Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol.30, No.7, pp. 1103-1109, 1993 (1148-9062/93 $6.00 + 0.00 Printed in Great Britain Pergamon Press Ltd

An Analytical Method for Determining Horizontal Stress Bounds from Wellbore Data C. P. TAN t D. R. WILLOUGHBYt S. ZHOU $ R.R. HILLIS:~

Several issues including control of stress-induced wellbore instability and design of deviated and horizontal wellbores require knowledge of the magnitude and orientation of the in-situ stress field. An analytical method has been developed to determine bounds to the in-situ horizontal stresses from wellbore data for cases where the explicit measurement of stresses is not available. The method has been used to determine the anisotropic stress state at depths between 1500 m and 4000 m in the immediate region of the Cossack and Wanaea fields in the North West Shelf of Australia. The validity of the method is demonstrated by the satisfactory comparison of the determined stress bounds with the minor and major horizontal stresses estimated directly from modified leak-off tests.

INTRODUCTION

The information required for analysis of stress- induced wellbore instability and design of deviated and horizontal wellbores includes the orientation and magnitude of the in-situ stress field. By assuming that the vertical stress is a principal stress, the orientation of the stress field is completely described by the orientation of the horizontal stresses which can be determined from breakout analysis, using four-arm caliper (dipmeter) data [1]. Although direct measurement of the in-situ stress magnitudes is possible using techniques based on hydraulic fracturing [2,3], such information is rarely available, making it necessary to resort to other methods to define the stress state. If such stress measurements are made, analytical methods are still required to provide a horizontal stress profile between the measured results.

Wellbore breakout and hydraulic fracture test data have been widely used by various researchers to constrain in-situ stress magnitudes [4-7]. Zoback et al. [4] proposed an analytical model for predicting the occurrence and shape of wellbore breakouts. Although the model could be used to estimate the principal horizontal stresses, it requires knowledge of breakout width and depth which are generally not available. Even when these data can be obtained (e.g. from a borehole televiewer), large errors could be incurred due to the time-dependent growth of breakout depth. It appears that in order to use breakout shapes as a method for determining horizontal stress magnitudes, it is necessary to observe breakout immediately after breakout occurs. Barton et al. [5] used wellbore breakout occurrence to determine the major horizontal stress bounds but the method requires hydraulic

t Australian Petroleum Cooperative Research Centre, CSIRO

Division of Petroleum Resources, Melbourne, Australia

* Department of Geology and Geophysics, University of Adelaide, Australia

1103

fracture test data to provide the estimated minor horizontal stress. Moos and Zoback [6], using the stress conditions for the occurrence or non-occurrence of breakout, provided a range of solutions for principal horizontal stresses based on the Mohr-Coulomb failure criterion for the stress concentration around a wellbore, combined with a model for frictional strength of crustal rock. Vernik and Zoback's model [7] took into account the polyaxial stresses around the wellbore and the role of pore pressure on rock strength but the model uses minor horizontal stresses determined from hydraulic fracture test data and it is difficult to implement the model computationally. Moreover, all the methods described above are applicable only to wellbores which are orthogonai to the principal in- situ stresses. In most cases this restricts their usage to vertical wellbores.

As part of a study to examine stress-induced wellbore instability in the North West Shelf of Australia, an analytical method has been developed to determine horizontal stress bounds from wellbore data.

DESCRIPTION OF THE METHOD

Horizontal stress bounds may be determined from a combination of standard leak-off test data, information relating to recent fault plane movements in the immediate region and estimation of the coefficients of active (minimum) and passive (maximum) earth pressure. The stress bounds determined are checked for consistency with the occurrence of compressive shear failure and the non- occurrence of hydraulic fracture observed in various wellbores in the immediate region. Based on the assumptions outlined below, the method is applicable to low-permeability, fine-grained material such as shales.

In the determination of the horizontal stress bounds, the leak-off pressure is assumed to have induced a hydraulic fracture. The fracture will occur when the effective tangential stress is less than the tensile strength of the formation (compression is taken as positive):

11()4 ROCK MECHANICS IN THE 1990s

G 0 - u < - IT[ ( 1 )

where o0 is total tangential stress (see Equation 4), u is pore pressure and T is effective tensile strength. The pressure in the wellbore that causes the fracture is assumed to not affect the pore pressure in the rock near the wellbore. This is reasonable for materials with a low permeability such as shales. Despite this restrictive assumption, the above equation has been found to give reasonable results when applied to calculating the major horizontal stresses from breakdown pressures measured by hydraulic fracture tests [3]. For more permeable materials or low-permeability materials fractured with a slow pumping rate, infiltration of drilling fluid from the wellbore into the rock [8] and time-dependency of the effect of the fluid infiltration [9] need to be taken into account.

Information relating to the relative magnitudes of the principal in-situ stress can be developed from consideration of fault plane movement. Since the method is concerned with the current stress state, care needs to be exercised with respect to the timing of fault activity. Given that fault plane movement is controlled by the nature of the in-situ stress field and pre-existing older structures, knowledge of recent movements provides a mechanism to establish the relative magnitudes of the principal stresses. If the vertical stress is a principal stress. then the other two principal stresses will be the major and minor horizontal stresses.

In a field of large lateral extent consolidating under a uniformly increasing vertical stress, a reduction of the effective vertical stress will not result in the same reduction in the effective horizontal stresses. In such cases the ratio of the effective major horizontal stress and effective vertical stress ( ~ /Ov) will increase until it approaches the value of Kp (coefficient of passive earth pressure = [1 + sint~'] / [l - sin00'] where ~' is the effective angle of internal friction). If the effective vertical stress is again increased, there may again be little change in the effective horizontal stresses and the ratio of the effective minor horizontal stress and effective vertical stress ( ~ / c~ ) may be reduced to a value of Ka (coefficient of active earth pressure = [1 - sin¢'] / [1 +sin¢']) . These relationships between the in-situ stresses and Ka and Kp can be expressed as:

~Jh > K a (2) or;

° ' H (3) t ~ < Kp

The Mohr-Coulomb failure criterion, which ignores the effect of the intermediate principal stress, is used to check the occurrence of compressive shear failure. The material is assumed to be reasonably described as linearly elastic and isotropic. There is considerable uncertainty and debate as to the influence of the intermediate principal stress on rock strength [10]. Although hollow cylinder

tests conducted mainly on sandstones [10,1 I] have shown that stress dependent elastic modulus has a significant effect on the stress distribution, it hag been observed through numerical modelling using elastic moduli obtained from consolidated undrained triaxial tests on a shale [12] that linear elastic modulus is a tolerable concept for this fine-grained material. Material anisotropy is only likely to be of great significance in the stability analysis of inclined wellbores in shales due to possible failure along bedding planes [13].

The equation for the total tangential stress at the wall of deviated wellbores under a generalised plane strain condition [13], obtained from the Kirsch equations and the superposition principle, is given by:

(50 =

where A =

B =

C = o(

0

AO'h + BtSH + C~v - Pin (4)

coso~{coscz(1 - 2cos20)cos 2 ~i + 2sin2 ~isin20 } + (1 + 2cos20) sin2[3 cos o~ {costx(l - 2 cos 20) sin2~ - 2 sin2[3 sin20} + (1 + 2cos20) cos213 (1 - 2 cos20) sin2ct is wellbore deviation angle measured from the vertical, is wellbore azimuth relative to the minor horizontal stress; is angle measured anticlockwise from the positive transformed x-axis to the radius of the point on the wellbore wall;

oh is total minor horizontal stress; OH is total major horizontal stress: Ov is total vertical stress; Pm is mud pressure.

The horizontal stress state which will result in hydraulic fracture, obtained from Equations 1 and 4, is given by:

PI + u - I T I - Coy O h ~. (5)

A + B N

where Pl is the leak-off pressure and N is the ratio of major and minor horizontal stresses (OH/Oh).

The horizontal stress bounds are expressed in terms of the minor horizontal stress normalised with respect to the vertical stress (O'h/Cv) and horizontal stress ratio, N. By plotting the lower and upper bounds of N and ola/t~v onto a (C~h/Ov, N) plane, the permissible horizontal stress bounds can be determined. Alternatively, the stress bounds can be determined from the equations for N and Ola/t~v bounds which have been derived for normal, strike- slip and reverse fault plane conditions.

Normal fault application

The stress state for this type of fault plane is given by:

(Sv tJv>t~H>t~ h i.e. ~ > N > I (6)

ROCK MECHANICS IN THE 1990s 1i05

The expression for the lower bound of the horizontal stress ratio is obtained either from the hydraulic fracture condition (Equation 5) and the relationship between the in- situ principal stresses and Ka (Equation 2) or from the fault plane condition (Equation 6):

+u +u

The upper bound of the horizontal stress ratio can be obtained using the fault plane condition (Equation 6) together with the value of minor horizontal stress given by either the relationship between in-situ principal stresses and Ka (Equation 2) or the case of Nl equal to 1:

N u = m i n i _-7¢~v , (A+B)av } (KaOv + u Pz + u -ITI- COy

Similarly, the lower bound of the normalised minor horizontal stress can be derived either from the relationship between in-situ principal stresses and Ka (Equation 2) or from the case of N l equal to 1:

( ~ h ) =maxlKa~.~ +u P/+u-ITI-CC~v /

The upper bound of the normalised minor horizontal stress is determined either from the hydraulic fracture condition (Equation 5) with N set to the upper bound value N u or from the intersection of the hydraulic fracture condition with the fault plane condition (Equation 6) curves:

(°e) u

Pz + u -]TJ- CGv (A + B N u )¢~v

Pl + u - [T[ - er v (B + C)

A¢Y v

if N u is left of the inter- section of the hydraulic fracture condition with the fault plane condition curves

otherwise

Strike-slip fault application

The strike-slip fault plane stress state is given by:

O" V tTH>C~v>Oh i.e. N> ~ > 1 (7)

The expression for the lower bound of the horizontal stress ratio is derived from the intersection of the hydraulic fracture condition (Equation 5) with the normal/strike-slip fault plane boundary (ah/av = l/N) curves:

N / = A~v Pl + u - ITI- Ov (B + C)

With the maximum value of ¢rh being limited by ¢~v (Equation 7), the upper bound of the horizontal stress ratio can be obtained by substituting ¢r h with Cry in Equation 5

which defines the horizontal stress state required for hydraulic fracture to occur:

Nu = ( P I + u-~'Tl-CC~Vc~v - A ) / B

The lower bound of the normalised minor horizontal stress can be determined from the normal/strike-slip fault plane boundary curves with N set to the upper bound value Nu:

l N u

Using the hydraulic fracture condition (Equation 5) with N set to the upper bound value Nu, the upper bound of the normalised minor horizontal stress can be obtained:

u (A+BNu)C~v

Reverse fault application

The stress state for this type of fault plane is given by:

(YV ¢~H>Crh>av i.e. N > I > - - (8)

ah

With the minimum value of ¢~h being limited by ¢~v (Equation 8), the lower bound of the horizontal stress ratio can be obtained from the hydraulic fracture condition (Equation 5) by substituting c~ v for ¢r h in the equation:

N l = / P I +u-lTl-C~Vo:v A / / B

The expression for the upper bound of the horizontal stress ratio is obtained from the hydraulic fracture condition (Equation 5) with ~h set to the limiting maximum value of ¢~H and the relationship between in-situ principal stresses and Kp (Equation 3):

I I N n = P I + u - I T I - C ~ S v - A '/B \ KpO" v + u I

From the fault plane condition (Equation 8), the lower bound of the normalised minor horizontal stress (¢yh/~v) t is equal to 1.

The upper bound of the normalised minor horizontal stress can be determined from the hydraulic fracture condition (Equation 5) with N set to the upper bound value Nu:

u (A+BNu)~v

The stress bounds determined above are checked for consistency with the occurrence of wellbore breakout, the most commonly occurring type of shear failure at the wall

1106 ROCK MECHANICS IN THE 1990s

[14], and the non-occurrence of hydraulic fracture at various locations in vertical wellbores in the immediate region. The occurrence of breakouts can be determined from breakout analyses using four-arm caliper logs [1]. For breakouts the minor and major principal stresses at the wellbore wall, as given by the Kirsch equations, are the radial stress (= mud pressure) and tangential stress respectively. By substituting the principal stresses at the wellbore wall into the failure criterion, the horizontal stress condition which will result in breakout can be obtained:

2(P m - u sin ~' + c ' cos ~') o h >

(1 - sin 0 ' ) (3N - 1)

where c' is effective cohesion. In addition, hydraulic fracture will not occur in

vertical wellbores if (refer to Equation 5):

Pm + u-IT] o h >

3 - N

The type of fault which exists in the immediate region will result in the following relationships (refer to Equations 6, 7 and 8):

Normal fault Strike-slip fault Reverse fault

Ov Ov Oh<-ff --if< oh< ov ~ > O v

By plotting the stress conditions for wellbore

breakout, non-occurrence of hydraulic fracture and the fault plane type representative of the region onto a (Oh, N)

plane, the horizontal stress bounds consistent with field

observations in the region can be defined. If any part of the permissible horizontal stress bounds lie outside the

stress bounds (defined by breakout, non-occurrence of hydraulic fracture and fault plane condition), then

refinement of the permissible stress bounds will be

required. Consistency with field observations can be

achieved by either increasing the lower bound or decreasing the upper bound minor horizontal stress.

The permissible stress bound checks described above for breakout can be similarly conducted for the two less common types of shear failure at the wellbore wall, toric and helical failures [14], by using the corresponding minor and major principal stresses at the wellbore wall which will result in the other types of shear failure.

APPLICATION TO THE C O S S A C K A N D

W A N A E A FIELDS

The method has been used to determine the anisotropic horizontal stress state at depths between 1500 m and 4000 m in the immediate region of the Cossack and Wanaea fields in the North West Shelf of Australia as part of stability analyses of proposed wetlbores [12]. The two fields are within 7 km of one another and, based on local knowledge, normal faults exist in the immediate region.

The standard leak-off test data, vertical stress, pore pressure and material tensile strength of four vertical wellbores (Cossack 1 and Wanaea 1. 2 and 3) are presented in Table 1. Locations in these four wellbores where breakout occurred, together with the mud weight used, other field-obtained stresses and relevant strength properties of the rock material are given in Table 2. The occurrence of breakouts were determined f rom

Table 1. Standard leak-off test and other relevant data used in determining the permissible horizontal stress bounds

Wellbore Depth Leak-off Vertical Pore Tensile Pressure Stress Pressure Strength

(mBRT) (MPa) (MP_a) (MPa) (MPa) Cossack 1 1862 29.0 38.9 18.8 1.0

2826 44.6 59.2 29.0 3.7 Wanaea 1 3361 58.4 71.0 44.0 3.7 Wanaea 2 1950 29.7 40.5 21 7 2.4

2468 37.3 51.8 32.5 1.5 Wanaea 3 1993 33.2 41.3 25.4 0.9

2615 41.6 54.5 29.9 3.7

Table 2. Selected wellbore breakout locations and other relevant data used to check the permissible stress bounds

Wellbore Depth Mud Vertical Pore Effective Effective Tensile Pressure Stress Pressure Cohesion Angle of Strength

Internal Friction

(mBRT) (MPa) (MPa) (MPa) (MPa) (°) (MPa)

Cossack 1 2860 30.3 59.8 28.9 12.7 31.0 3.7 2875 30.5 60.2 29.0 12.7 31.0 3.7

Wanaea 1 2218 28.1 46.3 22.4 5.3 31.0 1.6 2815 31.5 58.9 30.0 12.7 31.0 3.7 3075 34.4 64.9 33.7 12.7 31.0 3.7 3210 35.9 68.4 35.1 8.2 31.0 2.4

Wanaea 2 2910 30.0 60.9 29.4 12.7 31.0 3.7 2985 31.6 62.7 32.2 t2.7 31.0 3.7

Wanaea 3 2142 29.8 44.6 21.6 3.0 31.0 0.9

ROCK MECHANICS IN THE 1990s 1107

breakout analyses using four-arm caliper logs [1]. In the tables, the vertical stress was determined from check shot and corrected density logs; the pore pressure is the average value based on shale resistivity and shale transit time measured in-situ. Since measured values relating to the strength properties of the materials in the Cossack and Wanaea fields are generally not available, estimates were obtained from an unconfined compressive strength (UCS)- smectite content correlation and triaxial testing on material from Wanaea 3 [12].

The unconfined compressive strength-smectite content correlation is based on data for three typical shales tested in a saturated state [15]. Consolidated undrained triaxial tests have been carried out on Wanaea 3 shale core and the effective angle of internal friction of 31 ° determined from the tests is assumed for the shale materials in the Cossack and Wanaea fields. Based on the extended Griffith criterion, the effective tensile strength (T) is estimated as T = UCS/12. Although an unconfined compressive strength test is normally conducted at a relatively fast rate, samples are usually partially saturated which will result in a "drained" condition. The effective cohesion (c') is derived from the Mohr circle relationship between unconfined compressive strength and effective angle of internal friction (~)') viz. c' = UCS(1 - sin~')/2cos~'.

1.4

1.3-

1 2 "

1.1

.¢

t~ 0 . 9

0.8

0.7

0.6

0.5

COSSACK 1 (1862 mBR~

1.4

1.3

1,2

1.1 >

ff 0 .9

0 .8

0 . 7

0 . 6 -

0 . 5 ,

The calculated permissible horizontal stress ratio bounds and bounds of the normalised minor horizontal stress determined for the various locations are given in Table 3. Based on the stress measurements by Rummel [16] and Stephansson et al. [17] which show that N and O'h/Gv are nearly constant below 2 kin, it is assumed that the determined horizontal stress ratio (N) bounds are constant with depth (1.23 - 1.34).

Typical ((;h/Ov, N) plots for the four wellbores are shown in Figure 1. Examination of the figures will show that it is not feasible to assume constant normalised horizontal stress bounds (~h/Ov) with depth. Hence a variable Oh/~v bound profile defined by the values in

Table 3. Horizontal stress ratio and normalised horizontal stress bounds determined from standard leak-off test data

Wellbore Depth N l N u ( (rh/Crv) l ( ~rlr/~V) u (roBe7)

Cossack 1 1862 1.14 1.54 0.65 0.73 Wanaea2 1950 1.23 1.46 0.69 0.73 Wanaea 3 1993 1.11 1.35 0.74 0.80 Wanaea 2 2468 1.23 1.34 0.75 0.78 Wanaea 3 2615 1.21 1.44 0,69 0.75 Cossack 1 2826 1.19 1.53 0.65 0.71 Wanaea 1 3361 1.13 1.35 0.74 0.80

- o /

- E ' f f 4

1.2

" , y

le honzontal stresses

~ , . (cE1%) \

, , , , , ~ ~ " ~ " ~ ' ~

1.4 1.6 1.8

N = GHIG h

W A N A E A 3 ( 2 6 1 5 m B R T ) s

v , ,

, /

- J /

m. 9 /

i | 1.2

\

m~p.L/ . , ..

x

al stresses

• " ~ ' ~ , . ~ ~ " Lower bound

1 . 4 1 . 6 1 . 8

N = OHIO" h

1.4

1.3

1,2

1.1

0.8

0.7

0.6

0.5

z o=

% c 0 ~'//..

f

/ / / /

i i 1.2

W A N A E A 2 (2468 m B R T ) \

\ \

h o n z o n t a l slress~s

1.4 1.6 1 8 2

N = GH/G h

1.4

1.3

1.2

1.1

:) .9

3.8

~.7

?.6

).5

3 / 3, '

eo a ~%, /

" i i 1.2

WANAEA 1 (3361 mBRT)

~ ' i Permissible horizontal stresses

: " ~ ; , L o w . r ~ ° ~ ~. - - , 4 < ~ , , . (o~/o)

! i ! | i 1.4 1.6 1,8

N -- O"H/O" h

Fig. I. Permissible horizontal stress bounds determined from standard leak-off test data

1108 ROCK MECHANICS IN THE 19905

Table 3 is adopted. The range of the lower and upper bound oh/•v values over the depth interval, however, are fairly small, 0.65 to 0.75 for (Oh/Gv)l and 0.71 to 0.80 for (6h/Gv)u. Values of oh/Gv at locations between the depths in the table can be obtained by linear interpolation.

The permissible horizontal stress bounds determined above are checked for consistency with field-observed wellbore breakout and the non-occurrence of hydraulic fracture in the four wellbores. Reference to typical plots presented in Figure 2 will show that the permissible horizontal stress bounds determined from leak-off test data are within the stress bounds based on wellbore breakout and non-occurrence of hydraulic fracture. No refinement of the permissible stress bounds is required and the stress bounds represent the limits of the in-situ stresses.

The stress bounds determined are compared with the horizontal stresses estimated directly from four modified leak-off tests, based on hydraulic fracture procedures, conducted recendy in the region. In the modified leak-off tests, two leak-off cycles were conducted for each test and the pressure decrease after the first shut-in is measured over an extended period of time. The pump pressure-time

WANAEA 1 (2218 mBRT)

l llillboi'e breakout, non-occurrence of E 50 I hy~raullcfracture and normal fault condition

" ~ Permlss~le horizontal s~,es.s cleter mined S

" 1 ~o.I "

N

I¢ 10-

,:2 ' ; ' ' ,:6 ,:8

N = GH/G h

WANAEA 2 (2985 mBRT) 70 I Horizontal Itmmee c~nilt~t with

I . wellborn breakout, non-ocoJrrence of 60 ~ hydraullo fracture and normal fault oondition

|~.~."~.~s,~ Permissible horizontal stresees determined Z

I - > , j , ~ ~ . . ~ . , . . ¢ ~ - ~6,, , ' -

.um01iion

~ 20 . . . .

Ig

.R

N = aH/G h

Fig. 2. Permissible horizontal stress bounds checked for consistency with the occurrence of breakout and

the non-occurrence of hydraulic fracture

2

curve of the modified leak-off test conducted in Cossack 3, which is typical for the four tests, is shown in Figure 3. The minor horizontal stress is approximated by the first shut-in pressure. An estimate of the major horizontal stress is made using the minor horizontal stress value, the crack re-opening pressure and a range of pore pressures determined from shale resistivity and shale transit time, based on the elastic theory of stress conditions for hydraulic fracture [12] with the tensile strength set equal to zero.

The determined stress bounds compare satisfactorily with both the minor and major horizontal stresses estimated directly from the four modified leak-off tests as shown in Figure 4.

10 -, ,

~ ' 4

'i'i 2 .................. i

0 i l 0 600 1200

Pump [ S~ut4n I pump

.~- . . . . . . . . . . . . . . . . 4

i i ......... , i , i ,

1800 2400 3000 3600 T i m e (s)

Fig. 3. Modified leak-off test results for Cossack 3 at 1931 mBRT

CONCLUSIONS

An analytical method has been presented which allows in-situ horizontal stress bounds to be determined from wellbore data. The determination of the stress bounds is applicable to inclined wellbores• The application of the method has been shown through the determination of the anisotropic stress state at depths between 1500 m and 4000 m in the immediate region of the Cossack and Wanaea fields. The validity of the method is demonstrated by the satisfactory comparison of the determined permissible horizontal stress bounds with the minor and major horizontal stresses estimated directly from four modified leak-off tests conducted recently in the region.

The analytical method provides an alternative approach which enables the in-situ horizontal stress bounds to be determined from wellbore data and is applicable to other areas where similar data is available. The stresses can only otherwise be measured downhole at high cost and with a risk of damaging the wellbore wall. For those wellbores in which stress measurements are conducted, the analytical method may be used to determine the stress bounds between the depths with measured results•

ROCK MECHANICS IN THE 1990s 1109

1400

m

£3

1800

2200

2600

3000

3400

3800

4200

- I "'-/A:'~ , ; ~ . . ~/./.2>':,- Cossack 3

Cossack 2 ". ~'/, " .'.(/~.>'~ - <~" <././>.

/, '..

:'a;o°:=:o°= =:::°7: " Minor horizontal stress (modified leak-off tests) " z/¢///~.~..// Major horizontal stress (modified leak-off tests) "" ~ . / / ) - . .

i i

I I I I I

20 30 40 50 60 70

Horizontal Stresses (MPa)

I I

80 90

Fig. 4. Comparison of the permissible horizontal stress profile determined for the Wanaea and Cossack fields with the minor and major horizontal stresses estimated directly from modified leak-off tests

I O0

R E F E R E N C E S

1. Hillis R. and Williams A.F. The stress field of the North West Shelf and wellbore stability, APEA J., 33, 373-385

(1993). 2. Kunze K.R. and Steiger R.P. Extended leakoff tests to

measure in-situ stress during drilling. Prec. 32nd U.S.

Symp. Rock Mechanics, 35-44 (1991). 3. Jeffrey R.G. Hydraulic fracture stress testing. APCRC

Unrestricted Report No.O03, CSIRO Div. of

Geomechanics, Australia (1992). 4. Zoback M.D., Moos D., Mastin L. and Anderson R.N.

Wellbore breakouts and in situ stress, J. Geophys. Res., 90, 5523-5530 (1985).

5. Barton C.A., Zoback M.D. and Burns K. L. In situ stress orientation and magnitude at the Fenton geothermal site, New Mexico, determined from wellbore breakouts, Geophys, Res. Lett., 15,467-470 (1988).

6. Moos D. and Zoback M.D. Utilization of observations of well bore failure to constrain the orientation and magnitude of crustal stresses: application to continental, deep sea drilling project, and ocean drilling program boreholes, J. Geophys. Res., 95, 9305-9325 (1990).

7. Vernik L. and Zoback M.D. Estimation of maximum horizontal principal stress magnitude from stress-induced wellbore breakouts in the Cajon Pass scientific research borehole, J. Geophys. Res., 97, 5109-5119 (1992).

8. Detournay E. and Cheng A. Influence of pressurization rate on the magnitude of the breakdown pressure, Prec. 33rd U.S. Symp. Rock Mechanics, 325-333 (1992).

9. Haimson B.C. and Zhao Z. Effect of borehole size and pressurization rate on hydraulic fracturing breakdown pressure, Prec. 32nd U.S. Syrup. Rock Mechanics, 191-i99 (1991).

10. McLean M.R. and Addis M.A. Wellbore stability analysis: A review of current methods of analysis and their field application. IADC/SPE Drilling Conf., Houston, Texas, 261-274 (1990).

11. Santarelli F.J. and Brown E.T. Performance of deep wellbores in rocks with a confining-pressure dependent elastic modulus. Prec. 6th Int. Cong. Rock Mechanics,

Montreal, 1217-1222 (1987). 12. Kingsborough R.H., Williams A.F. and Hillis R.R.

Borehole instability on the North West Shelf of Australia. Prec. SPE Asia Pacific Conf., Perth, Western Australia, 653-662 (1991).

13. Aadnoy B.S. and Chenevert M.E. Stability of highly inclined boreholes. SPE Drilling Engg. (Dec.), 364-374 (1987).

14. Guenot A. Borehole breakouts and stress fields, Int. J.

Rock Mech. Min. Sci., 26, 185-195 (1989). 15. Steiger R.P. and Leung P.K. Predictions of wellbore

stability in shale formations at great depth. Rock at Great Depth (Vol. 3), Elf Aquitaine, Pau, 1209-1218 (1989).

16. Rummel F. Stresses and tectonics of upper continental crust - a review, Prec. Int. Syrup. Rock Stress and Rock Stress Measurements, O. Stephansson (ed.), Lulea, Centek Publ., 177-186 (1986).

17. Stephansson O., Savilahti T. and Bjarnason B. Rock mechanics of the deep borehole at Gravberg, Sweden, Rock at Great Depth, Maury & Fourmaintraux (eds.), 863-870 (1989).