Peng S., Zhang J., Engineering Geology for Underground Rocks
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6.5 Parametric analyses and application for borehole stability 145
local stress states. In this case the x-direction (
T
= 0q) is the local far-field
maximum stress direction and the y-direction (
T
= 90q) is the local far-field
minimum one.
In the following analyses, double-porosity, time, fracture, mud weight
and inclination effects are considered for inclined borehole problems. For
each analysis, only one specific parameter is allowed to be varied.
6.5.1 Double-porosity effects
The pore pressure at t = 100 s and
T
= 0q for the single-porosity model as
well as the one for the dual-porosity model are compared in Fig. 6.10. The
difference is evidenced by the increase in pore pressure in the matrix for
the dual-porosity media due to the associated large fracture compliance. A
similar comparison is made for Terzaghi’s effective radial stresses (Fig.
6.11). Although the pore pressure induced by dual-porosity effect in-
creases, the tensile stress does not increase, but decreases (Fig. 6.11). This
is because the total radial stress has a larger increment than the pore pres-
sure for the dual-porosity medium (Zhang 2002). Therefore, the effective
compressive stress, which equals to the difference of the total stress and
the pore pressure, increases and the effective tensile stress decreases,
which reduces the potential for borehole spalling (tensile failure).
0
2
4
6
8
10
1 1.5 2 2.5 3
Radial distance (r/R)
Pore pressure (MPa)
Dual-porosity, s = 0.1 m
Single porosity
$
0
T
Fig. 6.10. Comparison of pore pressure for single- and double-porosity models in
the local far-field maximum stress direction at t = 100 s.
146 6 Double porosity poroelasticity and its finite element solution
-2
0
2
4
6
8
10
12
14
1 1.5 2 2.5 3
Radial distance (r/R)
Radial stress (MPa
)
Dual-porosity, s = 0.1 m
Single porosity
$
0
T
Fig. 6.11. Comparison of effective radial stress for single- and double-porosity
models in the maximum stress direction at t = 100 s.
6.5.2 Time-dependent effects
In Figs. 6.12 through 6.14, the pore pressure distributions around the well-
bore are presented at three different time domains. In the near-field, pore
pressure concentrations occur at smaller times (for t = 10 s in Fig. 12 and t
= 100 s in Fig. 13) around the minimum stress direction (
T
= 90q), and the
pore pressures decrease as time increases, as shown in Figs. 6.12 - 6.14.
The pore pressure in the near field is larger as
T
increases, which is due to
Skempton’s effect (Detournay and Cheng 1993), because at 0q a larger far-
field compressive normal stress prevails (Fig. 6.4). Note that non-
monotonic pressure distributions and pressure peaks are found at a small
distance inside the wall at small times, which is attributed to the poroelas-
tic effect. At larger times this effect disappears. At large distances, the pore
pressure approaches asymptotically the far-field pore pressure value of 10
MPa, and as time increases this poroelastic effect becomes negligible.
6.5 Parametric analyses and application for borehole stability 147
-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3
Distance from hole center (m)
0.0
0.1
0.2
0.3
Distance (m)
t = 10 s
Fig. 6.12. Pore pressure distribution around the wellbore at t = 10 s.
-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3
Distance from hole center (m)
0.0
0.1
0.2
0.3
Distance (m)
t = 100 s
Fig. 6.13. Pore pressure distribution around the wellbore at t = 100 s.
148 6 Double porosity poroelasticity and its finite element solution
-0.3 -0.2 -0.1 -0.0 0.1 0.2 0.3
Distance from hole center (m)
0.0
0.1
0.2
0.3
Distance (m)
t = 1000 s
Fig. 6.14. Pore pressure distribution around the wellbore at t = 1000 s.
The Terzaghi’s effective radial stress distributions for four different
times are plotted in Figs. 6.15 and 6.16. The results clearly show a tensile
region near the wellbore at small times. The tensile radial stress magnitude
in the near field is much larger in the minimum stress direction (
T
= 90q)
than that in the maximum stress direction (
T
= 0q). This is due to higher
pore pressure in the minimum stress direction (refer to Figs. 6.12 and
6.13). It is evident that the radial stress has a very strong time effect; i.e.,
the tensile stress reduces significantly as time elapses. Tensile stress disap-
pears as time is large enough.
6.5 Parametric analyses and application for borehole stability 149
-4
-2
0
2
4
6
8
10
12
1 1.5 2 2.5 3
Radial distance (r/R)
Radial stress (MPa)
t = 10 s
t = 100 s
t = 1000 s
t = 10000 s
q 0
T
Fig. 6.15. Effective radial stress in the maximum stress direction for different
times.
-8
-6
-4
-2
0
2
4
6
8
10
1 1.5 2 2.5
3
Radial distance (r/R)
Radial stress (MPa)
t = 10 s
t = 100 s
t = 1000 s
t = 10000 s
q 90
T
Fig. 6.16. Effective radial stress in the minimum stress direction for different
times.
6.5.3 Mud weight effects
Rock failure, such as spalling and breakout, is likely to take place when a
borehole is drilled with air or without sufficient mud pressure to support it
before a casing is placed. However, too high mud pressures may induce
150 6 Double porosity poroelasticity and its finite element solution
borehole instability by tensile fracturing leading to unacceptable mud
losses. In the following analysis, the borehole true vertical depth is as-
sumed to be 1000 m, and four values of mud density are examined, i.e.,
U
w
= 0.006,
U
w
= 0.01,
U
w
= 0.02 and
U
w
= 0.025 MN/m
3
corresponding to mud
pressures of
p
w
= 6, p
w
= 10, p
w
= 20, and p
w
= 25 MPa. Other parameters
remain the same, as in previous analyses.
Due to the mud pressure, there are no tensile radial stresses inside the
borehole except for the case of a very small mud pressure value (e.g. p
w
=
6 MPa), as shown in Fig. 6.17; instead, a non-monotonic stress distribution
appears for higher mud pressures. Furthermore, the increasing mud
pressure causes an increase of the compressive radial stress; this is due to
the fact that the mud weight acts on the wellbore wall as an additional
radial stress component.
-2
0
2
4
6
8
10
12
14
1 1.5 2 2.5 3
Radial distance
(
r/R
)
Radial stress (MPa)
Pw = 6 MPa
Pw = 10 MPa
Pw = 20 MPa
Pw = 25 MPa
q 90
T
Fig. 6.17. Effective radial stress along the minimum stress direction at t = 100 s
for different mud pressures.
Figure 6.18 shows the response of the effective compressive tangential
stresses along the maximum stress direction (
T
= 0q) at t = 100 s. It is
obvious that the compressive tangential stress decreases as the mud
pressure increases, which reduces the high stress concentration around the
wellbore. It is noted that the effective tangential stress becomes tensile
along the local maximum stress direction for high mud pressures (e.g. p
w
=
25 MPa). This illustrates that the borehole will fail in tension or fracturing
when the tensile tangential stress exceeds the formation tensile strength.
6.5 Parametric analyses and application for borehole stability 151
-4
0
4
8
12
16
20
24
1 1.5 2 2.5 3
Radial distance (r/R)
Tangential stress (MPa)
Pw = 6 MPa
Pw = 10 MPa
Pw = 20 MPa
Pw = 25 MPa
q 0
T
Fig. 6.18. Effective tangential stress along the maximum stress direction at t = 100
s for different mud pressures.
6.5.4 Fracture stiffness effects
Figure 6.19 presents the pore pressure responses for different fracture stiff-
nesses along the minimum stress direction (
T
= 90q). It shows that the
pore pressure increases as the fracture stiffness becomes increasingly
smaller. When the stiffness is large enough, representing nearly no
deformation in the fractures, the pore pressure no longer varies with frac-
ture stiffness.
The comparisons of the effective radial stresses indicate that the com-
pressive stress increases, while the tensile stress reduces as the stiffness
decreases, as shown in Fig. 6.20. The radial tensile stress becomes insig-
nificant when the fracture stiffness is very small.
152 6 Double porosity poroelasticity and its finite element solution
0
2
4
6
8
10
12
14
1 1.5 2 2.5 3
Radial distance (r/R)
Pore pressure (MPa)
Kn = 4.821GPa/m
Kn = 48.21GPa/m
Kn = 482.1GPa/m
Kn = 4821GPa/m
q 90
T
Fig. 6.19. Pore pressure around the wellbore in the minimum stress direction at t
= 100 s for different fracture stiffnesses.
-4
-2
0
2
4
6
8
10
11.522.53
Radial Distance (r/R)
Radial stress (MPa)
Kn = 4.821GPa/m
Kn = 48.21GPa/m
Kn = 482.1GPa/m
Kn = 4821GPa/m
q 90
T
Fig. 6.20. Effective radial stress around the wellbore in the minimum stress direc-
tion at t = 100 s for different fracture stiffnesses.
6.5 Parametric analyses and application for borehole stability 153
6.5.5 Borehole inclination effects
To assess the effect of borehole inclination, the pore pressure responses at
t = 100 s,
T
= 90q for four different inclination angles, 0q, 40q, 70q and 90q
are examined in Fig. 6.21. Note that a hole inclination angle of
M
z
= 0q
represents a vertical hole. It shows that the inclination decreases the pore
pressure magnitude in this example. This results from the extra pore pres-
sure generated by Skempton’s effect. At
M
z
= 0q the relevant local far-field
compressive stresses at
T
= 0q are
V
x
= 29 MPa and
V
y
= 20 MPa, which
are changed to
V
x
= 25 MP and
V
y
= 20 MPa at
M
z
= 90q due to the borehole
inclination (refer to Fig. 6.22).
Figure 6.23 explores the influence of hole inclination on radial stresses
at t = 100 s,
T
= 0q. The inclination causes a reduction in the compressive
radial stress and an increase in the tensile radial stress. This trend is more
pronounced at the azimuthal angle
T
= 0q, because the local far-field total
stress varies from
V
x
= 29 MPa (effective stress of 19 MPa) at a hole
inclination
M
z
= 0q to
V
x
= 25 MPa (effective stress 15 MPa) compared to a
hole inclination M
z
= 90q (Zhang and Roegiers 2005); however, at an angle
of
T
= 90q, the local far-field total stress does not change (
V
y
= 20 MPa)
from a hole inclination
M
z
= 0q to
M
z
= 90q (refer to Figs. 6.4 and 6.22).
0
2
4
6
8
10
12
11.522.53
Radial distance (r/R)
Pore pressure (MPa)
Inclination angle 0 degree
Inclination angle 40 degrees
Inclination angle 70 degrees
Inclination angle 90 degrees
q 90
T
Fig. 6.21. Pore pressure distribution at t = 100 s,
T
= 90q for different borehole
plunges,
M
z
.
154 6 Double porosity poroelasticity and its finite element solution
V
y
= 20 MPa
V
z
= 25 MPa
p
ma
=p
fr
=10 MPa
T
T
V
z
= 29 MPa
p
ma
=p
fr
=10MPa
V
y
= 20 MPa
V
x
= 29 MPa
V
x
= 25
(a) Hole inclination
M
z
= 0°
(b) Hole inclination
M
z
= 90°
Fig. 6.22. State of stress in the local coordinate systems after 0q and 90q inclina-
tions.
Figures 6.24 and 6.25 demonstrate the effect of borehole inclination on
the effective tangential and axial stresses at the azimuthal angle of
T
= 90q.
It is clear that at larger inclination angles the tangential stress decreases
and the axial stress increases because the inclination from
M
z
= 0q to
M
z
=
90q causes the local far-field total stress to vary from
V
x
= 29 MPa,
V
z
= 25
MPa to
V
x
= 25 MPa,
V
z
= 29 MPa (see Fig. 6.22).
From the above analyses the effects of borehole inclination on wellbore
pressures and stresses depend strongly upon the far-field stress magni-
tudes. By the same token, for the given boundary and far-field stress condi-
tions, the borehole inclination can be optimized in order to avoid high
stress concentrations.