alternative method usually is employed (Bredehoeft et al., 1976). During the second (and subsequent) cycles of fluid injection, the pressure
versus time record is monitored and then analyzed carefully to identify a change in slope of the
rising pressure that signals the reopening of the
fracture. The elastic model suggests that the reopening pressure, P r , is that necessary to increase
the circumferential stress just slightly above zero:
(6.114)
This interpretation of the pressure versus time
record implies that the in-situ tensile strength is:
(6.115)
The final measure of pressure used to estimate
the in-situ stresses is that just sufficient to hold the
developed fracture open against the least compressive horizontal stress, S h . The relationship
between this pressure and stress is found from a
different solution to the elastic boundary value
problem, because the geometry now is that of a
crack, not a circular hole. The opening, ⌬u, of a
crack in an elastic body is proportional to the difference between the internal pressure, P, that
forces the walls apart, and the remote compressive stress, here taken as S h , that pushes them
together (Pollard and Segall, 1987):
(6.116)
Thus, the pressure and the stress must be just
about equal as the crack starts to open, or as it
closes. Just as the fracture surfaces come together,
a second knee in the pressure record indicates the
shut-in pressure, P s , and this is interpreted as equal
in magnitude to the least compressive stress:
(6.117)
Given the depth of overburden, D, and its
average unit weight, g*, and the two pressures, P s
and P r , read from the pressure versus time record,
one can calculate all three in-situ principal stresses
as:
(6.118)
This interpretation presumes that a vertical fracture has propagated in a plane that is perpendicS H ϭ 3P s Ϫ P c ϩ T ϭ 3P s Ϫ P r
S h ϭ P s
S V ϭ g*D
For ⌬u ϭ 0,P ϭ P s ϭ S h
⌬u ϰ P Ϫ S h
T ϭ P c Ϫ P r
For ϭ 0,P ϭ P r ϭ 3S h Ϫ S H
ular to the direction of S h , so the fracture orientation determines the orientation of S H and S h . The
other principal stress is presumed to be vertical.
Alternatively, one can use the breakdown pressure, P c , and a measure of the tensile strength, T,
in the determination of S H .
The second wellbore procedure provides a
direct determination of the orientation of the insitu stresses, S H and S h . As with the hydraulic fracturing method, the simplest interpretations
depend upon a vertical wellbore and the presumption that one of the principal stresses is vertical. A variety of instruments, including the
borehole camera, dipmeter, acoustic televiewer,
and electrical resistance microscanner are capable of measuring the shape of the wellbore
(Amadei and Stephansson, 1997, p. 308). Although
the drilling bit is designed to cut a cylindrical
hole, the records from these instruments demonstrate that sections of some wellbores are not
cylindrical, but instead have systematic increases
in radii along two diametrically opposed zones
(Fig. 6.35). These zones are referred to as wellbore
breakouts because it is inferred that the hole was
enlarged by the breakage of rock, due to a local
stress concentration, and the subsequent spalling
of the rock fragments into the wellbore.
The geometry of the zones of broken rock
associated with a wellbore breakout suggest that
these are not a result of a single fracture extending perpendicular to the wellbore, as in the
hydraulic fracturing procedure (Fig. 6.31). Instead,
it has been proposed that a set of shear fractures
oriented oblique to the wellbore (Fig. 6.35), or a set
of opening fractures oriented parallel to the wellbore, is responsible for the fragmentation of the
rock (Zoback, 1985; Zheng et al., 1989). In either
case the stress concentration induced by drilling
the hole is held responsible for the fracturing.
Once the rock is fractured, flow of the drilling
fluid carries the fragments away, leaving the open
breakout.
The Kirsh solution for the elastic boundary
value problem of a circular hole subject to internal pressure and remote biaxial compressive
stresses provides the equations necessary for an
elementary analysis of the stress concentration
that may cause breakouts. To assure that
hydraulic fractures have not initiated, and that
238
FORCE, TRACTION, AND STRESS
versus time record is monitored and then analyzed carefully to identify a change in slope of the
rising pressure that signals the reopening of the
fracture. The elastic model suggests that the reopening pressure, P r , is that necessary to increase
the circumferential stress just slightly above zero:
(6.114)
This interpretation of the pressure versus time
record implies that the in-situ tensile strength is:
(6.115)
The final measure of pressure used to estimate
the in-situ stresses is that just sufficient to hold the
developed fracture open against the least compressive horizontal stress, S h . The relationship
between this pressure and stress is found from a
different solution to the elastic boundary value
problem, because the geometry now is that of a
crack, not a circular hole. The opening, ⌬u, of a
crack in an elastic body is proportional to the difference between the internal pressure, P, that
forces the walls apart, and the remote compressive stress, here taken as S h , that pushes them
together (Pollard and Segall, 1987):
(6.116)
Thus, the pressure and the stress must be just
about equal as the crack starts to open, or as it
closes. Just as the fracture surfaces come together,
a second knee in the pressure record indicates the
shut-in pressure, P s , and this is interpreted as equal
in magnitude to the least compressive stress:
(6.117)
Given the depth of overburden, D, and its
average unit weight, g*, and the two pressures, P s
and P r , read from the pressure versus time record,
one can calculate all three in-situ principal stresses
as:
(6.118)
This interpretation presumes that a vertical fracture has propagated in a plane that is perpendicS H ϭ 3P s Ϫ P c ϩ T ϭ 3P s Ϫ P r
S h ϭ P s
S V ϭ g*D
For ⌬u ϭ 0,P ϭ P s ϭ S h
⌬u ϰ P Ϫ S h
T ϭ P c Ϫ P r
For ϭ 0,P ϭ P r ϭ 3S h Ϫ S H
ular to the direction of S h , so the fracture orientation determines the orientation of S H and S h . The
other principal stress is presumed to be vertical.
Alternatively, one can use the breakdown pressure, P c , and a measure of the tensile strength, T,
in the determination of S H .
The second wellbore procedure provides a
direct determination of the orientation of the insitu stresses, S H and S h . As with the hydraulic fracturing method, the simplest interpretations
depend upon a vertical wellbore and the presumption that one of the principal stresses is vertical. A variety of instruments, including the
borehole camera, dipmeter, acoustic televiewer,
and electrical resistance microscanner are capable of measuring the shape of the wellbore
(Amadei and Stephansson, 1997, p. 308). Although
the drilling bit is designed to cut a cylindrical
hole, the records from these instruments demonstrate that sections of some wellbores are not
cylindrical, but instead have systematic increases
in radii along two diametrically opposed zones
(Fig. 6.35). These zones are referred to as wellbore
breakouts because it is inferred that the hole was
enlarged by the breakage of rock, due to a local
stress concentration, and the subsequent spalling
of the rock fragments into the wellbore.
The geometry of the zones of broken rock
associated with a wellbore breakout suggest that
these are not a result of a single fracture extending perpendicular to the wellbore, as in the
hydraulic fracturing procedure (Fig. 6.31). Instead,
it has been proposed that a set of shear fractures
oriented oblique to the wellbore (Fig. 6.35), or a set
of opening fractures oriented parallel to the wellbore, is responsible for the fragmentation of the
rock (Zoback, 1985; Zheng et al., 1989). In either
case the stress concentration induced by drilling
the hole is held responsible for the fracturing.
Once the rock is fractured, flow of the drilling
fluid carries the fragments away, leaving the open
breakout.
The Kirsh solution for the elastic boundary
value problem of a circular hole subject to internal pressure and remote biaxial compressive
stresses provides the equations necessary for an
elementary analysis of the stress concentration
that may cause breakouts. To assure that
hydraulic fractures have not initiated, and that
238
FORCE, TRACTION, AND STRESS