Propuestas

Following Babcock’s initial study. Bell and Gough (1979) found that the azimuthal distribution o f breakouts in Alberta is radically unlike that o f joint sets at the surface. Four concentrations o f Joint directions are mappable on the surface and only one significant concentration o f subsurface breakout directions is apparent at depth. They went on to argue that the process o f drilling a vertical borehole through rocks bearing unequal horizontal stresses could concentrate the stresses so as to produce sub-surface breakouts parallel to the minimum stress direction. Equations derived by Kirsch (1898) and given in Timoshenko and Goodier (1984), relating the principal far field horizontal stress to a circumferential stress acting on the circumference o f a hole in a plate were utilised by Bell and Gough (1979) and Gough and Bell (1981, 1982). These authors suggested that, providing that the plate is

infinitely large compared to the hole and is under uniaxial compressive stress S, (Figure

4.2a); at distance r from the centre of a circular hole of radius a, the stress components are:

Oo - — Tre ~ - 2 1 -

2

3a cos 20

s

_{1 +}

7

2

1

-

2a^

3a cos 2 0 [4.1] sin 2 0

where 0 is measured from the direction of S and and are respectively, the radial,

tangential and shear stresses. At the hole wall r = a, and vanish, as they must at a

free surface, and the tangential stress is:

00 = S - 2S cos 20 [4.2]

which varies between a maximum compressive stress of 35 on the diameter at right angles to the applied stress, and maximum tensile stress -5 on the diameter parallel to 5.

l U U U U U U

m

S

(a)

Figure 4.2: (a) Stress near a small circular hole in a large plate under uniaxial compression. The curves show the radial variation of transverse stress, cr^, along the diameter transverse to the applied compression 5; (b) A small circular hole in a large plate under biaxial compression. (After Bell and Gough, 1979).

Bell and Gough (1979) suggested that the distribution of stresses around a borehole can induce two possible modes of failure (Figure 4.3); compressive shear failure at the wall in

the vicinity 0 = 90° and 270° and tensile failure at 0 = 0° and 180° involving the

propagation of cracks parallel to the applied stress.

A closer approximation to the geological situation is displayed in Figure 4.2b where unequal

compressive stresses 5 and s, where 5 > 5, are applied orthogonally to a plate with a small

hole in it. From superposition it follows that, at the hole boundary, r = a\

Oq = s + s - 2 ( S - s) cos 26 [4.4]

It was therefore proposed that oq varies between the maximum value (35 - s) at points m

and n and the minimum ( 3 5 - 5) at points p and q. The concentration of tangential stress

near m and n is illustrated in Figure 4.3.

Figure 4.3: Mechanism of a breakout in a borehole. The trajectories (solid lines) give the directions of the principal stresses in a uniaxially loaded plane with a circular hole. The magnitude of the compressional stresses is indicated by the density of the trajectories. The

accumulation of compressional stresses in the area of 6 = 90° and 270° leads to conjugate shear

failure (dashed lines) so that the borehole is elongated by brittle deformation. In the area of

0 = 0° and 180° the stresses become tensile and fractures develop parallel to S. (Adapted from

Bliimling et aL, 1983).

Where measurements have been made, for example McGarr and Gay (1978), the horizontal

principal stresses in the upper crust (% and where Ujj > a^) range from equality

(ur = (Th) (0 “ 4(7^^ or more. In situations where the horizontal stresses are

approximately equal, Oq = 2(Jr (Bell and Gough, 1979). In a borehole drilled under such

conditions, the horizontal stress acting on the borehole wall is doubled and spalling may

occur, but without a preferential orientation. When ur = or more, large, consistently

oriented breakouts are developed throughout the borehole (Bell and Gough, 1979).

In a layered sequence of sedimentary rocks Bell and Gough (1979) suggested that some beds will behave elastically and subsequently fail due to brittle fracture, whereas others will undergo some form of anelastic or plastic deformation. They concluded that well-aligned breakouts will form preferentially in the most competent rocks. The region of failure is predicted to be roughly triangular in cross-section and enclosed by flat, conjugate shear planes oriented at a constant angle to the azimuth of the far-field horizontal principal

stresses. In the words o f Zoback et al. (1985) the breakouts would have the appearance o f pointed ‘dog ears’ on opposite sides o f the borehole.

A study by Brown et al. (1980) in the Cotton Valley sandstone. East Texas, concluded that

the consistency in orientation o f elliptical boreholes is due to in situ stresses within the

rock. The authors went on to state that the direction o f hydraulically induced fractures could

be predicted from breakout analysis. However, Brown et al. (1980) were incorrect in

assuming that the direction o f such fractures would be parallel to the minimum stress direction, as Hubbert and Willis (1957) demonstrated that hydraulically induced fractures would be oriented normal to the least horizontal stress.

Gough and Bell (1982) proposed that a linear Mohr-Coulomb failure criterion could provide a first order explanation for both the location and geometry o f localised compressive shear failure. In this hypothesis, the vertical stress is assumed to approximate the overburden pressure Epgd (where p is the density, g is the gravitational constant and d is the depth of each rock layer) and is essentially constant over a depth range o f several hole diameters so that the stress near the wall is essentially plane stress (Gough and Bell, 1982).

The stresses at the wellbore wall close to the ends o f the diameter PQ (at right angles to ajj)

in Figure 4.4 were modelled by Gough and Bell (1982). At P and Q, = 0 and Uq = 3ajj

- At Q’ (a point very close to Q; or P’ very close to P) the horizontal stress difference

is nearly 3ajj - this stress difference falls rapidly as the distance from the centre o f the

borehole (r) increases (Gough and Bell, 1982). It has been proposed that, at approximately four borehole radii from the centre o f the borehole, the stresses are essentially equal to the far-field applied stress (Springer, 1987).

The theory o f brittle shear fracture according to a linear Mohr-Coulomb failure criterion and illustrated in Figure 4.5 led Gough and Bell (1982) to the following predictions for the development o f borehole breakouts:-

1. Fracturing will commence at the wellbore wall (where r = a).

2. The shear strength tq at a = 0 is highly significant, any non-zero Oq must cause fracture

if Tq is small.

3. Shear fracture will produce breakouts o f limited depth.

4. Tensile fractures may occur at the ends o f the diameter parallel to <jjj but are unlikely

M axim um H orizontal Stress (o ,,) Minimum Horizontal Stress ( a j Qo ^ f -► >•

Maximum Horizontal Stress (o„)

0

( 3 o „ - o J

Figure 4.4: Principal stresses near the maximum stress concentration produced by a circular

hole in rock under biaxial stress. A points P and Q, At Q’, very near

Q, the stress difference is nearly compared with far from the hole. (After Gough

and Bell, 1982).

. --- 1.04 .— \ --- 1.09 a 1.25 a

Figure 4.5: Mohr circles for four points at, and near the wellbore wall, along a line perpendicular to the maximum horizontal stress. For rock with the fracture criterion shown, shear fracturing will produce breakouts with well grouped azimuths. (Adapted from Gough and Bell, 1982).

In general, fractures intersecting the wellbore wall will make an angle \l/ with (Gough

and Bell, 1982); for example, when the coefficient o f internal friction of the rock = 1,

\J/ = 22.5°. Gough and Bell (1982) proposed that fractures would initiate at Q and lead to

outward migration of the stress maximum (Figure 4.6).

Consequently, the spalling will migrate away from the centre of the borehole as long as the fracture planes continue to intersect it. Further fracturing may occur, for instance at N (Figure 4.6), but the fractures will no longer intersect the borehole, and in a perfectly homogeneous, isotropic rock volume, spalling was predicted to terminate near point M

(Figure 4.6). Bell and Gough (1983) continued the use of Mohr-Coulomb failure criteria to predict that, ‘with typical rock parameters’, breakouts will initially extend the hole to

between 8 and 1 0% of its original diameter and that appreciable initial shear strength is

necessary to produce tightly grouped breakout azimuths throughout a wellbore section.

Figure 4.6: Shear fractures leading to a breakout. For /x=l and \p=22.5°. Fractures such as

those through point N will not intersect the borehole and breakout will not occur. (Adapted from Gough and Bell, 1982).

Hottman et al. (1979) assumed that failure initiates at the wellbore surface, as opposed to

at some distance behind the surface and modelled failure by comparing octahedral shear stress (T^gJ and the effective confining pressure near the wellbore (P^ - Pp) where

T o e , -

i

+ ("e -

" «r)^

P c -P p - - P . [4.5b]

and Pg is the confining pressure, Pp is the pore pressure and Oq, and are, in turn,

functions of the three principal stresses. The model presented in Figure 4.7 was proposed following the drilling of a series of boreholes off the coast of Alaska where the stress

regime is thought to be > Sy. Breakouts were observed to develop in portions

of the wellbore where pore pressure was abnormally high (Hottman et al., 1979). This

increase in pore pressure was proposed to have initiated a lowering of the wellbore strength and caused failure to occur. An increase in the drilling fluid density (mud weight) was seen to add support to the wellbore wall, thereby decreasing the tendency for the borehole to fail

For the propagation of a tensile crack, the elastic solutions proposed by Gough and Bell

(1982) predict that the minimum tangential stress at the borehole wall is tensile if >

3(7^. However, the nucléation of a tensile fracture is controlled not only be the magnitude

of the externally applied stress but also by the existence of a suitable flaw to initiate crack growth (Griffith, 1925). The exact location of tensile failure within a general region of tension cannot be categorically predicted from consideration of the stress distribution alone.

Rock Strength Curve UNSTABLE (borehole breakouts) increasing 2 STABLE (no breakouts)

Effective Confining Pressure (P, - P^)

Figure 4.7: Octahedral shear stress versus effective confining pressure for borehole breakouts

occurring in the Gulf of Alaska. (Adapted from Hottman et al., 1979).

The interest in breakout formation, theory and applications grew throughout the early

1980’s (Zoback and Zoback, 1980; Gough and Bell, 1981; Plumb, 1982; Zoback et al.,

1982; Bell and Gough, 1983; Bliimling et al., 1983; Cheatham, 1984) culminating in a

paper by Fordjor et al. (1983) who realised that the ‘insignificant regression of breakout

azimuths on depth supports the view that the orientation data represent stress in the lithosphere rather than in the sediments alone’. This followed the proposal by Zoback and Zoback (1980) that stress patterns within provinces delineated in the conterminous United States can be attributed to the direction of present plate motion and residual thermal and dynamic effects.

4.2.2.2 Model Two (Zoback et al., 1985)

A second predictive model of the location and geometry of wellbore instabilities and

breakout initiation was developed by Zoback et al. (1985). This model, as with previous

models (Bell and Gough, 1979; Gough and Bell, 1982), considered the stress distribution around a circular hole in a biaxial stress field again adopting a linear Mohr-Coulomb failure criterion using the modified Griffith criterion of McClintock and Walsh (1962) for failure of a material with closed cracks, the faces of which obey Amonton’s law for surficial sliding.

Zoback et al. (1985) again used the Kirsch (1898) equations in order to model breakouts in cylindrical holes in a thick, homogeneous, isotropic elastic plate subjected to effective

biaxial stresses {S and s):

1 * - j ) ( i - i ! L ■ 1 . 3 * ' ) cos ^4 2 0 . ^2 7 ^ - -^(5 - i) 1 + 3 ^ ^ COS 20 - . ^ - 3 ^ sin 2 0

a. - 1 (5 + 5) 1 + —

- Ir.Ç -

1 +

ms 9/9 -

K.6]

where is the radial stress, Oq is the circumferential stress, is the tangential shear

stress, R is the radius of the hole, r is the distance from the centre of the hole, d is the

azimuth measured from the direction of 5 and AP is the difference between the fluid pressure in the borehole and that in the formation (a positive value indicates excess pressure in the borehole).

Near the wellbore, the rotation of the maximum and minimum principal stress azimuths results in markedly curved potential shear surfaces (Figure 4.8a). The magnitude of shear and effective normal stresses along these potential failure surfaces varies as a function of

r and 6 (Zoback et at., 1985).

Oh

(SX

-Oh

Figure 4.8: (a) The orientation of potential shear failure surfaces adjacent to a wellbore under

the following conditions: = 45MPa, = 30MPa, AP = 0, and /x = 1.0; (b) Area in which

failure is expected where the cohesive strength of the rock (t q) = 12.5MPa. The other variables

The region where compressive shear failure is expected to occur can be predicted from the extended Griffith criterion (Griffith, 1925; McClintock and Walsh, 1962). This criterion considers the extension o f closed cracks which have a finite frictional strength in a biaxial stress field. In this context, potential failure surfaces around a wellbore are considered as cracks with a coefficient o f sliding friction (/i), subjected to a shear stress and an effective

normal stress. Zoback et al.y stated that:

"As discussed by Paterson (1978) and Jaeger and Cook (1979), the McClintock and Walsh (1962) analysis is equivalent to the Coulomb criterion in which the failure envelope has a

slope equal to /z and an intercept Tq equal to the cohesive strength o f the rock".

The region o f failure can then be computed in terms o f a simple Mohr’s circle. Failure will occur where the radius o f the circle (r'), given by:

2

(aff-Cr) ^ - 2

2

[4.7]

is greater than or equal to the distance (d) from the centre o f the circle to the failure line

given by [equation 4.8] (Zoback et aL, 1985).

d -

( 1

+ a_r)

[4.8]

Assuming that the Navier-Coulomb criterion (a^ = Tq - fiOg) applies, the maximum value

o f cohesive strength at which the material will fail is given by equation 4.9.

1 2n2 To - ( 1 + /z^) r ₂ 1 2 r CQ-Or ₂ 2 - / i 2 [4.9]

For most rocks, ji varies between 0.6 and 1.0 (Byerlee, 1978) and Tq can vary from several

megapascals to a few tens o f megapascals. It is possible to substitute appropriate values into the above equations to enable the prediction o f the size o f the initial region in which the ratio o f shear stress to normal stress on the potential shear surfaces is large enough to cause failure.

As illustrated in Figure 4.8a, the modelled shear surfaces are curved and failure was

predicted (Zoback et al., 1985) to occur as crescent shaped zones resembling breakouts

fractures (Figure 4.9) have been documented from boreholes drilled in the tuffs beneath the Nevada Test Site (Bell, 1990).

Two particular aspects of the geometry of the broken out borehole illustrated in Figure 4.8b; the angular extent of the breakout (2<^y), and the maximum depth of the breakout (ry -

R) were noted by Zoback et al. (1985) and it was proposed that a knowledge of ry and

II was sufficient to determine the ratio of the principal stresses acting perpendicular to the

borehole axis (Figure 4.10).

Figure 4.9: A wellbore breakout within a volcanic tuff beneath the Nevada Test Site. The rugged surface of the breakout is consistent with an origin due to spalling of the wall rock following the growth of intersecting fractures (Redrawn from a photograph in Bell, 1990).

The effect of increasing the ratio of the horizontal principal stresses was inferred by Zoback

et al. (1985) to make the modelled breakouts much larger for given values of /x and tq

(Figure 4.11). Similarly, for a given stress ratio and Tq, much smaller breakouts result for

larger values of fi, especially where the stress ratio is large. In general, the breakouts were

modelled to deepen as the stress ratio increases (Zoback et al., 1985).

From the model proposed by Zoback et al. (1985) the region of initial spalling and breakout

formation is seen to be less extensive where there is excess fluid pressure in the borehole (AP is positive). It was proposed that the strong influence of AP on the size and shape of

breakouts was due to the change in normal stress on potential failure planes near the well bore. It was concluded that positive AP increases normal stresses on those planes and inhibits failure, whereas negative AP lowers normal stresses and promotes failure (Figure 4.12).

3-1

^ = 0.6

1.00 1.04 1.08 1.12 1.16

Distance from centre o f borehole (r) Initial Radius (R)

Figure 4.10: The relationship between the ratio of the horizontal principal stresses and the maximum depth and width of breakouts. The curves correspond to breakouts with various

values of the half-width, where ^ = 0.6 and AP = 0. (Adapted from Zoback et al., 1985).

The cross-sectional shape of the breakouts produced by the mathematical model of Zoback

et al. (1985) were shown to be more similar to natural breakouts than the idealized ‘dog

ear’ shaped breakouts suggested by the Gough and Bell analysis.

Oh = 10 MPa Oh = 15 MPa a , = 10 MPa Oh =20 MPa Oh = 10 MPa Oh =30 MPa ft = 0.5 | L l = 1.0

Figure 4.11: The theoretical size of the areas in which the compressive shear strength of the rock is exceeded by the concentrated stresses. For the values of the effective compressive principal stress and the coefficient of friction shown, the contours define the size of the initial failure zone for a given value of the cohesive strength of the rock where AP = 0. (After Zoback gf a/., 1985).

Using an ultrasonic borehole televiewer in geothermal wells in Auburn and the Nevada Test

Site, Zoback et al. (1985) compared the breakout shape predicted by their analysis with the

geometry o f the spalled well bores. The profiles observed ranged from angular, pointed breakouts to deeper, flat-bottomed breakout geometries. However, there is no guarantee that all the damaged material has been removed from the borehole wall and the profiles measured may therefore not have been fully developed.

Figure 4.12: The effect of wellbore fluid pressure AP on the size of breakouts. The contours

define the size of the initial failure zones for Tq = lOMPa, when = 22MPa, = 1 IMPa,

and / 4 = 0.6; (a) No excess wellbore pressure (AP = 0); (b) Excess pressure in the wellbore

of 2.5MPa (AP = 2.5MPa); (c) Wellbore pressure which is 2.5MPa less than the formation

pore pressure (AP = -2.5MPa). (After Zoback et at., 1985).

The application o f a simple elastic failure model to the problem o f breakout growth was,

however, questioned by Zoback et al. (1985) who conceded that, following initial spalling,

subsequent changes in stress concentration around the non-circular hole were important. It was concluded that inelastic deformation and time dependant failure processes such as subcritical crack growth around the wellbore are important in breakout development

(Zoback et al., 1985).

It is not currently known if the process o f breakout formation is the instantaneous response o f the rocks forming the borehole wall to the change stress conditions following drilling and subsequent removal o f the rock or if the process is, in fact time dependant. What has become clear throughout the course o f this study, is that fewer breakouts are detected by

In document Evaluación de la gobernanza de las TICS dentro de la estrategia organizacional utilizando el marco de referencia ISO/IEC 38500 para empresas del sector público Caso de estudio: “Empresa pública metropolitana de transporte de pasajeros de Quito – EPMTPQ – (página 74-77)