure and collapse pressure will decrease, leading to a gradual narrowing of the safe drilling window. The tensile breakdown pressure is sensitive to elastic anisotropy. (4) With increasing anisotropy of the Poisson's ratio, the breakdown pressure and collapse pressure remain basically unchanged, which indicates - unlike the Young's modulus anisotropy - that the influence of Poisson's ratio anisotropy on wellbore stability is very limited.

6.3. Model limitations

Conflicts of interest

There are still many shortcomings and areas for improvement in the research of mechanically anisotropic shales. For example, in the calculation of breakdown pressure and collapse pressure, shale is commonly considered as a transverse isotropic medium for layered sediments, which ignores the influence of structural weakness such as sedimentary bedding. Secondly, the establishment of the transverse isotropic shale borehole stress distribution model can only calculate the stress distribution at the borehole wall, and cannot calculate and analyze the stress distribution away from the borehole. Therefore, the model can only be used to analyze the instability initiated at the borehole wall. In addition, artifacts further away from the borehole wall, such as the presence of structural weaknesses, may cause shear failure and instability of the borehole when various stresses reach failure conditions. Establishing a calculation model for solving the stress distribution away from the borehole wall remains a challenging task, was recently solved for cases where the wellbore remains aligned with the plane of isotropy in transverse isotropic shale (Weijermars et al., 2019). Thirdly, pore pressure is considered constant and thus simply subtracted from the total stress to give the effective stress. Any time-dependent coupling between pore fluid diffusion and solid deformation is ignored. In case such coupling were to occur, the dynamic increase of pore pressure at the borehole wall under the action of seepage may decrease the actual breakdown pressure, and the calculated breakdown pressure will be higher than the actual value. Fourthly, the stable mud weight window is established using a modified Hoek-Brown failure criterion, and the effect of the intermediate principal stress is

The authors declare no conflict of interest. Acknowledgement This research was supported by the China Postdoctoral Science Foundation funded project (2018M640289). Nomenclature Ex, Ey, Ez, Eh and Ev are Young's moduli in different orientations of formation rock, MPa υxy, υyz, υxz, υh and υv are Poisson's ration in different planes of formation rock, dimensionless number Gxy, Gyz, Gxz, Gh and Gv are shear moduli in different planes of formation rock, MPa σH is maximum horizontal principal stress, MPa is minimum horizontal principal stress, MPa σh σv is vertical stress, MPa is the far-field stress tensor under the principal stress coσp ordinate system σg is the far-field stress distribution tensor under the geographic coordinate system αpg is the azimuth of maximum horizontal principal stress, (°) βpg is the angle between the direction of σv and Zg-axis, (°) 992

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σb

is the far-field stress distribution tensor under the borehole coordinate system σxx,b, σyy,b, σzz,b, τxy,b, τxz,b, τyz,b are the far-field stress under the borehole coordinate system, MPa αbg is the azimuth of borehole axis, (°) is the deviation angle of the borehole axis, (°) βbg ℜe is real component of an imaginary number Φ’i is derivatives of the stress analytical functions zi, μi and λi are complex numbers a is the radius of the borehole, m; pw is fluid pressure in the well, MPa is stiffness matrix of borehole rock, MPa Mij Pp is the pore pressure, MPa Ks is bulk modulus of solid phase, MPa σci is the uniaxial compressive strength of the rock, MPa m and s are material constants of rock mass, and s = 1 for intact rocks σt is the tensile strength of rock, MPa A and B are constants βm is the inclination angle of the bedding when the sample uniaxial compressive strength is minimum, ° β is the inclination angle of the bedding when uniaxial compression test is conducted, °

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