The stress concentration around a vertical well drilled parallel to the vertical principal stress, Sv, in an isotropic, elastic medium is described by the Kirsch equations (Kirsch 1898); see also Jaeger and Cook (1979). As illustrated in Figure 6.1 (taken from Kirsch’s original paper), the creation of a cylindrical opening (like a wellbore) causes the stress
trajectories to bend in such a way as to be parallel and perpendicular to the wellbore wall because it is a free surface which cannot sustain shear traction. Moreover, as the material removed is no longer available to support far-field stresses, there is a stress concentration around the well. This is illustrated by the bunching up of stress trajectories at the azimuth of Shmin, which indicates strongly amplified compressive stress. In contrast, the spreading out of stress trajectories at the azimuth of SHmax
indicates a decrease in compressive stress.
Mathematically, the effective stresses around a vertical wellbore of radius R are described in terms of a cylindrical coordinate system by the following:
σrr = 1
2(SHmax+Shmin−2P0)
1− R2 r2
+ 1
2(SHmax−Shmin)
×
1− 4R2 r2 + 3R4
r4
cos 2θ+ P0R2
r2 (6.1)
σθθ = 1
2(SH max+Shmin−2P0)
1+ R2 r2
−1
2(SH max−Shmin)
×
1+3R4 r4
cos 2θ− P0R2
r2 −σT (6.2)
τrθ = 1
2(SH max−Shmin)
1+ 2R2 r2 −3R4
r4
sin 2θ (6.3)
whereθis measured from the azimuth of SHmaxandP is the difference between the mud weight in the wellbore and the pore pressure, P0.σTrepresents thermal stresses arising from the difference between the mud temperature and formation temperature (T). This will be ignored for the moment but is considered below. It can be shown that for any reasonable amount of elastic anisotropy, the stress concentration around a vertical well is not changed in any significant way (Lekhnitskii 1981). Hence, while anisotropic rock strength induced by weak bedding planes can have an important effect on wellbore failure (as described below), elastic anisotropy generally does not.
There are several important points about these equations that are illustrated in Figure 6.2 for the following parameters:
r SHmax=90 MPa
r SHmaxorientation is N90◦E (east–west) r Sv=88.2 MPa (depth 3213m)
r Shmin=51.5 MPa r Pp=Pmud=31.5 MPa
First, the stress concentration varies strongly as a function of position around the wellbore and distance from the wellbore wall. Also, the stress concentration is sym- metric with respect to the direction of the horizontal principal stresses. For an east–west direction of SHmax, Figure 6.2a shows thatσθθ (the so-called effective hoop stress) is
a .
100
50 150
R
c .
1 1.1 1.2 1.3 1.4 1.5
20 40 60 80 100 120 140
58.5 MPa
Normalized radial distance
1 1.1 1.2 1.3 1.4 1.5
20 40 60 80 100 120
20 MPa
Normalized radial distance
SHmax
0
SHmax
Shmin Shmin
sθθ (MPa)
sθθ (MPa)sθθ (MPa)
Figure 6.2. (a) Variation of effective hoop stress,σθθaround a vertical well of radius R subject to an east–west acting SHmax. Note thatσθθvaries strongly with both position around the wellbore and distance from the wellbore wall. Values of stress and pore pressure used for the calculations are described in the text. (b) Variation ofσθθwith normalized distance, r/R, from the wellbore wall at the point of maximum horizontal compression around the wellbore (i.e. at the azimuth of Shmin). At the wellbore wall,σθθis strongly amplified above the values of SHmaxand Shminin accordance with equation (6.2). At r/R=1.5, the hoop stress is approximately 30% greater than the effective far-field stressσHmaxthat would be present at that position in the absence of the well. (c) Variation ofσθθwith normalized distance, r/R, from the wellbore wall at the azimuth of SHmax, the point of minimum horizontal compression around the wellbore. At the wellbore wall,σθθis close to zero. At r/R=1.5, the hoop stress is slightly greater than the effective far-field stressσhminthat would be present at that position in the absence of the well.
strongly compressive to the north and south, the azimuth of Shmin, or 90◦ from the direction of SHmax.σθθ decreases rapidly with distance from the wellbore wall at the azimuth of Shmin(Figure 6.2c) as given by equation (6.2). Note that at a radial distance equivalent to∼1.5 wellbore radii, the value ofσθθis about 50% greater than the far-field value ofσHmax(58.5 MPa), whereas it is almost three times this value at the wellbore wall.
In marked contrast, at the azimuth of SHmax the hoop stress is only slightly above zero because of the relatively large difference between SHmaxand Shmin(equation 6.2).
Under such circumstances, the wellbore wall can go into tension which would lead to the formation of drilling-induced tensile wall fractures (Aadnoy 1990; Moos and Zoback 1990) because the tensile strength of rock is so low (Chapter 4). At the azimuth of SHmax, the hoop stress increases rapidly with distance from the wellbore wall. Note in Figure 6.2b that at r=1.5R, the value ofσθθis slightly greater than the far-field value of σhmin which would be equivalent to Shmin−Pp=20 MPa (≡51.5−31.5 MPa).
Hence, drilling-induced tensile wall fractures are restricted to being extremely close (∼several mm to cm) to the wellbore (Brudy and Zoback 1999), unless the pressure in the wellbore is sufficient to extend the fracture away from the wellbore as a hydrofrac (see below).
Note that the stress components described in equations (6.1)–(6.3) are independent of elastic moduli. For this reason, the manner in which stresses are concentrated does not vary from formation to formation. Moreover, the stress concentration around a wellbore is independent of R, the wellbore radius.
Because stresses are most highly concentrated at the wellbore wall, if either com- pressive or tensile failure is going to occur, it will initiate there. Figure 6.3a shows the variation ofσθθ,σzzandσrrat the wellbore wall for the same far field stresses used in Figure 6.2. Note the extremely large variations inσθθwith position around the well.σzz
varies in a similar manner but the variations are much more subdued. The average value ofσzzis the same as the far-field vertical effective stress of 56.7 MPa (88.2−31.5 MPa).
In Figures 6.2a and 6.3a it is obvious that compressive failure of the wellbore wall is most likely to occur in the area of maximum compressive hoop stress (at the azimuth of Shmin) if the stress concentration exceeds the rock strength (Bell and Gough 1979;
Zoback, Moos et al. 1985). The zone of compressive failure around the well is shown in Figure 6.3c assuming a Mohr–Coulomb failure criterion and C0=45 MPa,µi=1.0.
The stress concentration exceeds the rock strength everywhere within the contour lines shown in Figure 6.3c on opposite sides of the hole. The breakouts have a finite width, wBO, the span of failed rock around the wellbore wall on one side, and initial depth, both of which depend on rock strength for a given stress state (Zoback, Moos et al.
1985). The colors in Figure 6.3c indicate the value of rock strength required to pre- vent failure. Hence, hot colors means it takes high strength to prevent failure (because the stress concentration is high) whereas cold colors mean even a low-strength rock will not fail (because the stress concentration is low). The contour line describes the
−20 0 20 40 60 80 100 120 140 160 0
20 40 60 80
Mohr diagram at 0o, 180o
Effective stress (MPa)
Shear stress (MPa)
b.
South
0 50 100 150
Required C0 a .
Stress at wellbore wall (MPa)
0 90 180 270 360
0 20 40 60 80 100 120 140 160
Angle around the hole (from south)
c .
WB O
sqq szz
srr
sqq
szz
srr
Figure 6.3. (a) Variation of effective principal stresses,σθθ,σrrandσzzaround a vertical wellbore as a function of azimuth. The far-field values of stress and pore pressure are the same as used for the calculations shown in Figure6.2. As discussed in the text, the variation ofσθθaround the wellbore is four times the difference between SHmaxand Shminin the far field (equation6.9). As the mud weight is assumed to equal the pore pressureσrr=0.σzzvaries around the well in the same manner asσθθbut without the extreme variation of values. (b) The three principal stresses at the wellbore wall at the point of maximum stress concentration (θ=0, 180◦) shown as a
three-dimensional Mohr diagram. Note that the strength of the rock is exceeded (a Mohr–Coulomb failure criterion is assumed, C0=45 MPa,µi=1.0) such that the rock on the wellbore wall is expected to fail. (c) The zone of compressive failure around the wellbore wall for the assumed rock strength is indicated by the contour line. This is the expected zone of initial breakout formation with a width given by wBO. Between the contour line and the wellbore wall, failure of even stronger rocks would have been expected (the scale indicates the magnitude of rock strength required to inhibit failure). Lower rock strength would result in a larger failure zone.
boundary between the zones where the stress concentration exceeds the strength (as defined above) or does not.
To better visualize why breakouts and tensile fractures around a wellbore are such good indicators of far-field stress directions, let us first simplify equations (6.1)–(6.3) for the stresses acting right at the wellbore wall by substituting r=R. In this case, the effective hoop stress and radial stress at the wellbore wall are given by the following equation:
σθθ =Shmin+SHmax−2(SHmax−Shmin) cos 2θ−2P0−P−σT (6.4)
σrr =P (6.5)
whereP is the difference between the wellbore pressure (mud weight, Pm) and the pore pressure. The effective stress acting parallel to the wellbore axis is:
σzz =Sv−2ν(SHmax−Shmin) cos 2θ−P0−σT (6.6) whereνis Poisson’s ratio. At the point of minimum compression around the wellbore (i.e. parallel to Shmin) atθ =0◦, 180◦, equation (6.4) reduces to
σθθmin=3Shmin−SHmax−2P0−P−σT (6.7)
whereas at the point of maximum stress concentration around the wellbore (i.e. parallel to SHmax) atθ =90◦, 270◦,
σθθmax=3SHmax−Shmin−2P0−P−σT (6.8)
such that the difference between the two is
σθθmax−σθθmin=4 (SHmax−Shmin) (6.9)
which corresponds to the amplitude of the sinusoidal variation of hoop stress around the wellbore shown in Figure 6.3a and helps explain why observations of wellbore failures so effectively indicate far-field stress directions. Fundamentally, the variation of stress around the wellbore wall amplifies the far-field stress concentration by a factor of 4.
Introduction to breakouts
To understand the zone of compressive failure (breakouts) that results from the stress concentration around the wellbore (Figure 6.2), one simply has to consider the fact that like in a rock mechanics experiment, the rock surrounding the wellbore is subject to three principal stresses defined by equations (6.4)–(6.6). If these stresses exceed the rock strength, the rock will fail. The stress state at the wellbore wall at the azimuth of Shmin(where the stress concentration is most compressive), is shown in Figure 6.3b using a three-dimensional Mohr diagram. This can then be compared to a failure law defining the strength of the rock. For this example, a Mohr–Coulomb failure law was used but, of course, any of the failure laws discussed in Chapter 4 could have been
considered. In the general region of the maximum stress concentration around the well (θ =0◦, 180◦), wherever the stress concentration exceeds the strength of the rock, failure is expected. Thus, the zone of compressive failure (initial breakout formation) within the contour line in Figure 6.3c indicates the region of initial breakout formation using the strength of materials concept first introduced in Chapter 3. The growth of breakouts after their initial formation is discussed later in this chapter.
The most reliable way to observe wellbore breakouts is through the use of ultrasonic image logs that were described in Chapter 5. As shown in Figure 6.4a, a standard unwrapped televiewer images breakouts as dark bands of low reflectance on opposite sides of the well. Interactive digital processing allows cross-sections of a well (such as that shown in Figure 6.4c) to be easily displayed (Barton, Tessler et al. 1991), which makes it straightforward to determine both the orientation and opening angle, wBO, of the breakouts. Breakouts form symmetrically on both sides of the well, but during routine data analysis, the orientations of the breakouts are documented independently (e.g. Shamir and Zoback 1992). The two out-of-focus zones on opposites sides of the well in the electrical image shown in Figure 6.4b also correspond to breakouts.
These result from poor contact between the wellbore wall and the pad upon which the electrode array is mounted. At any given depth, the azimuth of maximum horizontal stress is 90◦ from the mean of the azimuths of the breakouts on either side of the well. As illustrated below, comprehensive analysis of breakouts in wellbores can yield thousands of observations, thus enabling one to make profiles of stress orientation (and sometimes magnitude) along the length of a well.
It is easily seen in the equations above that if we raise mud weight,σθθ decreases (andσrrincreases) at all positions around the wellbore. This is shown in Figure 6.5a for P=10 MPa (compared to Figure 6.3a). As a point of reference, at a depth of 3213 m, this is equivalent to about a 10% increase in excess of hydrostatic pressure. Two phe- nomena are important to note. First, with respect to compressive failures, by increasing the mud weight, the zone of failure is much smaller in terms of both wBOand breakout depth (the dashed lines indicate wBOin Figure 6.3). This is shown in Figure 6.5b which was calculated with exactly the same stresses and rock strength as Figure 6.3c, except for the change in Pm. This is because asP increases,σθθdecreases andσrrincreases such that the size of the Mohr circle (Figure 6.3b) decreases markedly in the area of the wellbore wall subjected to most compressive stress. This demonstrates why increasing mud weight can be used to stabilize wellbores, a subject to be considered at length in Chapter 10.
Introduction to drilling-induced tensile fractures
The second point to note about wellbore failure is that as P increases and σθθ
decreases, the wellbore wall can locally go into tension atθ=90◦, 270◦and contribute to the occurrence of drilling-induced tensile fractures. This is illustrated in Figure 6.5a.
WBO
270 90
180 c. 0
Figure 6.4. (a) Wellbore breakouts appear in an ultrasonic borehole televiewer image as dark bands on either side of a well because of the low-amplitude ultrasonic reflections off the wellbore wall.
(b) Breakouts appear as out-of-focus areas in electrical image data because of the poor contact of the electrode arrays on the pads of the tool where breakouts are present. (c) A cross-sectional view of a well with breakouts can be easily made with televiewer data making determination of the azimuth of the breakouts and wBOstraightforward. Note the drilling-induced tensile fracture in the left image located 90◦from the azimuth of the breakouts, just as expected from the Kirsch equations. From Zoback, Barton et al. (2003). Reprinted with permission of Elsevier.
0 90 180 270 360 0
50 100 150
Angle around the hole (from south)
Stress at wellbore wall (MPa)
50 Required C0 0
WBO a .
b.
sqq
szz
srr
Figure 6.5. (a) Stress concentration at the wellbore wall and (b) zone of compressive failure around the wellbore (similar to Figure6.3) when the mud weight has been raised 10 MPa above the mudweight. Figure6.3b compares the width of breakouts for the two cases. Note that raising the mud weight decreases the size of the breakouts considerably. The area in white shows the region where tensile stresses exist at the wellbore wall.
In Figure 6.5b, the area around the wellbore wall in which tensional stresses exist is at the azimuth of the maximum horizontal stress and is shown in white. As noted above, under normal circumstances, drilling induced tensile fractures are not expected to propagate more than a cm from the wellbore wall. Thus, the formation of drilling- induced tensile fractures will not lead to a hydraulic fracture propagating away from the wellbore (which could cause lost circulation) unless the mud weight exceeds the least principal stress. In the case of deviated wells, this is somewhat more complicated and is discussed in more detail in Chapter 8.
Because drilling-induced tensile fractures do not propagate any significant distance away from the wellbore wall (and thus have no appreciable effect on drilling), wellbore image logs are essentially the only way to know if drilling-induced tensile fractures are present in a well. This can be seen quite clearly in the two examples of electrical image logs in Figure 6.6 (from Zoback, Barton et al. 2003). As predicted by the simple theory discussed above, the fractures form on opposite sides of the wellbore wall (at the azimuth of SHmax, 90◦ from the position of breakouts) and propagate along the axis of the wellbore. As discussed in Chapter 8, the occurrence of axial drilling-induced tensile fractures is evidence that one principal stress is parallel to the axis of the wellbore. Note that in the televiewer image shown in Figure 6.4, there are tensile fractures on opposite sides of the wellbore wall that are 90◦from the midpoints of the wellbore breakouts. In other words, this well was failing simultaneously in compression and tension as it was being drilled. However, because the tensile fractures do not affect the drilling process, and because the breakouts were not excessively large (see Chapter 10) there were no problems with wellbore stability. In Figure 6.6c. the orientations of SHmaxdetermined from breakouts and drilling-induced tensile fractures in a section of a well are shown (the orientation of breakouts were rotated 90◦ as they form at the azimuth of Shmin).
Note that the breakouts and tensile fractures form 180◦apart, on opposite sides of the well and the breakouts and tensile fractures form 90◦apart, exactly as predicted on the basis of the simple theory described above.
To illustrate how robust drilling-induced tensile fractures are as stress indicators, a stress map of the Visund field in the northern North Sea is shown in Figure 6.7 (after Wiprut and Zoback 2000). In the Visund field, an extremely uniform stress field is observed as a function of both depth and position in the oil field. Drilling-induced tensile fractures were observed in five vertical wells (A−E). The depth intervals logged are shown in white in the lower right corner of the figure and the intervals over which the tensile fractures were observed is shown by the black lines. The rose diagrams show the orientation and standard deviation of the drilling-induced tensile fractures observed in wells A–E as well as a compilation of the 1261 observations made in all of the wells.
Note that numerous observations in each well indicate very uniform stress with depth (standard deviations of only∼10◦). As these observations come from depths ranging between 2500 m and 5200 m and from wells separated by up to 20 km, a spatially uniform stress field is observed.
0 60 120 180 240 300 360 5400
5600
5800
6000
6200
6400
6600
6800
7000
7200
SHmax (TC) SHmax (BO)
SHmax
Depth [feet]
c.
azimuth
Figure 6.6. (a) and (b) Electrical image logs showing drilling-induced tensile fractures that are on opposite sides of the wellbore (in the direction of maximum horizontal stress) and parallel to the axes of these two vertical wells (indicating that Svis a principal stress). (c) Example of a well in which the same stress orientation is indicated by breakouts and tensile fractures. From Zoback, Barton et al. (2003). Reprinted with permission of Elsevier.
62oN
6oE 5oE 4oE 3oE 2oE 61oN
4 3 2
1 5
0
Km
N O RWAY
B e r g e n
N
A
E D
C B
A - C E N T R A L FAU LT
H I G H R E F L E C T I V I T Y D U E TO G A S I N T H E B R E N T R E S E RVO I R
R E D U C E D R E F L E C T I V I T Y D U E TO G A S L E A K AG E
SHmax
SHmax
Well D C B
A E
D C
B A
E a.
b.
SHmax = 101°± 10°
SHmax = 107°± 11°
SHmax = 97°± 11°
SHmax = 102°± 10°
SHmax = 97°± 9°
SHmax = 100°± 10° n = 254
n = 221
n = 90 n = 1261
n = 618 n = 78
Data from all wells
Figure 6.7. Drilling-induced tensile fractures in five wells in the Visund field in the northern North Sea indicate a remarkably uniform stress field both spatially and with depth (after Wiprut, Zoback et al.2000). The rose diagrams illustrate how uniform the tensile fracture orientations are with depth in each well and the field as a whole. The length of each well logged with an electrical imaging device is shown in white in the diagram in the lower right. The drilling-induced tensile fractures are shown by the vertical black lines.