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Journal of Petroleum Science and Engineering

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Wellbore stability analysis in transverse isotropic shales with anisotropicfailure criteriaYuwei Lia,b,c,d, Ruud Weijermarsb,∗a Department of Petroleum Engineering, Northeast Petroleum University, Daqing, 163318, ChinabHarold Vance Department of Petroleum Engineering, Texas A&M University, College Station, TX, 77843, USAc Institute of Unconventional Oil & Gas, Northeast Petroleum University, Daqing, 163318, Chinad PetroChina Dagang Oilfield Company, Tianjin, 300280, China

A R T I C L E I N F O

Keywords:Transverse isotropyBorehole stabilityShaleBreakdown pressureCollapse pressure

A B S T R A C T

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

1. Introduction

The drilling of stable wells in unconventional shale plays is tech-nically challenging, because shales commonly are well-bedded andpossess pervasive anisotropy with mechanical properties displayingtransverse isotropy. Rather than approximating well behavior using anover-simplified isotropic model, the pre-drill and real-time stabilityanalysis of shale wells need to take into account the transversely iso-tropic elastic properties. An appropriate wellbore stability model forshale needs to quantify the resulting stress concentrations near the welland evaluate failure conditions, especially for horizontal wells in theshale target zone.

Improved borehole stability control is a key factor for the effectivedevelopment of shale oil and gas fields. Not only will drilling time beshortened, but the required hydraulic fracture treatment would be lessprone to borehole collapses. Borehole instability problems still occurduring the drilling and completion of shale wells (Chen et al., 2003;Akhtarmanesh et al., 2013; Zhang, 2013; You et al., 2014), one of the

reasons being that the prediction of borehole tensile breakdown andcompressive collapse pressures are not accurate enough to managedrilling fluid pressure effectively. In addition, shale is chemically active(Ekbote and Abousleiman, 2006) and drilling fluid seepage in shaleformations will greatly affect the borehole stability, especially whendrilling with water-based muds. Shale exhibits a high degree of me-chanical (elastic) anisotropy, notably higher than in sandstone. If theinfluence of anisotropic mechanical characteristics on the stress dis-tribution and rock strength around the wellbore can be modelled moreaccurately, the actual drilling of wells in shale will encounter fewerwellbore stability problems. Due to the difference of mechanical para-meters in different directions (Fig. 1a&b), the stress concentrationsaround boreholes in anisotropic shales attain higher values than inisotropic formations (Li et al., 2015), which is one factor why shalewells are more prone to borehole instability. At present, the modelsused to calculate the stress around boreholes in transversely isotropicformations include single-porosity and multi-porosity poroelasticmodels. Multi-porosity poroelastic models are more complicated than

https://doi.org/10.1016/j.petrol.2019.01.092Received 26 April 2018; Received in revised form 17 January 2019; Accepted 28 January 2019

∗ Corresponding author.E-mail address: r.weijermars@tamu.edu (R. Weijermars).

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

Available online 29 January 20190920-4105/ © 2019 Elsevier B.V. All rights reserved.

T

models using single-porosity poroelastic model. Not only the fluidpressure in the primary pores, but also the fluid pressure in the otherpore types such as fractures, will affect the stress distribution aroundboreholes. Abousleiman and Cui (1998) present closed-form solutionsfor the pore pressures and stress fields for the inclined borehole and thecylinder using the single-porosity poroelastic model. Their researchshowed that the anisotropic material coefficients play an important rolein calculating the in-plane stress fields.

Many previous studies have considered the key factor of anisotropyin 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 collapsepressure and breakdown pressure quantitatively by combining thetraditional linear elastic borehole stress distribution model with weakplane strength theory, and analyzed the relationship between theborehole stability and a weak surface under different slope angles. Yanet al. (2013) found that shale gas reservoirs are mechanically aniso-tropic and borehole instability therefore is prone to occur; a boreholestability model in shale was created based on transverse isotropictheory and single plane of weakness theory. Liu et al. (2014) set up acalculation model for a horizontal open-hole well in a rock comprisingmultiple-weak-planes to analyze the factors affecting the collapse vo-lume, which was used in a field application. Liu and Yu (2016) analyzedthe collapse pressure of Bonan shale (Jiyang Depression, China) withthe consideration of tri-axial stresses, and they established a more ac-curate prognosis model for the prediction of collapse pressure based onthe weak plane strength theory.

Considering the occurrence of each weak plane for variable bore-hole azimuth, borehole stress conditions and in-situ stress, Chen et al.(2015) established a collapse pressure prediction model for horizontalwells in shale formation with multiple weak planes. Liu et al. (2015)presented an in-situ stress model of three-dimensional fluid-solid cou-pling using percolation mechanics, and included the breakdown pres-sure according to the maximum tensile stress theory. Li et al. (2015)established a 3D numerical model to improve borehole breakdownpredictions of horizontal borehole in laminated shales by incorporatingthe anisotropic elastic deformation and hydro-mechanical coupling ef-fects. Ma et al. (2016) stated that mechanical-chemical coupling is themost important factor of borehole stability for horizontal drilling inshale gas reservoirs. Their effective stress tensor around a borehole isbased on a pore pressure propagation model coupled with a stressdistribution model using failure criteria for shale with mechanicalweakness planes. In order to predict the breakdown pressure more ac-curately, Shan et al. (2017) proposed a staged finite element modelapproach to obtain the stress distribution around a horizontal borehole.Zhang et al. (2017a) modelled the stress field around a borehole inanisotropic shale and analyzed the effects of the elastic parameters'anisotropic ratios on the well circumferential stresses. Zhang et al.

(2017b) proposed a numerical model to investigate the impact of fluidpressure on the breakdown. They came to the conclusion that shaledeformation, permeability anisotropy and fluid properties have sig-nificant impacts on the tensile breakdown pressure.

The studies cited above advanced the spectrum of concurrent the-oretical methods developed for borehole stability analysis in shaleformations. However, when calculating the stress distribution aroundthe borehole in anisotropic rocks, the impact on the effective stressneeds to be considered together with the changes in the breakdown andcollapse pressures due to anisotropic strength characteristics. In thepresent study, a novel borehole stability analysis model for transverseisotropic shale formations is established by considering the impact oftransverse isotropic properties of shale on the stress concentrationstogether with the changes in tensile breakdown and shear collapsecriteria. The model presented below for wellbore stability is new be-cause the failure criteria not only consider the mechanical character-istics of shale when calculating the stress distribution, but also considerthe strength anisotropy when judging wellbore rock failure.

2. Stress distribution around anisotropic boreholes

2.1. Model properties

When drilling in conventional reservoirs such as sandstone, weoften assume that the rock responds as an isotropic medium (Fig. 1a).For such isotropic cases, the elastic engineering constants are equal inall directions:

= = =E E E Ex y z (1)

= = =xy yz xz (2)

=+

G E2(1 ) (3)

In contrast, for transverse isotropic shale formations (Fig. 1b), theelastic engineering constants are no longer isotropic, but become tensorquantities:

= = =E E E E E;x y zh v (4)

= = =;xz yz xyv h (5)

=+

= =+ +

= =G E G G G E EE E E

G G G2(1 )

; ;2

;xyx z

x z xz zyz xzh

h

hh v v

(6)

where Ex, Ey, Ez, Eh and Ev are the Young's moduli in different directionsof the formation for a conveniently oriented coordinate system(Fig. 1b); υxy, υyz, υxz, υh and υv are Poisson's ratios; Gxy, Gyz, Gxz, Gh andGv are the corresponding shear moduli.

There are significant differences between the isotropic mechanicaland the transverse isotropic mechanical properties [cf. Eqs. (1)–(6)].Depending on whether the assumed model is an isotropic (Fig. 1a) ortransverse isotropic one (Fig. 1b), the stress response to penetration bya borehole will be different for the same drilling parameters. Our aim isto apply a model to calculate the stress in transverse isotropic shaleformations when perforated by a cylindrical drill hole (Fig. 1b), andestablish the safe drilling window using anisotropic failure criteria.

2.2. Coordinate transformations

Shale formations are mainly developed using horizontal and ex-tended reach wells, and there is a certain angle between the direction ofthe borehole axis and the geographic coordinate system or the stresscoordinate system. Coordinate system transformations are required totransform the far-field in-situ stresses from a geographic coordinatesystem to a borehole coordinate system (Jaeger et al., 2009).

First, the far-field in-situ stress distribution is transformed from theprincipal stress coordinate system to the geographic coordinate system

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

Y. Li and R. Weijermars Journal of Petroleum Science and Engineering 176 (2019) 982–993

983

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

= × ×R Rg pgT

p pg (7)

= =R0 0

0 00 0

;cos cos sin cos sin

sin cos 0cos sin sin sin cos

pg pg pg pg pg

pg pg

pg pg pg pg pg

p pg

H

h

v

(8)

Next, the far-field in-situ stress tensor σg under the geographic co-ordinate system, is transformed to the stress tensor σb under the bore-hole coordinate system (Fig. 2b). The stress transformation relationshipare as follows:

= × × =R Rxx b xy b xz b

yx b yy b yz b

zx b zy b zz bb bg g bg

T, , ,

, , ,

, , , (9)

=Rcos cos sin cos sin

sin cos 0cos sin sin sin cos

bg bg bg bg bg

bg bg

bg bg bg bg bg

bg

(10)

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

2.3. Stress distribution algorithms

When the coordinate system transformation is completed, we candirectly apply the far-field in-situ stress to calculate the stress dis-tribution around the wellbore in the borehole coordinate system. Thecalculation method above is based on the theoretical works ofLekhnitskii (1981), Amadei (1984), Serajian and Ghassemi (2011) andLu et al. (2015). The far-field stress tensor components, σij,b, willmodify near the borehole when the drill penetrates the rock. The in-duced stress distribution around the borehole can be calculated bytaking into account the anisotropic perturbations due to the hole's

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

=

a a a a a aa a a a a aa a a a a aa a a a a aa a a a a aa a a a a a

0 0 0

0 0 0

0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

E E E

E E E

E E E

G

G

G

11 12 13 14 15 1621 22 23 24 25 2631 32 33 34 35 3641 42 43 44 45 4651 52 53 54 55 5661 62 63 64 65 66

1

1

1

1

1

1

x

xyy

xzz

xyy y

yzz

xzz

yzz z

yz

xz

xy

(11)

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

R

R

R

R

R

= + = + + += + = + + += = + += + = + + += = + +

= + + + +

e µ z µ z µ ze z z ze µ z µ z µ ze µ z µ z µ ze z z z

a a a a a

2 [ ( ) ( ) ( )]2 [ ( ) ( ) ( )]2 [ ( ) ( ) ( )]2 [ ( ) ( ) ( )]2 [ ( ) ( ) ( )]

( )

x xx b x i xx b

y yy b y i yy b

xy xy b xy i xy b

xz xz b xz i xz b

yz yz b yz i yz b

z zz b a x i y i yz i xz i xy i

, , , 12

1 1 22

2 2 3 32

3 3

, , , 1 1 2 2 3 3 3

, , , 1 1 1 2 2 2 3 3 3 3

, , , 1 1 1 1 2 2 2 2 3 3 3

, , , 1 1 1 2 2 2 3 3

,1

31 , 32 , 34 , 35 , 36 ,33

(12)

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

= +

+ + +( )

z i p µ µ

i ip i µ µ

( ) ( )( )

( )( 1) ( ) ( )]

µxy b yy b w

xy b xx b w yz b xz b

1 11

2 1, , 2 2 3 3

, , 2 3 , , 3 3 2

za11 2

12

(13)

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

Y. Li and R. Weijermars Journal of Petroleum Science and Engineering 176 (2019) 982–993

984

= +

+ + +( )

z i p µ µ

i ip i µ µ

( ) ( )( )

( )(1 ) ( ) ( )]

µxy b yy b w

xy b xx b w yz b xz b

2 21

2 1, , 1 3 3 1

, , 2 3 , , 3 1 3

za22 2

22

(14)

= +

+ + +( )

z i p µ µ

i ip i µ µ

( ) ( )( )

( )( ) ( )( )]

µxy b yy b w

xy b xx b w yz b xz b

3 31

2 1, , 2 1 1 2

, , 1 2 , , 2 1

za33 2

32

(15)

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:

= + +µ µ µ µ µ µ( ) ( )2 1 2 3 1 3 1 3 3 2 (16)

= =e i, 1,2,3ii (17)

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

= + =z a µ i(cos sin ), 1,2,3i i (18)

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):

+ =µ µ2 0552

45 44 (19)

+ + + =µ µ µ µ2 (2 ) 2 0114

163

12 662

26 22 (20)

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

= =aa a

ai j, , 1,2,4,5,6ij ij

i j3 3

33 (21)

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

= = =l µl µ

l µl µ

l µl µ

( )( )

;( )( )

;( )( )1

3 1

2 12

3 2

2 23

3 3

2 3 (22)

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

= +l µ µ µ( ) 2i i i2 552

45 44 (23)

= + + +l µ µ µ µ( ) ( ) ( )i i i i3 153

14 562

25 46 24 (24)

The stress distribution around the borehole given by Eq. (12) can besolved according to the above computational schedule.

2.4. Pore pressure

Because shale is a porous medium, the effective stresses σ′j thatimpacts on the borehole stability can be determined from the totalstress of Eq. (12) by subtracting the pore pressure Pp as follows (Shao,1998; Tan et al., 2015):

=

=

=

+ +

+

+ +

p

p

p

(1 )

(1 )

(1 )

x xM M M

K p

y yM M

K p

z zM M M

K p

32

3

3

s

s

s

11 12 13

13 33

11 12 13(25)

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

= ++

M E E EE E E( )

(1 )( 2 )x z x

z z x11

v2

h h v2 (26)

= ++

M E E EE E E( )

(1 )( 2 )x z x

z z x12

h v2

h h v2 (27)

= =M M E EE E E2

x z

z z x13 23

v

h v2 (28)

=ME

E E E(1 )

2z

z z x33

2h

h v2 (29)

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

2.5. Conversion from Cartesian to polar coordinates

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

== +== == +

p psin cos sin 2

0sin 2 ( ) cos 2

r w p

x y xy

z z

z rz

r y x xy

2 2

12 (30)

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

3. Anisotropic borehole stability model

Significant leak-off in the matrix will occur when the pressure of thedrilling mud column in the borehole is greater than the tensile strengthof borehole rock. Both the tensile and compressive strengths of trans-verse isotropic shale are also anisotropic material properties, whichmeans that both the tensile and compressive strengths of the boreholerock are different in different directions of the borehole radial.

3.1. Tensile failure

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

= + +m sii

1 3 c3

c

0.5

(31)

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 dependon the properties of the rock mass. For example, a rock mass subjectedto strong disturbance, will have m=0.001, hard and complete rockmass, m=25. For a completely broken rock mass, s=0, and com-pletely in tact rock mass, s=1 (Liu et al., 2013).

Rock tensile strength can be established by conducting a BrazilianTest. 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 σ1has the relationship: σ1=−3σ3. By combining Eq. (31), the criterionfor empirical tensile failure can be expressed as:

= + +m s3 ii

3 3 c3

c

0.5

(32)

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

=m s4 0t

i

t

ic

0.5

c (33)

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

= +m m s32 1024 16t ic

2 0.5

(34)

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

Y. Li and R. Weijermars Journal of Petroleum Science and Engineering 176 (2019) 982–993

985

= A B [ cos 2( )]i mc (35)

where A and B are constants, their unit is MPa; A and B are determinedby the mechanical properties of the rocks. They can be calculated byequation (35) using the experimental results of the uniaxial compres-sive strength test. βm is the inclination angle of the bedding when thesample uniaxial compressive strength is minimum; β is the inclinationangle of the bedding when uniaxial failure test is conducted, β= θwhen at the borehole wall. Equation (35) is an empirical correction, andthe authors suggest that this equation should be used when drillingthrough a shale formation with bedding development. Moreover, theformation depth should not be too deep such that the influence ofthermal stress can be ignored.

To investigate tension failure along the inclination angle of thebedding, we substitute Eq. (35) into Eq. (34) and the correspondingmechanical model is established. This yields an equation for the tensilestrength of layered shale:

= +A B m m s{ [cos 2( )]}32 1024 16t m

2 0.5

(36)

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

= +A B m m s{ [cos 2( )]}32 1024 16m

2 0.5

(37)

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

3.2. Compressive shear failure

For compressive strength of a layered shale, the empirical Hoek-Brown failure criterion was initially proposed by Hoek and Brown(1980) for intact rocks, and theoretical calculation methods of com-pressive 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 istoo small to maintain the stress balance, and shear failure is initiatedunder the condition of compressive stress.

According to the shear failure criterion of Eq. (31), and because ofthe stress equivalences σ′θ= σ1 and σ′r= σ3, one can substitute Eq. (35)into Eq. (31), and we get:

= + +A B mA B

{ [cos 2( )]}[cos 2( )]

1r mm

0.5

(38)

Combining Eqs. (38) and (30), the shear collapse pressure whendrilling 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 cor-rectly in order to determine the tensile breakdown pressure and theshear collapse pressure for wellbore stability analysis. Therefore, wefirst verify the calculation result of circumferential stress obtained bythe stress model of Section 2. The circumferential stress profile com-puted with our model (Fig. 3) is compared to that of Serajian andGhassemi (2011), using the same input parameters for a nearly iso-tropic test case used in their study (Table 1). The circumferentialstresses calculated by our model are consistent with the reference

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

4.2. Synthetic case study

We assume that the horizontal well extends along the direction ofthe horizontal maximum principal stress (Fig. 4), and the in-situ stresscoordinate system is consistent with the geographical coordinatesystem. Under this assumption, the borehole coordinates Xb, Yb and Zbare in the same direction as native, far-field stresses σv, σh and σH, usedin Eq. (8), respectively. The rock mechanical parameters and in-situstress data used for our synthetic case study calculations are shown inTable 2.

The circumferential stress variation with the angle around theborehole 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 themaximum compressive stress of 36.23 MPa occurs at 0° +π. The resultsindicate that the borehole rock is prone to tensile rupture in the verticaldirection (90° +π). The synthetic example assumes Andersonian,normal fault stress conditions. Normal faulting Andersonian stress re-gimes in rocks with isotropic elastic properties will also form verticalcracks when the borehole pressure exceeds the critical strength(Thomas and Weijermars, 2018). However, in transverse isotropic shalethe stress concentrations around the wellbore are higher as follows fromthe comparison in Fig. 5b. The impact of the modulus anisotropy ex-pressed by the ratio Ev/Eh was also evaluated with several discretevalues (0.2, 0.6, 1, 2 and 6), using Pw=50.48MPa. The results ofFig. 5b show that with transverse isotropy in the horizontal plane as inFig. 4, which occurs when 0< Ev/Eh < 1, the tangential stress max-imum occupies a broader region with a lower maximum stress ascompared to the isotropic case Ev/Eh= 1. Reversely, when the trans-verse isotropy is parallel to the vertical direction (rotated 90° from thatshown in Fig. 4), which occurs when 1< Ev/Eh<∞, the tangentialstress peaks at higher values as comared to the isotropic case (Fig. 5b).

4.3. Tensile strength variation

The anisotropy of the tensile strength of the wellbore is calculatedusing Eq. (36). The tensile strength of the borehole wall rock is not aconstant value, but varies with the tangential position around the well

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

Y. Li and R. Weijermars Journal of Petroleum Science and Engineering 176 (2019) 982–993

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(Fig. 6a). The tensile strength changes cyclically, reaching a minimumvalue of 1.35MPa at 90°, and maximum of 19.71MPa at 0°. The var-iation is relatively easy to explain: When the well circumference angleis 90°, the tensile failure of the borehole needs to overcome the

compaction caused by the minimum horizontal principal stress. Whenthe well circumference angle is 0°, tensile strength of borehole rock iscaused by the compression in the direction of vertical, maximumprincipal stress. The vertical stress is much larger than the minimumhorizontal principal stress, and the deviatoric stress is compressionalwhereas the horizontal principal deviatoric stress is tensional. Thetensile strength of the anisotropic shale is smallest in the direction ofthe minimum horizontal principal stress. When the circumferentialangle of the borehole is 90°, the tensile strength is at a minimum. Leak-off will occur when the effective circumferential stress magnitude isequal to or greater than the tensile strength (Fig. 6b). Finally, accordingto the data in Table 2, the tensile breakdown pressure and shear col-lapse pressure are calculated to be 50.48MPa and 30.37MPa, respec-tively.

4.4. Sensitivity analysis: impact of in-situ stress anisotropy coefficient onwellbore stability

In order to analyze the influence of stress anisotropy on boreholestability, we define the stress anisotropy coefficient:

=K v

h (39)

In the sensitivity analysis, we keep σh constant, increase σv con-tinuously, and increase Kσ from 1.0 to 1.4 with incremental increases of0.1. The collapse pressure and breakdown pressure vary with the stressanisotropy coefficient as shown in Fig. 7. The effect of stress anisotropyon rock collapse pressure and breakdown pressure was analyzed byusing Eqs. (30), (37) and (38). The collapse pressure and the breakdown

Table 1Input parameters for the verification model (after Serajian and Ghassemi, 2011).

No. Parameters Symbol Unit Value Applied in Eqs

1 maximum horizontal principal stress σH psi 0.0085×106 (7), (8)2 minimum horizontal principal stress σh psi 0.008×106 (7), (8)3 vertical stress, MPa; σv psi 0.01× 106 (7), (8)4 pore pressure pp psi 0.0061×106 (25)5 Young's modulus in orientation of Xg-axis Ex psi 14.00001×106 (11)6 Young's modulus in orientation of Yg-axis Ey psi 14.00001×106 (11)7 Young's modulus in orientation of Zg-axis Ez psi 1.4× 106 (11)8 Poisson's ration in planes of XgYg υxy / 0.2 (11)9 Poisson's ration in planes of YgZg υyz / 0.19999 (11)10 Poisson's ration in planes of XgZg υxz / 0.19999 (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.

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

Table 2The parameters for the using of model calculation.

No Parameters Symbol Unit Value Applied equations

1 maximum horizontal principal stress σH MPa 38.4 (7), (8)2 minimum horizontal principal stress σh MPa 36.2 (7), (8)3 vertical stress, MPa; σv MPa 45.2 (7), (8)4 pore pressure pp MPa 18.1 (25)5 Young's modulus in orientation of Xg-axis Ex GPa 25.1 (11)6 Young's modulus in orientation of Yg-axis Ey GPa 25.1 (11)7 Young's modulus in orientation of Zg-axis Ez GPa 19.3 (11)8 Poisson's ration in planes of XgYg υxy / 0.19 (11)9 Poisson's ration in planes of YgZg υyz / 0.17 (11)10 Poisson's ration in planes of XgZg υxz / 0.17 (11)11 radius of the borehole a m 0.1 (18)12 bulk modulus of solid phase Ks GPa 69 (25)13 material constant A MPa 72.7 (36), (37), (38)14 material constant B MPa 63.4 (36), (37), (38)15 material constant m MPa 0.79 (36), (37)16 material constant s MPa 0.45 (36), (37)17 inclination angle of the bedding when the sample uniaxial compressive strength is minimum βm ° 45 (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.13–14according to Liu et al. (2013), and No. 15–17 according to Shi et al. (2016).

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pressure both decrease when the stress anisotropy coefficient increases(Fig. 7). The tensile breakdown pressure decreases from 61.71 to43.65MPa, and the collapse pressure for shear failure decreases from35.96 to 27.07MPa. Consequently, the safe drilling fluid densitywindow gradually narrows when the stress anisotropy increases. Ourconclusion is that the anisotropy of in-situ stress has a significant in-fluence on the stability of the wellbore, which is why any local increasein the anisotropy of stress generally prompts drilling engineers to paycloser attention to wellbore stability analysis and control.

The reason for the drastic reduction in the breakdown pressure ismainly the change in the circumferential stress around the wellbore(using Eq. (37)). Figs. 8 and 9 show that the circumferential wellborestress σθ reaches the shear breakdown pressure at different in-holepressures when the stress anisotropy coefficient Kσ changes. For ex-ample, when the stress anisotropy coefficient increases, the boreholecircumferential stress anisotropy will increase accordingly. When thestress anisotropy coefficient Kσ increases from 1.0 to 1.4, the maximumvalue of borehole circumferential stress increases from 0.11 to58.35MPa: An increase of nearly 530 times. Increase of the cir-cumferential stress due to elastic anisotropy makes the wellbore moreprone to tensile failure and thus promotes instability of the borehole.

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 strengthof borehole rock varies with the circumferential angle (b) Leak-off occurs wheneffective circumferential stress magnitude is equal to the tensile strength.

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

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4.5. Sensitivity analysis: impact of circumferential stress anisotropycoefficient on wellbore stability

In an alternative approach, we analyze the effect of the in-situ stressanisotropy on the borehole circumferential stress using the

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

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

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

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circumferential stress anisotropy intensity normalized by the cir-cumferential stress anisotropy for the isotropic elastic continuum case:

= =K, 1.0, 1.1, ...1.4.K

K Kmax min

max1.0

min1.0 (40)

The variation of circumferential stress anisotropy according to Eq.(40) is included in Fig. 10 (left-hand scale). An increase in the aniso-tropy of the in-situ stress, translates to the circumferential stress ani-sotropy intensity of the borehole increasing linearly. The anisotropycoefficient of in-situ stress Kσ increases from 1.0 to 1.4, and the cir-cumferential stress anisotropy ΔKσ increases from 1.0 to 48.31. Thecorresponding breakdown pressure decreases linearly, which againshows that and increase of the anisotropy of the formation will increasethe circumferential stress anisotropy. The stress concentration on theborehole will be more pronounced for increased stress anisotropy. As adirect consequence, the wellbore will be more prone to tensile failure.

Fig. 11 visualizes the influence of in-situ stress anisotropy on thecollapse pressure of borehole. When the anisotropy of in-situ stressincreases, the difference between the circumferential stress and theradial stress increases, so that shear failure can occur under the lowermud weight pressure, and the collapse pressure decreases with the in-crease of anisotropy of in-situ stress. The main reason of the wellborecollapse is that the pressure of the drilling fluid cannot maintain thestress balance of the borehole rock, and does not preclude the cir-cumferential stress and radial stress difference from reaching the cri-tical condition of shear failure.

Next, we define the intensity ratio of circumferential stress and ra-dial stress difference as:

= =K( )( )

, 1.0, 1.1, ...1.4.K

Kr max

r max1.0 (41)

The variation of the intensity ratio (of the circumferential stress andradial stress difference) is calculated (according to Eq. (41)), andplotted in Fig. 12. When the stress anisotropy increases, the cir-cumferential stress and radial stress difference increases linearly. Forexample, when the stress anisotropy coefficient Kσ increases from 1.0 to1.4, ΔΔKσ increases from 1.0 to 8.72 (Fig. 12, black line, left scale). Thecorresponding borehole collapse pressure decreases linearly (Fig. 12,red line, right scale). The results of Fig. 12 show that due to the en-hanced anisotropy of formation stress, the shear collapse pressure de-creases. This means that with the increase of the anisotropy of theformation stress, the mud weight pressure used to ensure borehole doesnot occur collapse is reduced. But the anisotropy of formation stressreduces the difference between the breakdown pressure (Fig. 7) andcollapse pressure, and the controllable range of drilling fluid density insafety drilling becomes smaller, which requires more precise control ofborehole stability to ensure safe drilling. Higher anisotropy of in-situstress is less favorable to wellbore stability due to a reduction in safedrilling margins.

The results show that due to the enhanced anisotropy of formationstress, a smaller hydrostatic pressure of drilling fluid can maintain thestress balance of the rock wall. Such stress balance of the borehole is

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.

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aimed for to make the borehole remains stable and easy to control bydensity of the fluid used in the drilling process However, the anisotropyof shale makes the difference between breakdown pressure and collapsepressure smaller, and the controllable range of drilling fluid density forsafe drilling becomes smaller, which reduces borehole stability.Therefore, the stronger the anisotropy of in-situ stress is, the moreunfavorable to the control of borehole stability.

5. Sensitivity analysis: impact of anisotropic elastic engineeringmodulus on wellbore stability

The differences in wellbore stability between transverse isotropicshale and isotropic rock are mainly reflected in the elastic moduli andPoisson's ratio. For the transverse isotropic shale these mechanicalparameters are different in horizontal and vertical directions. In orderto better understand and explain the influence of the special mechanicalproperties of transverse isotropic shale on borehole stability, the re-spective effects of the elastic moduli and Poisson's ratio on collapsepressure and breakdown pressure of borehole are separately evaluatedbelow.

5.1. Elastic moduli anisotropy and wellbore stability

We define the elastic modulus anisotropy coefficient:

=K EEE

v

h (42)

For our sensitivity analysis we keep Eh constant and change Ev, andmake KE decrease from 1.0 to 0.6 (at increments of 0.1) to analyze theinfluence of the change of elastic modulus in different directions on theborehole stability. When the elastic modulus anisotropy of transverseisotropic shale increases (Fig. 13.), then both the breakdown and col-lapse pressures will decrease. The breakdown pressure is more affectedby the anisotropy of elastic modulus. The difference between thebreakdown and collapse pressures decreases when the elastic modulusare more anisotropic and the safe drilling window narrows.

5.2. Poisson's ratio anisotropy and wellbore stability

Similarly, we define the Poisson's ratio anisotropy coefficient:

=K v

h (43)

We keep υh constant and change υv and use Kυ with stepwise de-creases from 1.0 to 0.6 to analyze the influence of the change ofPoisson's ratio in different directions on the borehole stability. Whenthe anisotropy of Poisson's ratio of transverse isotropic shale increases(Fig. 14), the collapse and breakdown pressures only show a slightdecrease, which impact is not very significant. Although the differencebetween breakdown pressure and the collapse pressure of the safetydrilling range is slightly decreased, the decrease is not significant whichimplies that the rock Poisson's ratio anisotropy has very limited influ-ence on the borehole stability.

6. Discussion

6.1. Principal insights

In this paper, the wellbore stability model for transverse isotropicshale is advanced by considering the various mechanics parameters inthe vertical direction and the horizontal direction as well as the ani-sotropy of tensile strength and shear failure strength in different di-rections of the borehole. The influence of the in-situ stress conditionsand the anisotropic characteristics of rock mechanics parameters on thecollapse pressure and breakdown pressure of borehole in drilling pro-cess is calculated and analyzed. The transverse isotropic mechanical

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

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

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

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properties have a significant effect on the borehole stability of shalewells. When the anisotropy of the elastic moduli increases, the impacton the borehole stability reduces the safe drilling window. Increasedanisotropy of the in-situ stress also has a significant impact on thestability. The larger the stress anisotropy, the more difficult will be thepreservation of wellbore stability, because the safe drilling window issubstantially narrower for strongly anisotropic in-situ stresses.

6.2. Practical application and model inputs

The present study analyzes borehole stability under the assumptionof transversely isotropic elasticity. Shale formations are known to betransverse isotropic. Measurements of the elastic stiffnesses on shalehave confirmed these are commonly transverse anisotropic (Laubie andUlm, 2014). Rather than measuring the stiffnesses under laboratoryconditions, sonic dipole logging tools can measure the five non-van-ishing stiffness tensor components (C11, C13, C, C44, and C55) directly inthe borehole (e.g. see Brooks et al., 2015; Aderibigbe et al., 2016). Themethodology to obtain such parameters for the sonic dipole tool havebeen detailed in Herwanger and Koutsabeloulis (2011). The stiffnessescan be translated using standard conversion expressions to obtain theelastic engineering constants (in our model Ex, Ey, Ez, Eh and Ev; υxy, υyz,υxz, υh and υv). Further inputs required for computing the magnitudeand orientation of the resulting anisotropic stress concentrations nearthe borehole are the local tectonic stress tensor components (as inTable 1).

Separately, a number of intrinsic rock parameters are required forquantifying the anisotropic strength, based on which the safe drillingwindow for a certain depth can be calculated. The acquisition of keyfailure strength parameters (Ks, A, B, m, s and βm) is currently onlypossible via laboratory tests (as per references cited in Table 2). Thelimited availability of such input parameters remains an obstacle forwider practical application of our model until such data becomeavailable for the specific rock formation studied. Our study highlightthe necessity to secure such experimental data of anisotropic failure as akey step for more accurate wellbore stability model analyses whendrilling shale formations.

6.3. Model limitations

There are still many shortcomings and areas for improvement in theresearch of mechanically anisotropic shales. For example, in the cal-culation of breakdown pressure and collapse pressure, shale is com-monly considered as a transverse isotropic medium for layered sedi-ments, which ignores the influence of structural weakness such assedimentary bedding. Secondly, the establishment of the transverseisotropic shale borehole stress distribution model can only calculate thestress distribution at the borehole wall, and cannot calculate and ana-lyze the stress distribution away from the borehole. Therefore, themodel can only be used to analyze the instability initiated at theborehole wall. In addition, artifacts further away from the boreholewall, such as the presence of structural weaknesses, may cause shearfailure and instability of the borehole when various stresses reachfailure conditions. Establishing a calculation model for solving thestress distribution away from the borehole wall remains a challengingtask, was recently solved for cases where the wellbore remains alignedwith the plane of isotropy in transverse isotropic shale (Weijermarset al., 2019). Thirdly, pore pressure is considered constant and thussimply subtracted from the total stress to give the effective stress. Anytime-dependent coupling between pore fluid diffusion and solid de-formation is ignored. In case such coupling were to occur, the dynamicincrease of pore pressure at the borehole wall under the action of see-page may decrease the actual breakdown pressure, and the calculatedbreakdown pressure will be higher than the actual value. Fourthly, thestable mud weight window is established using a modified Hoek-Brownfailure criterion, and the effect of the intermediate principal stress is

neglected as in previous studies (Wang and Shen, 2017; Leea andPietruszczakb, 2017). However, the intermediate principal stress mayimpact the rock failure process (Mohammad and Pinnaduwa, 2016). Abroad range of failure criteria was compared under isotropic stress as-sumptions (Wang and Weijermars, 2019). The effect of anisotropicfailure criteria is first advanced in the present study.

7. Conclusions

This paper considers the geo-mechanical impact of a layered shaleas a transverse isotropic elastic medium, and accounts for the associatedanisotropy of its tensile strength and shear failure strength. A wellborestability model that accounts for both the enhanced stress anisotropyand change in failure criteria due to the elastic anisotropy shale pro-vides an important supplement to, and improvement over, isotropicWBA methods. The following systematic insights were obtained byapplying the model to a synthetic case study, expanded with a sensi-tivity analysis to various key parameters:

(1) The strength of horizontal shale is intrinsically anisotropic, which iswhy the anisotropy of tensile and compressive strengths (onbreakdown pressure and collapse pressure, respectively) should beconsidered in the wellbore stability analysis.

(2) Separately, the in-situ stress anisotropy significantly influences thestability of any borehole. When the in-situ stress anisotropy in-creases, the difference between the circumferential stress and radialstress also increases. The breakdown pressure and collapse pressurewill then decrease, and the safe drilling window decreases gradu-ally, which requires close attention in wellbore stability analysis toensure safe drilling conditions.

(3) With increasing anisotropy in the elastic moduli, both the break-down pressure and collapse pressure will decrease, leading to agradual narrowing of the safe drilling window. The tensile break-down pressure is sensitive to elastic anisotropy.

(4) With increasing anisotropy of the Poisson's ratio, the breakdownpressure and collapse pressure remain basically unchanged, whichindicates - unlike the Young's modulus anisotropy - that the influ-ence of Poisson's ratio anisotropy on wellbore stability is verylimited.

Conflicts of interest

The authors declare no conflict of interest.

Acknowledgement

This research was supported by the China Postdoctoral ScienceFoundation funded project (2018M640289).

Nomenclature

Ex, Ey, Ez, Eh and Ev are Young's moduli in different orientations offormation rock, MPa

υxy, υyz, υxz, υh and υv are Poisson's ration in different planes of for-mation rock, dimensionless number

Gxy, Gyz, Gxz, Gh and Gv are shear moduli in different planes of for-mation rock, MPa

σH is maximum horizontal principal stress, MPaσh is minimum horizontal principal stress, MPaσv is vertical stress, MPaσp is the far-field stress tensor under the principal stress co-

ordinate systemσg is the far-field stress distribution tensor under the geographic

coordinate systemαpg is the azimuth of maximum horizontal principal stress, (°)βpg is the angle between the direction of σv and Zg-axis, (°)

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σb is the far-field stress distribution tensor under the boreholecoordinate system

σxx,b, σyy,b, σzz,b, τxy,b, τxz,b, τyz,b are the far-field stress under the bore-hole coordinate system, MPa

αbg is the azimuth of borehole axis, (°)βbg is the deviation angle of the borehole axis, (°)ℜe is real component of an imaginary numberΦ’i is derivatives of the stress analytical functions

zi, μi and λi are complex numbersa is the radius of the borehole, m;pw is fluid pressure in the well, MPaMij is stiffness matrix of borehole rock, MPaPp is the pore pressure, MPaKs is bulk modulus of solid phase, MPaσci is the uniaxial compressive strength of the rock, MPam and s are material constants of rock mass, and s=1 for intact rocksσt is the tensile strength of rock, MPaA and B are constantsβm 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 com-

pression test is conducted, °

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