Peng S., Zhang J., Engineering Geology for Underground Rocks
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7.1 Wellbore stability while drilling 165
Borehole instabilities are the main cause of drilling difficulties, resulting
in an expensive loss of time, sometimes in a loss of part of or even whole
boreholes. Wellbore instabilities make logging very difficult to perform
and to interpret (Maury and Sauzay 1987). A bad condition of the borehole
wall alters artificially the annulus zone corresponding to the depth of in-
vestigation of most of the logging tools. The shape of the borehole can be
strongly modified giving an elongated hole in one direction, diameter re-
duction in the other direction and also almost circular cavings in places. In
the Cusiana field in Colombia, even though some measures to prevent
borehole instability were taken, extensive breakouts in fissile and naturally
fractured shales – of up to 44" in 12
1
/
4
" hole – occurred (Willson et al.
1999). Approximately 10% of the well costs in the Cusiana field were
spent coping with bad holes, mainly because of abnormally high tectonic
stresses induced by an active thrust-faulting environment (Addis et al.
1993, Last and McLean 1995). In addition to the cost associated with
borehole instability while drilling, borehole stability also has a substantial
impact on reservoir productions (Bradley 1979).
There are several stages in the life of a well, i.e., drilling, completion,
stimulation, flow tests, production, and depletion. Borehole instabilities
can be encountered in all these stages (Ramos et al. 1996). In the drilling
stage, the main concerns are to determine the mud composition and density
(mud weight) which will maintain the integrity of the well, without the loss
of drilling fluids. During the completion and stimulation stage, the reser-
voir must be connected to the well via perforations. This operation could
fail if the rock adjacent to the cemented casing is non-brittle. Prior to full
production, downhole tests include open-hole logging, fluid sampling,
build-up, drawdown, injection, and deliverability tests. It is possible to in-
duce wellbore failure and casing collapse during these processes (Peng et
al. 2007). As hydrocarbons are depleted, the drained region compacts
which could induce solids production, casing damage, surface subsidence
and wellbore failure. All these stages in the life of a well, integrated bore-
hole stability analyses are important to ensure the reservoir economical
production and minimize the costly problems induced by the wellbore in-
stabilities.
In order to predict wellbore stability, stress components and distribu-
tions near the wellbore due to drilling perturbation need to be analyzed.
Applying rock failure criteria wellbore stress and rock strength can, then,
be compared to determine rock failures, such as shear failure, tensile fail-
ure etc. Finally, a mud weight range can be determined to avoid wellbore
shear, compressive, and tensile failures. This mud weight range is the safe
mud weight window for drilling.
166 7 Wellbore/borehole stability
7.2 Wellbore stability – elastic solution
7.2.1 Vertical borehole in an isotropic stress field
The plane strain solution (i.e. strain in z-direction
H
z
= 0) can be applied to
a wellbore loaded on its external boundary by an isotropic stress,
V
, and on
its internal boundary by a drilling mud pressure, p
m
(Fig. 7.5). This is
known as Lame’s problem.
In cylindrical coordinates, the stress components at a distance r from the
wellbore are given by:
°
°
°
°
¯
°
°
°
°
®
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
0
1
1
2
2
2
2
2
2
2
2
zrzr
vz
m
m
r
r
Rp
r
R
r
Rp
r
R
TT
T
WWW
VV
VV
VV
(7.1)
where R is the wellbore radius; r is the distance from the wellbore center;
V
r
,
V
T
and
V
z
are the radial, tangential and vertical stress components, re-
spectively; W
r
T
, W
rz
, and W
T
z
are the shear stress components;
V
and
V
v
are the
horizontal and vertical stresses in the far-field stress field, respectively.
Fig. 7.5. A borehole in an isotropic stress field.
7.2 Wellbore stability – elastic solution 167
At the wellbore wall, r = R, one obtains:
°
¯
°
®
vz
m
mr
p
p
VV
VV
V
T
2
(7.2)
It can be seen that stress concentrations around the wellbore occur due
to drilling. The radial displacement or well convergence, can be computed
using the following equation:
r
R
G
p
u
m
r
2
2
V
(7.3)
where G is the shear modulus.
7.2.2 Vertical borehole in an anisotropic stress field
Kirsch’s problem refers to a borehole subjected to a bi-axial stress state
V
H
,
V
h
(
V
H
>
V
h
) (Fig. 7.6). The solution for the various stress components is
expressed as follows:
°
°
°
°
°
°
°
¯
°
°
°
°
°
°
°
®
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
0
2sin
32
1
2
)(
2cos
2
)(4
2cos
3
1
2
)
(
1
2
)(
2cos
34
1
2
)(
1
2
)(
4
4
2
2
2
2
2
2
4
4
2
2
2
2
4
4
2
2
2
2
rzz
hH
r
hH
vz
m
hHhH
m
hHhH
r
r
R
r
R
r
R
r
R
p
r
R
r
R
r
R
p
r
R
r
R
r
R
WW
T
VV
W
T
VV
QVV
T
VVVV
V
T
VVVV
V
T
T
T
(7.4)
where
V
H
and
V
h
are the maximum and minimum far-field horizontal
stresses;
T
is defined in Fig. 7.6.
In particular, at the borehole wall, the tangential and radial stress com-
ponents can be expressed as follows:
168 7 Wellbore/borehole stability
mr
mhHhH
p
p
V
T
V
V
V
V
V
T
2cos)(2)(
(7.5)
The maximum and minimum tangential stress components at the bore-
hole wall can be obtained as follows:
°
¯
°
®
$
$
09at3
0at3
TVVV
TVVV
T
T
mhH
mHh
p
p
(7.6)
It can be concluded from Fig. 7.6 and Eq. 7.6 that the maximum tangen-
tial stress occurs in the minimum far-field horizontal stress direction
(
$
09
T
), and the minimum tangential stress is in the maximum far-field
horizontal stress direction (
$
0
T
). That is the reason why the shear failure
first occurs at the borehole wall of the minimum horizontal stress direction.
H
V
p
m
h
V
h
V
r
T
H
V
Fig. 7.6. A vertical borehole in an anisotropic stress field.
7.2.3 Inclined borehole in an anisotropic stress field
For an inclined borehole, the stress components can be calculated by refer-
ring to Eq. 7.4 by considering far-field stress change due to the hole devia-
tion. This introduces shear stress components of far-field stress to the hole.
In a cross section perpendicular to the axial direction of the hole, the stress
components in the far-field need to be calculated. The wellbore stresses
7.2 Wellbore stability – elastic solution 169
then can be calculated. Bradley (1979) gave the stress distribution around a
borehole located in an arbitrary far-field stress field:
°
°
°
°
°
°
°
°
°
°
°
°
¯
°
°
°
°
°
°
°
°
°
°
°
°
®
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
2
2
2
2
4
4
2
2
2
2
2
2
2
2
4
2
4
4
2
2
2
2
4
4
2
2
4
4
2
2
2
2
1)cossin(
1)cossin(
32
12cos2sin
2
2sin42cos)(2
2sin
3
1
2cos
3
1
2
)(
1
2
)(
2sin
34
1
2cos
34
1
2
)(
1
2
)(
r
R
r
R
r
R
r
R
r
R
r
R
r
R
p
r
R
r
R
r
R
r
R
p
r
R
r
R
r
R
r
R
r
R
yzxzz
xzyzrz
xy
yx
r
xyyxzzz
mxy
yxyx
mxy
yxyx
r
TWTWW
TWTWW
TWT
VV
W
TQWTVVQVV
TW
T
VVVV
V
TW
T
VVVV
V
T
T
T
(7.7)
where
V
and
W
with subscript of r and
T
are the normal and shear stresses in
a cylindrical coordinate system with the z-direction parallel to the drilling
direction; while
V
and
W
with subscript of x, y and z are the normal and
shear stresses in a Cartesian coordinate system which has the same z-axis
as the cylindrical system, with the z-direction parallel to the drilling direc-
tion;
T
is the azimuthal angle measured from the x-axis.
The conversion of this Cartesian coordinate system’s stresses from in-
situ principal stresses can be performed through coordinate transformation
(refer to the previous chapter in Sec. 6.3.4).
The stresses at the wellbore wall can be written as:
170 7 Wellbore/borehole stability
°
°
°
°
¯
°
°
°
°
®
0
0
)cossin(2
]2sin42cos)(2[
2sin42cos)(2
rz
r
yzxzz
xyyxzzz
xyyxmyx
mr
p
p
W
W
TWTWW
TWTVVQVV
TWTVVVVV
V
T
T
T
(7.8)
7.2.4 Vertical borehole considering pore pressure effects
The effective stress components around a vertical borehole with considera-
tion of pore pressure and temperature effects are expressed as:
°
°
°
°
°
°
°
°
°
°
¯
°
°
°
°
°
°
°
°
°
°
®
¸
¸
¹
·
¨
¨
©
§
c
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
c
¸
¸
¹
·
¨
¨
©
§
¸
¸
¹
·
¨
¨
©
§
c
0
2sin
32
1
2
)(
2cos)(
2
)(2cos
3
1
2
)(
1
2
)2(
)(2cos
34
1
2
)(
1
2
)2(
4
4
2
2
2
2
2
2
4
4
2
2
2
2
4
4
2
2
2
2
rzz
hH
r
ThHpvz
Tpm
hH
phH
pm
hH
phH
r
r
R
r
R
r
R
p
r
R
pp
r
R
r
R
p
r
R
pp
r
R
r
R
r
R
p
WW
T
VV
W
QVTVVQVV
VT
VV
VV
V
T
VV
VV
V
T
T
T
(7.9)
where p
p
is the pore pressure;
V
T
is the thermal stress induced by the dif-
ference (ǻT ) between the mud temperature and the formation temperature.
The tangential thermal stress can be calculated by the following equation:
7.2 Wellbore stability – elastic solution 171
Q
D
V
'
1
TE
T
T
(7.10)
where Į
T
is the thermal constant; E is the Young’s modulus; and Ȟ is the
Poisson’s ratio.
The effective stress at the wellbore wall can be written as:
°
¯
°
®
c
c
c
ThHpvz
hHTmphH
pmr
p
pp
pp
QVTVVQVV
TVVVVVV
V
T
]2cos)(2[
2cos)(2
(7.11)
If the temperature effect is negligible, the effective tangential and radial
stress components can be expressed as:
°
¯
°
®
q
c
q
c
c
0,3
90,3
min
max
TVVV
TVVV
V
T
T
mpHh
mphH
pmr
pp
pp
pp
(7.12)
where ș is defined in Fig. 7.6, and ș = 90° is the minimum far-field direc-
tion.
Assuming that the effective maximum tangential and radial stresses
are the principal maximum and minimum stresses, the Mohr-Coulomb
failure criterion can be written as follows, when the hole is stable:
r
qUCS
V
V
T
c
d
c
max
(7.13)
Inserting Eq. 7.12 into Eq. 7.13, the minimum mud pressure, p
m
, can be
calculated:
1
)1(3
t
q
pqUCS
p
phH
m
V
V
(7.14)
Introducing
)sin1()cos2(
I
I
cUCS , )sin1()sin1(
I
I
q ,
the minimum mud pressure, p
m
, can be rewritten as the following form:
IIIVV
sincos)sin1)(3(
2
1
phHm
pcp t
(7.15)
where
I
is the angle of internal friction; c is the cohesion.
The minimum mud pressure is also called collapse pressure or shear
failure pressure, which is the minimum mud pressure to prevent wellbore
172 7 Wellbore/borehole stability
from shear failures. The minimum mud pressure gradient is named the
minimum mud weight, or shear failure gradient (SFG).
It can be seen that the minimum mud weight depends highly on the pore
pressure. Therefore, precise prediction of the pore pressure is of vital
importance for determining drilling mud weight.
Figure 7.7 plots the minimum mud pressure and different failure areas
for a vertical borehole at depth of 2000 m. The minimum effective tangen-
tial (hoop) stress and effective radial stress are also plotted in the same fig-
ure to determine the rock splintering/spalling (when radial stress is less
than tensile strength) and drilling-induced tensile failure (tangential stress
exceeds tensile strength).
Figure 7.8 shows a relationship of the mud pressure and wellbore fail-
ures. It can be seen from Figs. 7.7 and 7.8 that when the mud pressure is
less than the pore pressure, the wellbore has splintering failure or spalling.
When the mud pressure is less than the shear failure gradient, the wellbore
has shear failure or breakout. If the mud pressure is too high, the drilling
induced hydraulic fractures are generated, which may cause drilling mud
losses.
-20
-10
0
10
20
30
10 15 20 25 30 35 40
Mud pressure (MPa)
Stress at wellbore (MPa)
Min MW
Tensile strength
Min hoop stress
Radial stress
Safe Mud weight range
Min. Mud pressure
Tensile failure
Splintering failure
Shear failure
Compressive failure
Pore pressure
-20
-10
0
10
20
30
10 15 20 25 30 35 40
Mud pressure (MPa)
Stress at wellbore (MPa)
Min MW
Tensile strength
Min hoop stress
Radial stress
Safe Mud weight range
Min. Mud pressure
Tensile failure
Splintering failure
Shear failure
Compressive failure
Pore pressure
Fig. 7.7. Mud pressure window for a wellbore drilled at depth of 2000 m with far-
field stress of
V
v
= 46 MPa,
V
H
= 40 MPa,
V
h
= 30 MPa and pore pressure of p
p
=
20 MPa. The rock strength parameters are UCS = 36 MPa,
I
= 30°, and tensile
strength of zero. The calculated safe mud pressure ranges from 21.3 to 30 MPa.
7.3 Wellbore stability – elastoplastic solution 173
Fig. 7.8. Schematic relationship of mud pressure (mud weight) and wellbore fail-
ure behaviors. To keep wellbore stability, the mud weight should be in appropriate
range (modified from Madge et al. 2006).
7.3 Wellbore stability – elastoplastic solution
An ideal case is considered, i.e. a plane strain circular infinite domain
loaded on its internal boundary by a mud pressure p
m
and at infinity by an
isotropic far-field stress
V
(Fig. 7.9) in an ideal elastoplastic material. In
Fig. 7.9 a failed/plastic zone with radius of R
p
is formed due to low mud
pressure. Assuming that the tangential stress (
V
T
) and radial stress (
V
r
) sat-
isfy
V
T
>
V
r
, an ideal plastic rock formation with a Mohr-Coulomb yield
criterion in effective stress form in this failed zone can be expressed as fol-
lows:
Fig. 7.9. A borehole in an ideal elastoplastic material.
174 7 Wellbore/borehole stability
)(
prfcp
pqp
c
V
V
V
T
(7.16)
where
V
c
c is the uniaxial compressive strength in the failed rock; p
p
is the
pore pressure;
)sin1()sin1(
fff
q
I
I
and
I
f
is the angle of internal
friction of the failed rock.
The equation of equilibrium is given by:
0
rdr
d
r
r
T
V
V
V
(7.17)
Substituting Eq. 7.16 into Eq. 7.17 and solving it, one obtains:
1
1
1
)1(
c
f
q
f
pfc
r
rC
q
pq
V
V
(7.18)
1
1
1
)1(
c
f
q
f
f
pfc
rqC
q
pq
V
V
T
(7.19)
Using the boundary condition (for r = R,
V
r
= p
m
), constant C
1
can be
expressed as:
f
q
f
pfc
m
R
q
pq
pC
c
1
1
1
)1(
V
(7.20)
Therefore, in the failed zone the stresses can be obtained (Roegiers
2002):
°
°
¯
°
°
®
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
c
c
¸
¹
·
¨
©
§
¸
¸
¹
·
¨
¨
©
§
c
c
1
1
1
)1(
1
)1(
1
)1(
1
)1(
f
f
q
f
pfc
mf
f
pfc
q
f
pfc
m
f
pfc
r
R
r
q
pq
pq
q
pq
R
r
q
pq
p
q
pq
VV
V
VV
V
T
(7.21)
In the elastic zone, the stress can be obtained from elasticity theory, i.e.:
°
°
¯
°
°
®
2
2
2
2
r
C
r
C
e
re
VV
VV
T
(7.22)