A wellbore stability model for formations with anisotropi crock strengths

Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

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

A wellbore stability model for formations with anisotropic rock strengths
Hikweon Lee a,n, See Hong Ong b, Mohammed Azeemuddin c, Harvey Goodman c
a
b
c

Geological Environmental Division, KIGAM, 92 Gwahang-no, Yuseong-gu, Daejeon 305-350, Republic of Korea
Baker Hughes, 191 St. George Terrace, Perth, WA 6000, Australia
Chevron Energy Technology Company, 1500 Louisiana, Houston, TX 77002, USA

a r t i c l e i n f o

abstract

Article history:
Received 22 August 2011
Accepted 15 August 2012
Available online 14 September 2012

Shale formations, due to the presence of laminations and weak planes, exhibit directional strength
characteristics. In most conventional wellbore stability analyses, rock formations are typically assumed
to have isotropic strength. This may cause erroneous results in anisotropic formations such as shales
which show strength variations with changing loading directions with respect to the plane of weakness.
Therefore a more complex wellbore stability model is required. We have developed such a model in

which the anisotropic rock strength characteristic is incorporated. Applying this model to two case
studies shows that shear failures occur either along or across the bedding planes depending on the
relative orientation between the wellbore trajectories and the bedding planes. Additionally, the extent
of failure region around the wellbore and the safe mud weights are significantly affected by the
wellbore orientation with respect to the directions of bedding plane and in-situ stress field.
& 2012 Elsevier B.V. All rights reserved.

Keywords:
breakouts
in-situ stress
anisotropic rock strength
drilling
wellbore trajectory
wellbore stability

1. Introduction
Drilling into the earth crust and replacing the heavier rock
materials with a lighter drilling fluid cause stress concentration
around the wellbore. The degree of stress concentration depends
on the wellbore orientation, the magnitude and orientation of insitu stresses, and the wellbore pressure (Bradely, 1979). When the
elevated stress exceeds the rock strength, the rock will fail
resulting in the development of wellbore breakouts (Zoback
et al., 1985). If excessive, the cavings produced by the spalling
of broken materials into the wellbore can cause drilling problems
such as pack-off, over-pulls, stuck-pipe and poor cementing, to
name a few.
In order to prevent these wellbore instability related problems,
a higher mud pressure is used to provide support against the
wellbore wall. Typically, the required mud weight is determined
by a wellbore stability analysis in which the rock material is

assumed to be linear elastic, homogeneous, and isotropic in
mechanical and strength behaviors (Bradely, 1979). This traditional stability analysis, however, may lead to erroneous results
particularly for a well drilled into thinly laminated rock formations such as shales and phyllites, and ignoring the strength
anisotropy associated with planes of weakness such as bedding
planes, foliations, or schistosity presented in these rocks.

n

Corresponding author. Tel.: ỵ82 42 868 3507; fax: ỵ82 42 868 3414.
E-mail address: (H. Lee).

0920-4105/$ - see front matter & 2012 Elsevier B.V. All rights reserved.
/>
The anisotropic strength behavior of various rock types has
been investigated by many researchers (Donath, 1964; Chenevert
and Gatlin, 1964; McLamore and Gary, 1967; Ramamurthy et al.,
1993; Niando et al., 1997; Ajalloeian and Lashkaripour, 2000; Tien
et al., 2006). Their works showed that the failure strength varies
as a function of the angle (c) between the weakness plane and the
axis of the major principal stress; attaining its maximum either at
c ¼01 or 901, and minimum when c is between 201 and 401. For
samples loaded at c ¼01 or 901, shear failure through the intact
rock matrix is dominant, while failure along the weakness plane is
more pronounced in samples loaded at c ¼301.
Aadnoy (1988) developed an analytical solution to study the
stability of inclined wellbores drilled into rock formations modeled as a transversely isotropic material. He showed that neglecting the anisotropic effects arising from the directional elastic
properties can result in errors in the wellbore stability analysis.
Ong and Roegiers (1993) proposed an analytical solution in
which a generalized three-dimensional anisotropic failure criterion was used in conjunction with the three-dimensional anisotropic stress model for stress distribution around the wellbore.
They indicated that wellbore stability is significantly influenced

by directional elastic properties, rock strength anisotropy, and insitu stress differentials.
Aoki et al. (1993) and Zou et al. (1996) also investigated
wellbore stability in anisotropic rock formations using numerical
methods. In particular, Aoki et al. (1993) developed a constitutive
model in which the induced pore-pressure due to material
deformation under undrained condition was included. They found
that the material anisotropy has a pronounced effect on the


110

H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

Nomenclature

sv
sH
sh

in-situ vertical stress
in-situ maximum horizontal principal stress
in-situ minimum horizontal principal stress
c
angle between the weakness planes and axis of the
major principal stress
as
Azimuth of in-situ minimum horizontal principal
stress
bs
deviation of in-situ vertical stress

aw
dip direction of the plane of weakness—1801
bw
dip angle of the plane of weakness
ab
wellbore azimuth
bb
wellbore deviation
y
wellbore circumferential angle
sexx, seyy, sezz, texy, texz, teyz normal and shear stresses in the global
coordinate system (Xe, Ye, and Ze)
sbxx, sbyy, sbzz, tbxy, tbxz, teyz normal and shear stresses in the
borehole coordinate system (xb, yb, and zb)

breakout shape and the determination of safe mud weights. In
addition, they noted that wellbore failures in laminated formations are attributed by shear failures through the rock matrix as
well as along the bedding planes.
Numerous wellbore instability problems have been reported
when drilling through thinly laminated rock formations (Vernik
and Zoback, 1990; Mastin et al., 1991; Last et al., 1995; Skelton
et al., 1995; Okland and Cook, 1998; Willson et al., 1999, 2003,
2007; Beacom et al., 2001; Edwards et al., 2003; Brehm et al.,
2006; Lang et al., 2011). From the observations of stress-induced
breakouts imaged, Vernik and Zoback (1990) and Mastin et al.
(1991) found that breakout shapes and orientations are strongly
affected by the degree of rock anisotropy and the angle between
the dip of foliation and the wellbore axis.
Similar observations were also made in many wells drilled into
thinly laminated rock formations (Willson et al., 1999, 2003,

2007; Beacom et al., 2001; Edwards et al., 2003; Brehm et al.,
2006). From these field observations, drilling normal to the
bedding planes were found to improve wellbore stability, while
wellbores that were nearly parallel to the bedding experienced
insurmountable instability issues (Last et al., 1995; Skelton et al.,
1995; Okland and Cook, 1998). In particular, Okland and Cook
(1998) suggested from field experiences that the ‘‘attack angle’’
defined by the angle between the wellbore axis and the bedding
planes needs to exceed 201 in order to avoid weak-plane related
wellbore instability problems. They observed that cavings from
problematic hole sections have two parallel surfaces rather than
the typical curved shape associated with shear failures (Edwards
et al., 2003; Willson et al., 2003; Brehm et al., 2006). This cavings
geometry indicated that shear failures were along the weakness
planes that were oriented unfavorably with respect to the in-situ
stress field and the wellbore trajectory.
In this paper, we will introduce a semi-analytical wellbore
stability model in which the rock strength anisotropy was
incorporated. We will describe the underlying assumptions, the
reference coordinate systems, and the failure criterions used in
the model. This will include stress transformations between the
defined coordinate systems and the different failure criteria
employed for the intact rock matrix and the planes of weakness.
Lastly, we will discuss the results of wellbore stability analyses
performed using the model developed for a hypothetical case as
well as a case with actual field data.

a
r
Pm

Pp

radius of wellbore
radial distance from the center of wellbore
mud pressure
formation pore pressure
a
Biot’s coefficient
u
Poisson’s ratio
s0 rr, s0 yy, s0 zz, try, trz, tyz Effective normal and shear stresses in
the cylindrical coordinate system (r, y, and z)
w
w
w
w
w
sw
xx, syy, szz, txy, txz, tyz normal and shear stresses in the weak
plane coordinate system (xw, yw, and zw)
tw
resultant shear stress acting on the plane of weakness
s0 w
effective normal stress acting on the plane of
weakness
Sw
intrinsic shear strength of the plane of weakness
mw
Friction coefficient of the plane of weakness
s0 1

maximum effective principal stress
s0 3
minimum effective principal stress
Si
intrinsic shear strength of intact rock matrix
mi
coefficient of internal friction of intact rock matrix

2. Wellbore stability model for anisotropic rock strength
2.1. Problem definition
Shales, due to the depositional environments, are intrinsically
anisotropic with regards to their mechanical behaviors. The
anisotropy considered here is fabric-related and resulted from a
partial alignment of plate-like minerals such as clay. The problem
to be considered is the stability of a wellbore drilled into a thinly
laminated anisotropic rock formation as shown in Fig. 1. It is
assumed that the plane of weakness is ubiquitously distributed in
the formation.
Drilling a wellbore with a given mud pressure into a formation,
which is fully saturated with a pore fluid and subjected to the
preexisting boundary stresses initially in static equilibrium, disturbs the state of stress in the vicinity of the borehole wall.
Although the wellbore is drilled into an elastically anisotropic
medium, the stress distribution around the borehole is determined by assuming that the rock formation is elastically isotropic.
In reality, the stress distribution around the wellbore is also
affected by directional mechanical properties (Aadnoy, 1988; Ong
and Roegiers, 1993) as well as fluxes driven by differences in

σzz
τzy
τzx

Pp

τyx
τyz

σyy

Pm
τxz
τxy
Weakness planes
σxx
Fig. 1. A schematic drawing showing a wellbore drilled into a laminated rock
formation subjected to boundary stresses.


H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

chemical potential, temperature gradient, and pressure gradient
between the drilling mud and the formation of interest (Gazaniol
et al., 1994; Lal, 1999; Tao, 2006). These factors, however, are not
taken into account in the calculation of the stress distribution
around the wellbore.

zb (Borehole down)

Z e(Down)

yb

θ
βb

2.2. Reference coordinate systems and rotation matrices
The process of developing the anisotropic wellbore stability
model involves multiple stress transformations between the
different reference coordinate systems. We defined five reference
coordinated systems; global coordinate system (GCS), in-situ
stress coordinate system (ICS), borehole coordinate system
(BCS), cylindrical coordinate system (CCS), and weak plane
coordinate system (WCS). Each of these and their mutual relationships are shown in Figs. 2–5. Based on the coordinate systems and
their relationships defined, we obtained rotation matrices that are
needed to transform stress components from one reference
coordinate system to the other.
Fig. 2 shows the relationship between GCS and ICS. The GCS is
defined with Xe in the North, Ye in the East, and Ze pointing
vertically down. The ICS is defined with xs in the minimum
horizontal principal stress (sh), ys in the maximum horizontal
principal stress (sH), and zs in the overburden (sv). Their relationship is defined by the two angles, stress azimuth (as) and stress
deviation (bs). It is typically assumed in many geomechanical
studies that the overburden is acting in the vertical direction, but
in reality it may not be vertical particularly in fields where the
surface topology is significantly changed, or the subsurface

111

xb
Top of Borehole
Ye (East)

αb

Xe (North)
Fig. 4. Relationship between global and borehole coordinate systems.

zb (Down),
(Down) zc
yb
a

θ

(r,θ)

Z e(Down), K

βs

xb

σv

zs

ys, J

αs

Fig. 5. Relationship between borehole and cylindrical coordinate systems.

σH

Ye (East)
I

xs

σh

Xe (North)

Fig. 2. Relationship between global and in-situ stress coordinate systems.

Ze (Down)
Weak plane
yw Horizontal plane
xw

zw
90-βw
Dip direction
βw
αw

Ye (East)

Xe (North)
Fig. 3. Relationship between global and weak plane coordinate systems.

geological structures such as folds and faults are complex. To

take into account the most general case, we introduce the angle
(bs) which is measured between Ze and zs. The rotation matrix for
the transformation of stress components from ICS to GCS was
derived by initially aligning the axes (xs–ys–zs) of ICS with the
axes (Xe–Ye–Ze) of GCS, and then applying two separate rotations
around the selected axes. The first rotation about the zs axis
rotates the xs–ys–zs axes counter-clockwise by an angle of as. The
resulting new frame would be referred to as the IJK axes. The
second step rotates the IJK axes about the J axis by a counterclockwise angle of bs producing the final frame as shown in Fig. 2.
These two rotations correspond with the rotation matrix
expressed in Eq. (1), representing the transformation of stress
components from ICS to GCS.
98
9
8
sinas 0 >
>
=>
=
< cosbs 0 sinbs >
< cosas
0
1
0
sinas cosas 0
E¼
>
>
>
>

;
: sinb 0 cosb ;:
0
0
1
s
s
2
3
cosas cosbs
sinas cosbs
sinbs
6
cosas
0 7
ẳ 4 sinas
1ị
5
cosas sinbs sinab sinbs cosbs
Fig. 3 shows a weak plane dipping towards the direction
aligned along the zw axis. Its configuration with respect to GCS
is defined by the dip angle (bw) measured between the horizontal
plane and the zw axis, and aw (dip direction—1801) measured
from the Xe axis (North) to the projection of the xw axis on the
horizontal plane. The rotation matrix for the transformation of


112

H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

stress components from GCS to WCS is derived by the same
approach as used to derive Eq. (1). The axes (xw–yw–zw) of WCS
are rotated by an angle of aw about the zw axis, and then the
rotated axes are rotated again by an angle of 90—bw about the yw
axis such that the xw axis is aligned to the normal of the plane of
weakness. The rotation matrix for transforming stress components in GCS to those in WCS is expressed by Eq. (2).
8
98
9
sinaw 0 >
>
< cosð90bw Þ 0 sinð90bb Þ >
=>
< cosaw
=
0
1
0
sinaw cosaw 0
Wẳ
>
>
: sin90b ị 0 cos90b ị >
;>
:
;
0
0
1

b
b
2
3
sinaw sinbw
cosbw
cosaw sinbw
6
sinaw
cosaw
0 7
2ị
ẳ4
5
cosaw cosbw sinaw cosbw sinbw
Fig. 4 shows the relationship between BCS and GCS. The
configuration of a borehole with respect to GCS is defined by its
azimuth (ab) measured counter-clockwise from the Xe axis
(North) to the xb axis aligned with the top of borehole, and
deviation (bb) measured between the Ze axis (Down) and the zb
axis (borehole axis). The transformation matrix of stress components between GCS and BCS is derived by the same approach as
Eq. (1) and is expressed by Eq. (3).
8
98
9
sinab 0 >
>
< cosbb 0 sinbb >
=>
< cosab

=
0
1
0
sinab cosab 0
B¼
>
>
>
>
: sinb 0 cosb ;:
;
0
0
1
b
b
2
3
sinab cosbb
sinbb
cosab cosbb
6 sina
cos
a
0 7
3ị
ẳ4
5
b

b
cosab sinbb sinab sinbb cosbb
Fig. 5 shows the relationship between BCS and CCS. The
transformation matrix is obtained by rotating the xb–yb–zb axes
by an angle of y around the zb axis. It is expressed in Eq. (4).
2
3
cosy siny 0
6
7
4ị
C ẳ 4 siny cosy 0 5
0
0
1

2.3. Transformation of stress components between reference
coordinate systems
The rotation matrices obtained in the previous section allow us
to transform stress components from a reference coordinate
system to the other. In-situ principal stresses can be transformed
to the different stress components in GCS using Eq. (5)
8 e
9
s
texy texz >
>
< xx
=
e

e
e
T
sics2ecs ¼ E  sics  E ẳ tyx syy tyz
5ị
>
>
: te
te se ;
zx

zy

zz

in which sics ¼{sh, 0, 0; 0, sH, 0; 0, 0, sv}, ET is the transpose of
matrix E. As shown in Fig. 2, the orientations of the in-situ
stresses with respect to GCS are defined by the angles as and bs,
which are used to derive the matrix E.
The stress components in GCS are transformed to those in BCS
using Eq. (6). The orientation of wellbore with respect to GCS is
defined by its azimuth (ab) and deviation (bb) both of which are
used to derive the matrix B.

secs2bcs ¼ B  sics2ecs  BT ¼

8 b
s
>
>

< xx

tbyx

>
>
: tb

zx

9

tbxy tbxz >
>
=
sbyy tbyz
>
>
tbzy sbzz ;

ð6Þ

Fig. 6. Normal and shear stresses acting on the plane of weakness.

The stress distribution around the wellbore for an elastic and
homogeneous formation in CCS is calculated using Eq. (7)
(Bradely, 1979)





sb ỵ sb ị
sb sb ị
a2
a2
a4
s0rr ẳ xx yy 1 2 ỵ xx yy 14 2 þ 3 4 cos2y
2
2
r
r
r


2
4
2
a
a
a
þ tbxy 14 2 þ 3 4 sin2y ỵ P m 2 aPp
r
r
r


b
b 
b
b 

2

s
ỵ
s
ị

s

s
ị
a
a4
s0yy ẳ xx yy 1ỵ 2  xx yy 1 ỵ3 4 cos2y
2
2
r
r


4
2
a
a
tbxy 1 ỵ 3 4 sin2yPm 2 aPp
r
r
a2
a2
cos2y4ntbxy 2 sin2yaP p

r2
r
"
#

sbxx sbyy ị
a2
a4
try ẳ
sin2y ỵ tbxy cos2y 1 ỵ2 2 3 4
2
r
r

h
i
2
a
trz ẳ tbyz siny ỵ tbxz cosy 1 2
r

h
i
a2
tyz ẳ tbxz siny ỵ tbyz cosy 1 ỵ 2
r

s0zz ẳ sbzz 2nðsbxx sbyy Þ

ð7Þ

in which a is the wellbore radius, r is the radial distance from the
center of the wellbore, Pm is the mud weight, a is the Biot’s
parameter of the formation, Pp is the formation pore-pressure, y is
the angle measured counter-clockwise from the xb axis. As stated
in the early section, it should be noted that for simplicity we
employed an isotropic stress model to calculate the stress
distribution around the wellbore although it is drilled into an
elastically anisotropic rock formation.
After calculating the effective stresses at each of the points
around the wellbore defined by the circumferential angle y and
the radial distance r, the state of stress at each of the points is
examined for shear failure across the intact rock matrix. This
procedure is described in detail in the next section (see Section
2.5). In addition, the stresses in CCS are projected onto the surface
of weak plane by using Eq. (8)

secs2wcs ¼ W  BT  C T  sccs  C  B  W T
9
8 w
s
twxy twxz >
>
=
< xx
w
w
w
¼ tyx syy tyz
>

;
: tw tw sw >
zx
zy
zz

ð8Þ


H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

113

in which sccs ¼{s0 rr, try, trz; try, s0 yy, tyz; trz, tyz, s0 zz} is obtained
from Eq. (7). Fig. 6 shows a normal stress (sw
xx ) and two shear
w
stresses (tw
and
t
)
projected
onto
the
surface
of the weak plane.
xy
xz
The two shear stress components are combined into a single
resultant shear stress on the plane of weakness by Eq. (9)

(Goodman, 1989). These stresses are examined for the condition
of shear slippage along the weak planes using a failure criterion to
be described in the next section (see Section 2.4).
q
2
w 2
9tw 9 ẳ tw
9ị
xy ị ỵ txz Þ

Duveau and Shao, 1998; Ajalloeian and Lashkaripour, 2000; Tien
et al., 2006).
Based on these observations, we employ in this study two
Mohr–Coulomb failure criteria to check for failure modes of
anisotropic materials; one for the intact rock matrix and the
other for the weak planes. The Mohr–Coulomb failure criterion for
the rock matrix is expressed by the following equation (Jaeger
and Cook, 1979):
q
s10 ẳ s30 ỵ2Si þ mi s30 Þð 1 þ mi 2 þ mi Þ
ð10Þ

2.4. Failure criteria and material properties

where, s0 1 and s0 3 are, respectively, the maximum and minimum
effective principal stresses that are mathematically equivalent to
the eigenvalues of sccs, Si is the intrinsic shear strength of intact
rock matrix, and mi is the coefficient of internal friction of intact
rock matrix. On the other hand, the Mohr–Coulomb failure
criterion for the plane of weakness is expressed by the following

equation (Jaeger and Cook, 1979):

Since anisotropic rocks have laminated textures such as beddings
or foliations, they exhibit a variation in both compressive strengths
and failure modes as a function of the loading angle between the
plane of weakness and the direction of the major principal stress.
Numerous literatures reported that the maximum and minimum
compressive strengths occur at the angle c ¼01 or c ¼901, and
c E301, respectively. Two failure modes were commonly observed;
one is slippage along the weak planes typically at c ¼ 301–601, and
the other is shear failure across the intact rock matrix typically at
c ¼01 or 901 (Donath, 1964; Chenevert and Gatlin, 1964; McLamore
and Gary, 1967; Ramamurthy et al., 1993; Niando et al., 1997;

tw ẳ Sw ỵ mw s0w

11ị

where tw and s w are the resultant shear and effective normal
stresses acting on the plane of weakness, respectively, Sw is the
intrinsic shear strength in the plane of weakness, and mw is the
coefficient of sliding friction in the plane of weakness.
The strength parameters S and m appeared in the failure
criteria can be determined from a series of triaxial compression
tests at various confining pressures. Additionally, in order to
0

Read inputs
Take a well Azimuth & a Deviation
(HAzi = 0 to 360 & HDev = 0 to 90)

Transform in-situ principal stresses from ICS to GCS
(see Eq. 5) and then from ECS to BCS (see Eq. 6)
Increase MW = 0 to x
Calculate stresses ((σrr τr θ τrz σθθ τθz σzz) at a
point (θ = 0 to 180) along the borehole wall
in cylindrical coordinate system (see Eq. 7)

Calculate max & min principal
stresses (σ1 & σ3 ) at the point

Project the stress components in
CCS on the plane of weakness with
stress transformation (see Eq. 8)

Failure across
intact rock
matrix? (see
Eq. 10)

Failure along
the weak
plane? (see
Eq. 11)

Go to
next MW

Yes
Go to
next point

& MW = 0

No
No
Store MW for
each of the
points
Get a Maximum MW

Go to
next MW
Yes
Go to
next point
& MW = 0

Store MW for
each of the
points
Get a Maximum MW

Go to next
HDev& HAzi

Compare MWs and get a
bigger MW
for the given HDev& HAzi
Fig. 7. A flow chart showing the computational algorithms for the determination of safe mud weights. HAzi and HDev refer to the wellbore azimuth and deviation,
respectively, and MW refers to the mud weight.

114

H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

derive the weak plane properties, these tests also need to be
carried out at different loading orientations (01o c o901). Plotting the strength data for each of the loading orientations on the
s1–s3 domain, and then linearly fitting the data will yield the
strength parameters. The cohesion and internal friction of an
intact rock matrix can be determined from triaxial compression
tests carried out on rock samples with c ¼ 0 and/or 901. With a
constant confining pressure, the axial loading is continued until shear
failure has taken place across the intact rock matrix (McLamore and
Gary, 1967; Duveau and Shao, 1998; Tien et al., 2006).
Similarly, the strength parameters of the weak planes can be
determined from the strength data obtained from triaxial compression tests, but with the loading orientations that would result
in the minimum strength (typically 301 r c r451). At these
orientations, rock failures occur mainly by sliding along the weak
planes (McLamore and Gary, 1967; Duveau and Shao, 1998; Tien
et al., 2006). If the full series of triaxial compression tests is not
feasible due to sample shortage and/or economical considerations, one can run the triaxial tests at c ¼01, 301, and 901 to
determine the strength parameters with the assumption that the
minimum strength occurs at the loading orientation of 301.
However, if this assumption is challenged, it is recommended to
run additional triaxial tests at c ¼451.
2.5. Computational algorithms for anisotropic wellbore stability
model
Based on the stress and strength/failure models described in
the preceding sections, a numerical solution has been developed
and implemented in a simulator. The latter can be used to study

the stability of a wellbore drilled into a dipping laminated rock
formation that exhibits strength anisotropy. Fig. 7 shows the
computational algorithms with which critical mud weights can be
obtained for any given wellbore azimuths and deviations. However, in the determination of mud weights while drilling, Eq. (7)
can be simplified by considering only the induced stresses by at
the borehole wall (r ¼a).
The simulator first reads the inputs that include the magnitude
and orientation of in-situ stresses, pore pressure, the mechanical
properties and strength parameters of intact rock matrix, and the
orientation (dip and dip direction) and strength parameters of
weak plane. With a given wellbore azimuth and deviation, the insitu stresses are transformed into stress components in GCS and
then in BCS using Eqs. (5) and (6). It then takes a mud weight
(MW) and calculates the effective stresses in CCS at a given point
of the borehole wall using Eq. (7).
The state of stress at the point is examined for two failure
criteria. First for shear failure across the intact rock matrix,
effective principal stresses (s01 and s03 ) acting on the point are
calculated, and the Mohr–Coulomb failure criterion for rock
matrix, Eq. (10), is used to check for failure. If the rock is failing
at the point, the mud weight is increased in a fixed increment and
the failure verification process goes through the same steps. This
iteration continues until the rock at the point become non-failing,
and the safe mud weight is recorded. Subsequently, the computations go to the next point and the entire safe mud weight
determination process repeats itself for all the point locations
defined by the circumferential angle y which changes from 01 to
1801 in a fixed increment. Once the checking for failure for all
points is completed, safe mud weights for all the points are
compared to locate the maxima. For the assessment of shear
failure along the plane of weakness, effective normal and resultant shear stresses on the plane are calculated using Eqs. (8) and
(9), which are then compared with the weak plane failure

criterion, Eq. (11), to determine the state of failure. The same
iteration scheme as for the intact rock matrix (isotropic strength)

case is applied to find the maximum safe mud weight for a given
wellbore azimuth and deviation.
Once the maximum safe mud weights are determined for both
cases, they are compared and the higher one is selected as the
critical mud weight for the given wellbore azimuth and deviation
to maintain stability. This procedure is repeated until the safe
mud weights for all the given wellbore azimuths and deviations
which, respectively, change from 01 to 3601 and 01 to 901 in fixed
increments, are obtained.
In addition, failure regions around a wellbore can be predicted
with a slight modification of the algorithms. The modification can
be made with given mud weight, wellbore azimuth and deviation
as constant inputs. A loop of computation, similar to that shown
in Fig. 7, is carried out with varying the angle (y) as well as the
radial distance (r) to check for failure around the wellbore. In this
case, the calculation is not limited to the case of r ¼a so that the
full solution (Eq. (7)) has to be used.

3. Wellbore stability analysis and discussion
3.1. Parametric wellbore stability analysis for a hypothetical case
Using the simulator developed, a parametric analysis has been
carried out using the input parameters provided in Table 1.
Among them are parameters that can be controlled during drilling
operations, which include mud weight and drilling direction
(wellbore azimuth and deviation). The analysis is conducted by
varying these two parameters to assess their effects on wellbore
stability.

3.1.1. Effects of drilling direction on failure region around the
borehole
Directional drilling has routinely been employed among others
to increase the length of exposed reservoir sections as well as
allow drilling to the reservoir where vertical access is difficult or
not possible. Drilling direction of a wellbore is commonly defined
by its azimuth (ab) and deviation (bb) (see Fig. 4). In this study,
the well azimuth is measured from North to East in the global
coordinate system, and the well deviation is measured from the
vertical. Shear failure regions around a wellbore for both rock
matrix and weak plane were examined for a constant mud weight
of 0.527 psi/ft (10.14 ppg) but with changing wellbore azimuths
and deviations. The failure regions are illustrated in Figs. 8 and 9.
In the figures, the areas shaded in green show rock matrix
failure regions and the areas shaded in red show regions of weak
plane failure. In the case of isotropic rock strength and under
normal faulting stress regime (Fig. 8a, Table 1), it is observed that

Table 1
Input parameters for a hypothetical case.
Well depth (ft)
Overburden stress (sv, psi/ft)
Maximum horizontal stress (sH, psi/ft)
Azimuth of minimum horizontal stress (deg.)
Minimum horizontal stress (sh, psi/ft)
Pore pressure (Pp, psi/ft)
Cohesion of rock matrix (Si, psi)
Coefficient of friction of rock matrix (mi)
Poisson’s ratio (u)
Biot’s parameter (a)

Cohesion of weak plane (Sw, psi)
Coefficient of friction of weak planes (mw, deg.)
Dip angle of weak plane (deg.)
Dip direction of weak plane (deg.)

8000
0.9
0.8
N01E
0.75
0.46
1015 (7 MPa)
0.58 (301)
0.35
1
435 (3 MPa)
251 (0.466)
151
1101


H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

Fig. 8. Failure regions around wellbores drilled along the sh (ab ¼01) and sH
(ab ¼901) directions with varying wellbore deviations. This analysis was performed with the input parameters shown in Table 1: (a) for an isotropic rock
strength case, (b) for the case in which anisotropic rock strength is taken into
account. The regions shaded with green represent the zones of shear failure across
the rock matrix, and the regions shaded with red represent the zones of shear
failure along the planes of weakness. Mud weight used was 0.527 psi/ft
(10.14 ppg).

the deviated wellbores drilled in the direction of sh (ab ¼01) are
more stable than those in the direction of sH (ab ¼901). The rock
matrix failure regions are much enlarged in slightly inclined
wellbores (bb ¼01–301) drilled along ab ¼901. For highly inclined
wellbores (bb ¼601 and 901), the regions are further extended into
the rock formations beyond the borehole wall in wellbores drilled
along ab ¼901 for the same well deviation. This is due to wellbores
drilled in the sH direction having larger stress differences under
the normal faulting regime (see Table 1).
When anisotropic rock strength is considered, shear failures
along the plane of weakness occur (Fig. 8b). It is interesting to
note from Fig. 8(b) that shear failures occur at four locations
around the top and bottom of the wellbore where the weak
planes intersect the wellbore wall at shallow angles. Such failures
could result in a ‘square borehole’ and had been observed in

115

Fig. 9. Failure regions around wellbores drilled along the directions of cross-dip
(ab ¼ 201), down-dip (ab ¼1101), and up-dip (ab ¼ 2901) with varying wellbore
deviations. This analysis was performed with the input parameters shown in
Table 1: (a) for isotropic rock strength case, (b) for the case in which anisotropic
rock strength is taken into account. The regions shaded with green color represent
the zones of shear failure across rock matrix, and the regions shaded with red
color represent the zones of shear failure along the planes of weakness. Mud
weight used was 0.527 psi/ft (10.14 ppg).

image logs (Mastin et al., 1991) and laboratory tests (Cook et al.,
1994; Okland and Cook, 1998). Additionally, this failure mode is

also the main source of blocky and platy cavings, which are
characterized by two parallel failure surfaces often observed
while drilling at a small acute angle to the bedding (Edwards
et al., 2003; Willson et al., 2003; Brehm et al., 2006). Fig. 9(a and
b) shows the failure regions of a wellbore drilled along up-dip
(ab ¼2901), down-dip (ab ¼1101), and cross-dip (ab ¼201) directions with varying deviations at the constant mud weight. It can
be seen that drilling along the up-dip direction improves wellbore
stability compared to that in the directions of down-dip and
cross-dip, an observation often experienced in the field (Last et al.,
1995; Skelton et al., 1995).


H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

3.1.2. Effects of drilling direction on critical mud weights
Altering drilling directions may require changing operational
mud weights in order to avoid wellbore instability problems.
There are two limit-bounds of mud weight; one is the lower
bound below which compressive failure (breakouts) of the wellbore will occur, while the other is the upper bound above which
tensile failure (fractures) will take place. Only the lower bound
critical mud weight will be discussed. In order to further demonstrate the influence of drilling direction on wellbore stability, the
lower bound mud weight required to maintain stability was

13

12.5

Mud Weight (ppg)

116

12

11.5

11

10.5

10

HAzi=0
HAzi=90
HAzi=20

0

15

30
45
60
75
Well Deviation (degrees)

90

14

Mud Weight (ppg)

12
10
8
6
4
HAzi=20
HAzi=110
HAzi=290

2
0

0

15

30
45
60
75
Well Deviation (degrees)

90

13

Mud Weight (ppg)

12.5

12

11.5

11
HAzi=290
HAzi=110
HAzi=20
HAzi=0
HAzi=90

10.5

10

Fig. 10. Polar plots of lower critical mud weights obtained with the input
parameters given in Table 1: (a) for isotropic strength case, (b) for the case in
which only shear failure along the planes of weakness is considered, and (c) for the
case in which (a) was combined with (b) by taking the greater of the mud weights
between (a) and (b) at the same wellbore azimuth and deviation. Mud weights
shown next to the color bar are in ppg. The symbol () represents the normal of
the plane of weakness.

0

15

30
45

60
75
Well Deviation (degrees)

90

Fig. 11. Lower critical mud weights as a function of well deviation obtained with
the input parameters given in Table 1: (a) for isotropic strength case, (b) for the
case in which shear failure along the planes of weakness is only considered, and
(c) for the case in which anisotropic rock strength is incorporated with the case of
isotropic rock strength.


H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

determined for the input parameters given in Table 1. Figs. 10 and
11 show the results of the analysis.
Fig. 10(a) shows a polar plot of lower bound mud weights for
the isotropic strength case in which the stress state around the
borehole was examined only for shear failure across the intact
rock matrix. Under the normal faulting stress regime, vertical or
slightly inclined wells require relatively less mud weight than
highly inclined or horizontal wells for wellbore stability controls.
This is more apparent from Fig. 11(a) which shows the variations
of lower critical mud weight with well deviations for the isotropic
strength case. Drilling along the sh direction (ab ¼01) requires less
mud weight than along the sH direction (ab ¼901) for the same
well deviation. The mud weight increases with increasing well
deviation for ab ¼901, whereas for ab ¼01 the mud weight
decreases and then increases with increasing well deviations.

The lowest mud weight occurs at the well deviation of about 281
for the give input parameters.
Fig. 10(b) shows a polar plot of lower critical mud weights for
the case in which the stress state around the borehole was
examined only for slippage along the weak planes using
Eq. (11). Compared to Fig. 10(a), a completely different mud
weight scheme is obtained. There is a circular area where the mud
weight required to maintaining wellbore stability is the lowest.
Note that in this polar plot the normal of the weak plane is
located in the center of the area. Fig. 11(b) shows that an up-dip
drilling along the well azimuth of 2901 requires no mud weight
within the range of well deviations from 01 to 281, and then
increased mud weights with increasing well deviation. Additionally, an up-dip drilling requires less mud weight than drilling in
the down-dip and cross-dip directions.
As described in Fig. 7, once the mud weights required for both
isotropic strength and slippage along the plane of weakness cases
are determined for a given well azimuth and deviation, they are
compared and then the larger of the two is selected as the
required mud weight to prevent both wellbore failure modes.
For example, for a wellbore drilled along the cross-dip direction
with a well deviation of 601, the critical mud weight required to
prevent rock matrix failure is about 11.2 ppg (Fig. 11a) whereas
the critical mud weight required to prevent weak plane shear
failure is 12 ppg (Fig. 11b). Therefore, the mud weight required to
maintain wellbore stability is 12 ppg (Fig. 11c). The same procedure is repeated for all well azimuths and deviations. Fig. 11(c)
shows the safe mud weights required to prevent both shear
failure modes. For well deviations less than 451, wellbores drilled
along the sh and cross-dip directions require less mud weight
than along the directions of sH, up-dip, and down-dip. However,
the converse is true for wells having deviations greater than 451

(Fig. 11c). In general, the case study described above indicates
that ignoring strength anisotropy in stability analyses for wells

117

drilled into laminated formations can lead to erroneous mud
weight recommendations during drilling, and thus can potentially
impact NPT and well costs. The above analysis also shows that by
simply changing the wellbore orientation can significantly
improve stability.
3.2. Application of the solution to a field case
A similar wellbore stability analysis had been carried out for
the case of Pedernales field, Venezuela. Table 2 shows the data
from Willson et al. (1999). As shown in the table, the state of insitu stress at the depth of interest is a strike-slip faulting regime
in which the maximum horizontal stress is the major principal
stress, and minimum horizontal stress is the least. The plane of
weakness is dipping toward North-West (3151) at a dip angle of
451.

Table 2
Input parameters for the case of Pedernales field, Venezuela (after Willson et al.,
1999).
Well depth (ft)
Overburden stress (sv, psi/ft)
Maximum horizontal stress (sH, psi/ft)
Azimuth of minimum horizontal stress (deg.)
Minimum horizontal stress (sh, psi/ft)
Pore pressure (Pp, psi/ft)
Cohesion of rock matrix (Si, psi)
Coefficient of friction of rock matrix (mi)

Poisson’s ratio (u)
Biot’s parameter (a)
Cohesion of weak plane (Sw, psi)
Coefficient of friction of weak planes (mw, deg.)
Dip angle of weak plane (deg.)
Dip direction of weak plane (deg.)

5500
0.98
1.2
N451E
0.92
0.468
1188
0.6 (311)
0.3
0.8
300
0.5 (26.61)
45o
315o

Fig. 12. Failure regions around wellbores drilled along the directions of cross-dip
(ab ¼ 451), up-dip (ab ¼1101), and down-dip (ab ¼ 3151) with varying wellbore
deviations. This analysis was performed with the input parameters given in
Table 2: (a) for isotropic rock strength case, and (b) for the case in which
anisotropic rock strength is taken into account. The regions shaded with green
represent the zones of shear failure across rock matrix, while the regions shaded
with red represent the zones of shear failure along the planes of weakness. Mud
weight used was 0.51 psi/ft (9.817 ppg).

118

H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

According to Willson et al. (1999), the drilling instability
problems in Pedernales field were mostly experienced in highly
inclined cross-dip wells. Fig. 12 shows the failure zones around
the wellbore drilled to the directions of cross-dip, up-dip, and
down-dip with varying deviations. The operational mud weight
used for this analysis was 9.82 ppg. It can be seen from Fig. 12
that up-dip wells (ab ¼1351) are much more stable than down-dip
and cross-dip wells (ab ¼3151 and 451). Failure regions of crossdip wells for the isotropic strength case do not significantly
change the width and depth with the variation of well deviation.
However, when the anisotropic rock strength is incorporated into
the analysis, the failure zones gradually enlarged and become
deeper with increasing well deviation.
Furthermore for the isotropic strength case, there is no difference
in the width and depth of failure zones between up-dip and

down-dip drillings. For both cases, the failure zones become
narrower and shallower with increasing well deviations. However,
when the anisotropic rock strength is taken into account, there is a
significant difference in the weak plane shear failure zones for the
range of deviation between 101 and 801. No weak plane slippage
appears in the up-dip wells, whereas the failure zones are formed at
four corners of down-dip and cross-dip wells.
Fig. 13 shows the polar plots of lower critical mud weight for the
field case. The isotropic strength case (Fig. 13a) suggests that highly

inclined wells drilled in the direction of sH are beneficial in terms of
wellbore stability controls. There are two options in the selection of
drilling direction along the sH direction. The well can be oriented
either along SE (1351) or NW (3151). To get an idea on which of the
two directions is stable, the anisotropic strength effect on wellbore
stability should be taken into consideration. As shown in Fig. 13(b),

17
16

Mud Weight (ppg)

15
14
Cross-dip
13

(αb = 45°)

12

Up- & Down-dip
(αb = 135° & 315°)

11

HAzi=45
HAzi=135
HAzi=315

10
9

0

15

30
45
60
Well Deviation (degrees)

75

90

17
16

Cross-dip
(αb = 45°)

Mud Weight (ppg)

15
14

Down-dip
13

(αb = 315°)

12
11
Up-dip
HAzi=45
HAzi=135
HAzi=315

10
9
Fig. 13. Polar plots of lower critical mud weights obtained with the input
parameters given in Table 2: (a) for isotropic strength case, and (b) for the case
in which anisotropic rock strength is incorporated with the isotropic rock strength
case. Mud weights shown next to the color bar are in ppg. The symbol ()
represents the normal of the plane of weakness.

0

15

(αb = 135°)

30
45
60
Well Deviation (degrees)

75

90

Fig. 14. Lower critical mud weights as a function of well deviations obtained with
the input parameters given in Table 1: (a) for isotropic strength case, and (b) for
the case in which anisotropic rock strength is incorporated with the isotropic rock
strength case.


H. Lee et al. / Journal of Petroleum Science and Engineering 96-97 (2012) 109–119

drilling in the direction of up-dip (ab ¼1351) requires less mud
weight compared to the down-dip direction (ab ¼3151). This is in
accordance with the field experience indicating that drilling along
the up-dip direction improved wellbore stability (Last et al., 1995;
Skelton et al., 1995; Okland and Cook, 1998).
It is further observed from Fig. 14, which shows the variation of
mud weight with well deviation along cross-dip, up-dip, and downdip direction, that drilling along the up-dip direction improves
stability compared to the other directions. In the case of cross-dip
drilling, the required mud weight significantly increases when the
well deviation is greater than 171. If the laminated rock formation is
extensively fractured, prudence should be exercised when using
such a high mud weight as significant mud pressure infiltration into
discontinuities can take place. This in turn will result in a loss of
mud pressure support against wellbore wall as well as potential
bulk strength reduction (reduction of normal effective stress and
lubricating effect of fracture faces), both of which can further
deteriorate the wellbore conditions.

4. Conclusions
A wellbore stability model has been developed incorporating

both isotropic and anisotropic rock strengths with the state of
stress around the borehole. Two shear failure modes were
considered in the model; failure across the intact rock matrix
and slippage along the weak planes. A stability analysis using the
model involves multiple transformations of stress components
between the different reference coordinated systems and their
projection onto the plane of weakness, and checking the state of
stress for the two types of failure around the borehole.
A sensitivity analysis using the model was performed for two
cases to evaluate the effects of in-situ stresses, weak planes, and
wellbore trajectory on the stability of wellbore drilled into thinly
laminated rock formations. When anisotropic rock strength is
included in the analysis, shear failure along the weak planes
typically occurs at four corners around the top and bottom of the
hole. These are also the locations where the weak planes intersect
the borehole wall at shallow angles. The extent of the failure
region depends on the relative orientation between wellbore
trajectory and bedding planes. In the analysis in which only the
bedding plane slippage is considered, it was found that there is a
circular area surrounding the bedding normal where no bedding
plane shear failure will occur even without any wellbore pressure
support. Additionally, the lower bound mud weight required to
maintain stability from compressive borehole failures was found
to be strongly dependent on the relative orientation between well
paths and bedding planes. Generally, an up-dip drilling is more
stable than down-dip and cross-dip drillings.

Acknowledgment
We would like to thank Dr. Pepin, Dr. Song, and Dr. Chang for
their valuable comments. This research was partially supported

by the basic research project of the KIGAM funded by the ministry
of knowledge economy of Korea.
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