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Geothermics 39 (2010) 250–259

Contents lists available at ScienceDirect

Geothermics

journa l homepage: www.e lsev ier .com/ locate /geothermics

oro-thermoelastic borehole stress analysis for determination ofhe in situ stress and rock strength

ingfeng Tao, Ahmad Ghassemi ∗

exas A&M University, College Station, TX 77843, USA

r t i c l e i n f o

rticle history:eceived 29 May 2008ccepted 8 June 2010vailable online 16 July 2010

eywords:

a b s t r a c t

The in situ stress state and rock strength are key parameters in a number of problems concerningpetroleum and geothermal reservoir development, particularly in well stimulation and optimum well-bore trajectory analyses. Inversion techniques utilized to determine the in situ stress and rock strengthbased on the observation of borehole failure and its analysis often assume elastic rock behavior. However,when drilling through high-pressure and high-temperature rocks, coupled poro-thermo-mechanical pro-cesses result in a time-dependent stress and pore pressure distribution around the borehole. In this work,

eothermalorehole breakout

n situ stressock strengthoroelasticityhermoelasticityhear failureellbore stability

the poro-thermoelastic effects on borehole failure are studied and their impact on wellbore stability andthe estimations of the in situ maximum horizontal stress and rock strength using wellbore failure data areinvestigated. It is shown that coupled poro-thermo-mechanical effects influence both failure mode andpotential. Also, when considering shear failure, neglecting heating and cooling effects will underestimateand overestimate rock strength, respectively. Therefore, for accurate assessment of wellbore stability andinversion of wellbore failure data, poroelastic and thermal factors should be considered.

. Introduction

The in situ stress state is an important parameter in reservoirevelopment. It is usually determined using a number of methodsEvans et al., 1999). The vertical stress (�v) can be determined usingensity measurements, and the minimum horizontal stress (�h) cane estimated using a leak-off test or by hydraulic fracturing. How-ver, there is no direct way to accurately determine the magnitudef the maximum horizontal stress (�H) and the in situ rock strength.he fact that the borehole failure occurrence depends on the initu rock stress and strength, and its location is governed by then situ stress and borehole orientation has made it possible to usereakouts analysis as a tool to constrain the horizontal in situ stressagnitudes (Bell and Gough, 1979; Zoback et al., 1985; Zoback andealy, 1992; Brudy et al., 1997) and the in situ rock strength (Peskand Zoback, 1995). Also, Djurhuus and Aadnoy (2003) developedn analytical method to determine the in situ stress direction fromorehole image logs; however this method requires the knowledge

f the magnitude of the in situ stress. Qian and Pedersen (1991)roposed a numerical inversion method for estimating the in situtress state according to breakout data for inclined wells.

∗ Corresponding author. Tel.: +1 979 845 2206; fax: +1 979 862 6579.E-mail address: ahmad.ghassemi@pe.tamu.edu (A. Ghassemi).

375-6505/$ – see front matter © 2010 Elsevier Ltd. All rights reserved.oi:10.1016/j.geothermics.2010.06.004

© 2010 Elsevier Ltd. All rights reserved.

Most wellbore stability analyses and breakout inversion tech-niques use an elastic stress analysis. An elastic model does notincorporate the coupled thermal and poro-mechanical processesthat play an important role in stability of boreholes in geother-mal reservoirs or high-temperature petroleum bearing formations.When rocks are heated or cooled, the bulk solid and the porefluid undergo a volume change. A volumetric expansion can resultin significant pressurization of the pore fluid depending on thedegree of containment and the thermal and hydraulic proper-ties of the fluid as well as the solid. When heated, water trappedin the pores may undergo pressure increases on the order of1.5 MPa/◦C for conditions typical of earth’s upper crust (Williamsand McBirney, 1979). The net effect is a coupling of thermal andporomechanical processes, which occur on various time scales andthe significance of their interaction depends on the problem ofinterest. For example, when drilling wells in high-pressure andhigh-temperature (HPHT) formations, strong coupling betweenthermal and poromechanical effects might develop that can sig-nificantly impact the stress/pore pressure distribution around thewellbore (Li et al., 1998) and thus, borehole failure and fractureinitiation. This is caused by the contrast in thermal and hydraulic

diffusivities of the rock and also because drilling through low-permeability rock (e.g. shales and many volcanic rocks) takes lesstime than the characteristic time (R2/cf), where R is the radius ofthe well and cf is the fluid diffusivity. Here we examine the roleof thermo-poroelastic effects on wellbore stability, and wellbore

Q. Tao, A. Ghassemi / Geothermics 39 (2010) 250–259 251

Nomenclature

a borehole radius [L]b material constant in Drucker–Prager failure crite-

rionB Skempton pore pressure coefficientc cohesioncft coupled thermal-fluid pressure coefficient

[ML−1 T−2 K−1]cm specific heat [L−2 T−2 K−1]cf fluid diffusivity [L2 T−1]cT thermal diffusivity [L2 T−1]C0 uniaxial compressive strength [ML−1 T−2]C1 variable defined in Eq. (A.15)C2 variable defined in Eq. (A.16)C3 variable defined in Eq. (A.17)D1 variable defined in Eq. (A.18)D2 variable defined in Eq. (A.19)G shear modulus [ML−1 T−2]hi linear heat flow in the ith direction [MT−3]k permeability of the solid matrix [L2]kT thermal conductivity of the solid matrix

[L MT−3 K−1]K0 modified Bessel function of second kind of order

zeroK1 modified Bessel function of second kind of order oneK2 modified Bessel function of second kind of order twomd material constant defined in Drucker–Prager failure

criterionp pore pressure [ML−1 T−2]p0 initial pore pressure [ML−1 T−2]pf formation pore pressure [ML−1 T−2]pm mud pressure in the borehole [ML−1 T−2]P0 isotropic compressive stress [ML−1 T−2]qi fluid velocity in the ith direction [L T−1]r radial distance to the center of the borehole [L]s Laplace variableS0 deviatoric stress [ML−1 T−2]t time [T]T temperature [K]Tf formation temperature [K]Tm mud temperature [K]u solid matrix displacement [L]

Greek letters˛ Biot’s coefficient˛f volumetric thermal expansion coefficient of the

pore fluid [K−1]˛s volumetric thermal expansion coefficient of the

solid matrix [K−1]ˇ variable defined in Eq. (A.12)ˇT variable defined in Eq. (A.14)ıij Kronecker delta� changeεij strain tensor of the solid matrix�cf failure orientation of the borehole� porosity�f frictional angle� wellbore inclination angle� constant (Eq. (A.7))ϕ internal friction angle viscosity of the fluid [ML−1 T−1] drained Poisson’s ratio

u undrained Poisson’s ratioω wellbore azimuth� defined in Eq. (A.10) m total mass density [ML−3]�1 effective maximum principal stress [ML−1 T−2]�3 effective minimum principal stress [ML−1 T−2]�h minimum horizontal principal stress [ML−1 T−2]�H maximum horizontal principal stress [ML−1 T−2]�ij stress tensor of the solid matrix [ML−1 T−2]�kk total principal stresses [ML−1 T−2]�v vertical stress [ML−1 T−2]� variable defined in Eq. (A.11)�T variable defined in Eq. (A.13)� change of pore volume

Subscripts0 initial�� tangentialf hydraulic or formationh minimum horizontalH maximum horizontali, j index of coordinatekk bulk valuer� shearrr radialT thermalv vertical

Superscriptsmax maximum′ effective stress

Over scripts∼ Laplace space

breakouts and its use to constrain the in situ stresses and rockstrength.

2. Theory of poro-thermoelasticity

When a rock consisting of an elastic solid matrix and fluid-filled pores is subjected to thermal perturbations in petroleum andgeothermal energy development, the temperature variations willcause the solid and fluid volumes to change, thereby disturbingstress and pore pressure equilibria.

2.1. Constitutive equations

The coupled constitutive equations of poro-thermoelastic mate-rial under non-isothermal conditions of conductive heat transporthave been developed by extending Biot’s poroelasticity theory(Biot, 1941, 1955) to non-isothermal conditions (McTigue, 1986;Kurashige, 1989):

εij = 12G

[�ij − v

1 + v�kkıij

]+ ˛(1 − 2v)

2G(1 + v)ıijp + ˛s

3ıijT (1)

� = ˛(1 − 2v)2G(1 + v)

�kk + ˛2(1 − 2v)2(1 + vu)2G(1 + v)(vu − v)

p − �(˛f − ˛s)T (2)

where εij and �ij are the change of strain and stress of the solidmatrix, respectively, p and � the change of pore pressure and porevolume, respectively, T the change of temperature, and ıij the Kro-necker delta. The material constants are the bulk shear modulus

2 othermics 39 (2010) 250–259

Gae

2

ia

q

h

wta

2

cb

G

wt

w

c

a

c

T

wtt

3d

atdcstps1pdca

m

Table 1Poro-thermoelastic parameters for shale.

Shear modulus G 760 MPaBiot’s coefficient ˛ 0.966Drained Poisson’s ratio 0.219Undrained Poisson’s ratio u 0.461Permeability k 1 × 10−20 m2

Porosity � 0.3Thermal expansion coefficient of solid ˛m 1.8 × 10−5 K−1

Thermal expansion coefficient of fluid ˛f 3.0 × 10−4 K−1

Thermal diffusivity cT 1.6 × 10−6 m2/s

Fig. 1 illustrates the effect of mud weight on stress distribu-tion 100 h after drilling (all results presented in the paper usethe geomechanics sign convention, i.e., compression positive); itshows that drilling with a 3 MPa overbalanced mud pressure (pw =

52 Q. Tao, A. Ghassemi / Ge

, Biot coefficient ˛, the drained and undrained Poisson’s ratios vnd vu, the initial porosity of solid �, and the volumetric thermalxpansion coefficients of the solid matrix ˛s and the pore fluid ˛f.

.2. Transport equations

The fluid and heat transport equations are obtained by neglect-ng thermal-osmosis (Ghassemi et al., 2009) and heat flow bydvection (Delaney, 1982):

i = − k

p,i Darcy′s law (3)

i = −kT T,iFourier′s law (4)

here q and h are the fluid and heat fluxes, respectively; k and he permeability of solid matrix and the fluid viscosity, respectively,nd kT the thermal conductivity of the fluid-saturated rock.

.3. Field equations

Following McTigue (1986), the field equations are obtained byombining the constitutive and transport equations with the forcealance, mass and heat conservation equations.

The Navier-type equation:

ui,jj + G

1 − 2vuj,ji − ˛p,i − 2G(1 + v)

3(1 − 2v)˛sT,i = 0 (5)

here u is the solid matrix displacement.The fluid diffusion equa-ion:

∂p

∂t− cf p,kk = cft

∂T

∂t(6)

here the hydraulic diffusivity coefficient cf is

f = 2kB2G(1 − v)(1 + vu)2

9(1 − vu)(vu − v)(7)

nd the thermo-hydraulic coupling coefficient cft is given by:

ft = ˛scf

k

[2(vu − v)

B(1 + vu)(1 − v)+ �

(˛f

˛s− 1

)](8)

he heat diffusion equation:

∂T

∂t− cT T,kk = 0 (9)

here cT = kT/( mcm) is thermal diffusivity, and m, and cm are theotal (fluid plus solid matrix) mass density and specific heat, respec-ively.

. Stress and pore pressure response around a boreholeuring and after drilling

When a well is drilled, the pore pressure and stress equilibriumre disturbed, which can cause borehole instability by breakouts orensile failure. The heating or cooling of the surrounding rocks byrilling mud also causes formation expansion or shrinkage, therebyausing possible borehole stability. The elastic solution for thetress concentration around the borehole known as the Kirsch solu-ion (Jaeger and Cook, 1979) cannot explain the transient stress andore pressure change. The time-dependent poroelastic solution oftress and pore pressure around a borehole (Carter and Booker,982; Detournay and Cheng, 1988) is needed to explain some tem-oral phenomena such as time-dependent borehole opening in the

irection of the minimum compressive in situ stress, a progressivelosure in the direction of the maximum compressive in situ stress,nd delayed borehole collapse within the elastic regime.

A poro-thermoelastic solution is necessary to include the ther-ally induced stress and pore pressure defined by Eqs. (5), (6)

Skempton’s coefficient B 0.915Fluid viscosity 3 × 10−4 cpFluid diffusivity c 6.0 × 10−8 m2/s

and (9). The solutions for stress and pore pressure are derived bycombining the boundary conditions (see Eqs. (A.1) and (A.2), andFig. A1 in Appendix A) at the borehole wall and at infinity withthe Navier type field, fluid diffusion, heat diffusion, and force bal-ance equations (Wang and Papamichos, 1994; Cui et al., 1997; Liet al., 1998). Using such procedures we have derived the coupledporo-thermoelastic solutions and used them here to analyze theoccurrence of borehole failure, and to study the use of breakouts toestimate in situ stress and rock strength. For completeness, thesesolutions are reported in Appendix A; they can be inverted into thetime domain using the method of Stehfest (1970). These solutionsassume that drilling the rock face is much faster than the diffusiveprocesses, as discussed above, and that the wellbore face is instan-taneously subjected to the new mud pressure and temperature thatare uniform throughout the solution time.

In addition to the in situ stress, both mud pressure and tempera-ture affect stress concentrations around the wellbore and thus rockfailure and breakouts. To illustrate the effects of those two factorswe first consider the problem of a wellbore in shale with parameterslisted in Table 1. The in situ stresses are assumed as: vertical stress�v = 21 MPa, maximum horizontal stress �H = 18 MPa, minimumhorizontal stress �h = 17 MPa and initial pore pressure p0 = 12 MPa.The assumed in situ stress is for a normal fault stress field, but thethermal and hydraulic results are applicable to other stress fieldsalso.

Fig. 1. Effects of mud pressure on the effective stresses in the direction of the min-imum horizontal stress under isothermal conditions at 100 h after drilling. Solidcurves: tangential stress, dashed curve: radial stress.

Q. Tao, A. Ghassemi / Geothermics 39 (2010) 250–259 253

1itptwotrtaaiwpttat

utpitsopfdttWsg

swbdintth

Table 2Poro-thermoelastic parameters for Westerly granite.

Shear modulus G 15,000 MPaBiot’s coefficient ˛ 0.44Drained Poisson’s ratio 0.25Undrained Poisson’s ratio u 0.33Permeability coefficient k 4 × 10−19 m2

Porosity � 0.01Thermal expansion coefficient of solid ˛m 2.4 × 10−5 K−1

Thermal expansion coefficient of fluid ˛f 3.0 × 10−4 K−1

Thermal diffusivity cT 5.1 × 10−6 m2/sSkempton’s coefficient B 0.81Fluid viscosity* 3 × 10−4 Pa sFluid diffusivity* c 7 × 10−5 m2/s

* The fluid viscosity is adjusted for a high temperature and thus the fluid diffusivityis changed.

tial variation has a different shape. The induced total thermoelasticstress is K˛s�T (Eq. (1)), where K is the solid bulk modulus. TheWesterly granite has a much higher bulk modulus (25 GPa) thanthe shale (1 GPa), and as a result, much larger stress is induced ingranite for the same temperature change. As the hydraulic diffusiv-

Fig. 2. Heating-induced pore pressure around a wellbore in shale.

5 MPa), the effective tangential stress in the direction of the min-mum horizontal stress decreases by 4.2 MPa at the wall relative tohe case pw = p0. While drilling with a 3 MPa underbalanced mudressure (pw = 9 MPa), the effective tangential stress in the direc-ion of the minimum horizontal stress increases by 4.2 MPa at theall. Therefore, increasing mud weight lowers the concentration

f effective tangential stress at the wellbore wall by reducing theotal stress and increasing the pore pressure. Note that the effectiveadial stress at the wellbore wall is not affected, as the change inotal radial stress and pore pressure are equal at the wall. In thisnd subsequent poroelastic analysis, we consider the stresses attime of 100 h, representing the long-time solution. This is done

n view of the fact that the compressive tangential stress at theellbore and its immediate vicinity, that is associated with mudressure diffusion develops rapidly; however, the concentration ofhe far-field stress around the wellbore is a transient process, withhe maximum stress concentration at the wall occurring some timefter the rock has been removed, delaying possible failure to laterimes (Detournay and Cheng, 1988; Cui et al., 1999).

To illustrate the impact of mud temperature, consider the sit-ation where the rock is continuously heated by a drilling fluidhat is 50 ◦C warmer than the formation. Fig. 2 shows the inducedore pressure around the wellbore for various times; a signif-

cant pressure increase is generated near the borehole at earlyimes. With increasing time, the peak of the induced pore pres-ure is reduced and moves away from the well. The magnitudef this pressure depends on the diffusivity and thermo-hydraulicarameter of the rock (cf = 6.0 × 10−8 m2/s and cft = 0.168 MPa/Kor this shale). For Westerly granite (McTigue, 1990) with a fluidiffusivity of 6.26 × 10−5 m2/s and cft of 0.41 MPa/K (the other poro-hermoelastic parameters for Westerly granite are listed in Table 2),he induced pore pressure is much lower, as shown in Fig. 3. As

esterly granite has a much higher fluid diffusivity compared withhale, the thermally induced fluid pressure dissipates faster in theranite and results in much lower pore pressure (Fig. 3).

Fig. 4 presents the thermally induced total tangential stress inhale. Heating creates a compressive tangential stress around theellbore caused by the tendency of the rock to expand near the

orehole wall. Away from it the magnitude of the induced stressecreases and changes its sign at some point inside the formation,

.e. it turns into a tensile stress. This is because as the materialear the borehole expands, it tends to pull apart the outer rock,hus inducing tension. The tensile zone gradually moves away fromhe borehole and diminishes in magnitude due to thermal andydraulic diffusion.

Fig. 3. Heating-induced pore pressure around a wellbore in Westerly granite.

Fig. 5 illustrates the thermally induced total tangential stressfor a wellbore in Westerly granite. It can be seen that not only themagnitude is much larger than in shale, but its peak value does notoccurs inside the rock because of larger fluid diffusivity and the spa-

Fig. 4. Heating-induced tangential stress for shale.

254 Q. Tao, A. Ghassemi / Geothermics 39 (2010) 250–259

iptt

lfpciit

4

fsAatdas

stfp

S

w

I

J

Haobar

pressure of pm ≤ 10 MPa. However, if the mud pressure is greaterthan or equal to 21 MPa, the rock will be hydraulically fractured.Consequently, to avoid any failure the safe drilling mud pres-sure predicted by an elastic material model is between 12.5 and21 MPa. But if poro-thermoelastic effects are considered, that pres-

Fig. 5. Heating-induced tangential stress for Westerly granite.

ty of the granite is much higher than that of the shale, the inducedore pressure dissipates faster. Therefore, there is no peak value ofotal tangential stress inside the rock resulting from the cumulatedhermally induced fluid pressure inside the rock.

From these results it can be seen that a wellbore in low-modulus,ow-permeability rock can experience a failure mode that is dif-erent from that of a high-modulus rock with a relatively higherermeability. The high pore pressure induced in the former (shale)an lead to tensile fracturing, whereas shear failure is more likelyn the latter (granite) case. As pointed out by Li et al. (1998), thesemportant phenomena cannot be captured by the conventionalhermoelastic approach (Zoback, 2010).

. Poro-thermoelastic wellbore stability analysis

Wellbore instability includes compressive failure, hydraulicracturing (tensile fracture along the radial direction) and radialpalling (tensile fracture perpendicular to the radial direction).ctually, tensile fracturing inside the formation may not be exactlylong the radial direction or perpendicular to it, but it is importanto distinguish between the two because they result in differentrilling problems; i.e. hydraulic fracturing causes lost circulationnd radial spalling leads to local failure of the formation and pos-ible enlargement of the wellbore.

In this section we consider the impact of poro-thermoelastictresses on wellbore failure. We use the Drucker–Prager failure cri-erion (Jaeger and Cook, 1979) to determine whether compressiveailure occurs. In doing so, we define a compressive shear failureotential (SFP):

FP =√

J2 − mdI1 − b (10)

here

1 = � ′rr + � ′

�� + � ′zz

2 = 16

[(� ′

rr − � ′��)2 + (� ′

rr − � ′zz)2 + (� ′

�� − � ′zz)2]

+ �2r� + �2

rz + �2�z (11)

ere md and b are coefficients that depend on the cohesion, c,

nd the angle of internal friction,ϕ. Compressive shear failureccurs when the maximum SFP value around the wellbore (i.e.orehole radius/radial distance to the center of the borehole, r/a,bout 1.0–1.5) is equal to or larger than zero, i.e. SPF ≥ 0. Theadial spalling occurs when the radial effective stress is less than

Fig. 6. Poro-thermoelastic effects on shear failure at t = 100 s.

zero; the radial spalling potential is defined by the radial effec-tive stress (RSP = � ′

rr). In the same way, the hydraulic fracturingpotential (HFP) is defined in terms of the effective tangential stress(HFP = � ′

��). Thus, hydraulic fracturing and radial spalling occur

when HFP < 0 and RSP < 0, respectively. The minimum RSP and HFPvalues around the wellbore (r/a ∼ 1.0–1.5) are taken as the RSP andHFP for the wellbore.

Consider a vertical wellbore at a depth of 1 km in shale) withparameters listed in Table 1 and in situ stress as: �v = 25 MPa,�H = 29 MPa, �h = 20 MPa, p = 10 MPa. (This is a strike-slip stressfield, but the method also can be applied to other stress fields.)Figs. 6–8 show the failure potentials for different modes as a func-tion of mud pressure at 100 s after the wellbore wall is exposedto drilling mud. The critically low mud pressure causing compres-sive failure and radial spalling, and the critically high mud pressurecausing hydraulic fracturing can be obtained from the figures todecide on the safe drilling mud weight (Table 3). The critical valuesare those at which the failure potential intersects the horizontalaxis. For the elastic material model, shear failure occurs if mudpressure pm ≤ 12.5 MPa, and radial spalling will appear for a mud

Fig. 7. Poro-thermoelastic effects on hydraulic fracturing at t = 100 s.

Q. Tao, A. Ghassemi / Geothermics 39 (2010) 250–259 255

Table 3Effects of the poro-thermoelastic stresses on safe drilling mud pressure.

Material model Loading condition Mud pressure (MPa)

Shear failure Hydraulic fracturing Radial spalling Safe drilling range

Elastic �T = 0 ◦C ≤12.5 ≥21.0 ≤10.0 12.5–21.0

setlcirta

5p

u1soamMs�rt(afo

5

mm

C0 = �1 −1 − sin ϕ

�3 (12)

where � ′1 and � ′

3 are the local effective maximum and minimum

Poroelastic �T = 0 ◦C ≤12.6

Poro-thermoelastic �T = −50 ◦C ≤11.5�T = +50 ◦C ≤16.1

ure will be different (Table 3). Heating increases the maximumffective principal stress in the tangential direction and reduceshe minimum effective principal stress in the radial direction (theatter could become tensile at short times), and so tends to causeompressive failure and radial spalling, and thus, increases the crit-cal low mud pressure required to prevent rock failure. Coolingeduces the effective tangential stresses, and therefore enhanceshe hydraulic fracturing, but increases the effective radial stress,nd thereby inhibits radial spalling.

. Determination of in situ stress and rock strength usingoro-thermoelasticity

It is common to estimate the in situ stress and the rock strengthsing observed wellbore breakouts using elastic (Peska and Zoback,995) or thermoelastic analysis (Zoback, 2010). In the followingections we study the impact of poro-thermoelastic mechanismsn such analysis. We use the data and Mohr–Coulomb criterionnd rock and stress data used in Peska and Zoback (1995) for aeaningful comparison of results. The stress data are for a Gulf ofexico region known as a zone of normal faulting, so the vertical

tress is the maximum principal stress, i.e. �h = 37.1 MPa < �H <

v = 43 MPa with a pore pressure of 29 MPa. The normal faultingegime also has significance for a number of geothermal reservoirshat are in a normal to strike-slip transitional state, e.g. Coso, CANygren, 2003; Sheridan et al., 2003; Davatzes and Hickman, 2006)nd Dixie Valley, NV (Hickman et al., 2000). However, the particularaulting regime assumed has no particular impact on the magnitudef the poro-thermoelatic effects.

.1. Elastic analysis

Similarly to Peska and Zoback (1995), assuming the rock for-ation to be homogeneous and isotropic elastic, and considering aud pressure equal to the formation pore pressure of 29 MPa, and

Fig. 8. Poro-thermoelastic effects on radial spalling at t = 100 s.

≥23.8 ≤16.0 16.0–23.8

≥23.3 ≤8.78 11.5–23.3≥24.4 ≤24.4 23.2–24.4

an internal coefficient of friction of 1.0, the compressive failure ori-entation of wells with different azimuth and inclination are shownin Fig. 9 using a lower hemisphere projection. For an arbitrarily ori-ented hole, the inclination with respect to vertical is given by theradial distance from the center, and its azimuth is measured clock-wise from the North. A vertical borehole is in the center of the circleand horizontal boreholes lie at the periphery. The short lines in thefigure represent the breakout orientation. For example, a boreholedeviated 60◦ from the vertical with a azimuth of 45◦ tends to failroughly along the upper left and lower right parts at an angle �cfclockwise from the bottom side of the borehole (Fig. 9).

Wells with an azimuth of N35◦E and a deviation angle of 32◦

from the vertical will tend to fail at �cf = 17◦ at the maximum prin-cipal stress of 40.0 MPa. For a tolerance of 5◦ for �cf (17◦ ± 5◦) i.e.for �cf in the range of about 12–22◦, those wells will fail only if themaximum principal stress is between 39.6 and 43.0 MPa (Peska andZoback, 1995). The magnitude of �H and azimuth of �h are plottedin Fig. 10 for wells that fit the breakout data within a tolerance of 5◦.The average azimuth of �h is between N36◦E and N49.5◦E, shownas a dashed curve in the figure, and is in agreement with Peska andZoback (1995).

According to Peska and Zoback (1995), if the regional principalstresses are known from another source, the upper limit of rockstrength can be estimated using the existence of a wellbore break-out. This is done by solving the Mohr–Coulomb failure criterion foruniaxial strength (Jaeger and Cook, 1979):

′ 1 + sin ϕ ′

principal stresses, respectively. C0 the minimum rock strength toprevent failure at a point can also be expressed using the cohesion

Fig. 9. Breakout directions (plotted using the “looking down the hole” conven-tion) for boreholes of various orientations, given �H = 40 MPa, �h = 37.1 MPa and�v = 43 MPa; Pm = 29 MPa; P0 = 29 MPa, �H is in the East–West direction, and �h is inthe North–South direction.

256 Q. Tao, A. Ghassemi / Geothermics 39 (2010) 250–259

Fima

s

C

tipu(t

5

ttTttsadatdb(2i

TMb

33.28 MPa. The heating case yields a 39.6–43.0 MPa range for themaximum horizontal principal stress, and 24.68–34.71 MPa for thecorresponding upper limit of rock strength (Fig. 12).

ig. 10. The magnitudes of maximum horizontal stress �H and the azimuth of min-mum horizontal stress �h for which a breakout can form at an angle 17◦ ± 5◦

easured from the bottom side of the borehole. The dashed curve is the averagezimuth of �h for a given magnitude of �H .

trength, c, and the internal friction angle, ϕ, i.e.,

0 = 2c cos ϕ

(1 − sin ϕ)(13)

Using an elastic model for stress concentration around the well,he predicted upper limit of rock strength for the previous examples 23.38 MPa for �H = 40 MPa and the minimum horizontal princi-al stress azimuth of N40◦E. Then, for a tolerance of 5◦ for �cf, thepper limit of rock strength must be between 22.08 and 32.40 MPaTable 4) for the breakout to occur. These results are consistent withhose of Peska and Zoback (1995).

.2. Poro-thermoelastic analysis

First, we consider the poroelastic case for the onset of forma-ion breakout by letting the drilling fluid and rock have the sameemperature. For the poroelastic case with parameters listed inable 1, and using the stress concentrations at a time of 100 h,he magnitude of the maximum horizontal principal stress andhe azimuth of the minimum horizontal principal stress are theame as those given by the elastic model. This is to be expecteds rock isotropy and homogeneity are assumed. Also, the pre-icted upper limits of rock strength, defined as the strength valuebove which failure would not have occurred, remain very closeo the elastic result of 23.97–34 MPa (using a tolerance of 5◦, as

one in this and all following cases). This is because the differenceetween mud and formation pore pressures was assumed to be zerobalanced drilling). If this difference is set to the typical value of.0 MPa (i.e. overbalance drilling with Pm = 31 MPa), then the max-

mum horizontal principal stress would fall in the 39.8–43.0 MPa

able 4aximum horizontal stress �H and the upper bound of rock strength C0 obtained

y inversion of breakouts using the long-time solution (t = 100 h).

Model and conditions �H (MPa) range C0 (MPa) range

Elastic �p = 0 MPa 39.6–43 22.08–32.40�p = 2 MPa 39.8–43 9.06–18.75

Poroelastic (�T = 0) �p = 0 MPa 39.6–43 22.29–31.82�p = 2 MPa 39.7–43 21.79–31.52

Poro-thermoelastic(�p = 0; pm = p0)

�T = 50 ◦C 39.6–43 24.68–34.71

�T = −50 ◦C 39.6–43 23.26–33.28

Fig. 11. Upper limit of rock strength (i.e., strength above with breakouts will notform) as a function of the maximum horizontal principal stress, �H , for the purelyelastic and poroelastic cases. The mud pressure is assumed as 31 MPa.

range using the elastic model, and in the 39.7–43.0 MPa range usingthe poroelastic model. However, the corresponding upper limit ofrock strength predicted by the elastic model is between 9.06 and18.75 MPa, whereas that predicted by poroelastic model would bein the 21.79–31.52 MPa range (Fig. 11). Thus, the upper bound ofrock strength obtained by a poroelastic analysis is significantlylarger than the elastic one, meaning that the rock strength maybe underestimated by an elastic breakout analysis.

If the temperature difference between the mud and formationand its coupling to pore pressure are also considered (a poro-thermoelatic analysis), the predicted upper limit of rock strengthwould be different from the elastic and poroelastic results. Twopossible cases are considered, one is cooling (�T = −50 ◦C, i.e. mudtemperature is lower than formation temperature) and the otherheating (�T = +50 ◦C). It is assumed that the mud pressure equalsthe pore pressure for both cases to isolate the poroelastic effect ofmud pressure. For the cooling case, the maximum horizontal prin-cipal stress is estimated to be in the 39.6–43.0 MPa range, and thecorresponding upper limit of rock strength is between 23.26 and

Fig. 12. Upper limit of rock strength as a function of the maximum horizontal stress,�H , for the poro-thermoelastic case using the long time solution (t = 100 h). The mudpressure is assumed as 29 MPa.

Q. Tao, A. Ghassemi / Geotherm

F�p

ietarblmecswf

torfzrc2eadop

6

tahwprw

gpaw

ig. 13. Upper limit of rock strength as a function of the maximum horizontal stress,H , for the poro-thermoelastic case using short time solution (t = 100 s). The mudressure is assumed as 29 MPa.

Again, the thermally induced pore pressure and tangential stressnside the rock is time-dependent and for this example the thermalffect is critical at short times. The upper limit of rock strength usinghe short-time solution (t = 100 s) is between 23.31 and 31.04 MPand between 44.79 and 65.69 MPa for the cooling and heating cases,espectively (Fig. 13). Hence, when the rock is heated, the upperound of rock strength obtained by a poro-thermoelastic analysis is

arger than that given by the poroelastic (and thus elastic) analysis,eaning that the rock strength may be underestimated if thermal

ffects are not considered. Note that in Fig. 13, the lines for theooling and isothermal cases are not straight. This is because formall �H values, the peak of failure potential occurs at the wellboreall, but as �H increases the peak moves inside the surrounding

ormation.It should be pointed out that in this work the impact of poro-

hermoelastic stresses has been studied based on the assumptionf rock isotropy and homogeneity. The inhomogeneous nature ofocks can lead to the formation of qualitatively different boreholeailure and thermal elongations parallel to the maximum hori-ontal principal stress orientation; these are suggested to be theesult of a pervasive, cooling-induced, tensile micro-cracking pro-ess prior to macroscopic failure localization (Berard and Cornet,003). The impact of rock elastic anisotropy has been consid-red by Vernik and Zoback (1990); poro-thermoelastic anisotropylso can radically change the pattern of pore pressure and stressistributions around the wellbore and thus would affect break-ut orientation and size, but is beyond the scope of the presentaper.

. Conclusions

Often elastic analysis is used to calculate the stress concen-ration and to assess rock failure around wellbores. However,n elastic model cannot capture the coupling between fluid andeat diffusion and rock deformation, which can be very importanthen drilling wells in geothermal fields, where a strong cou-ling between thermal and poro-mechanical effects might develop,esulting in large stress/pore pressure redistributions around theellbore.

Heating increases pore pressure and induces compressive tan-ential and radial stresses, whereas cooling reduces the poreressure and causes tensile tangential and radial stresses. The rel-tive magnitude of these induced stresses and pore pressure varyith rock mechanical and transport properties. The impact of these

ics 39 (2010) 250–259 257

effects on the stability of a wellbore has been considered usingthe Drucker–Prager criterion for compressive failure, and a tensilefailure criterion. The results show that poro-thermo-mechanicaleffects influence both the failure mode and potential; cooling tendsto prevent compressive failure and radial spalling, whereas heat-ing tends to enhance failure in compression and can cause tensilefailure by excessive increase of pore pressure.

Also, poro-thermoelastic effects on the estimations of the in situmaximum horizontal stress and rock strength using wellbore fail-ure data were studied. For typical overbalanced drilling, examplecalculations indicate that poroelastic effects only slightly reducethe estimated range of �H, but the range of uniaxial compressiverock strength, C0, is increased compared to the elastic model, mean-ing the latter model underestimates rock strength. However, inunderbalanced drilling, the elastic model would overestimate thatstrength.

When the formation is heated while drilling, the upper boundof rock strength (i.e. rock strength needed to suppress break-outs) is larger than that predicted by elastic analysis, so that therock strength will be underestimated even more if temperatureincreases are not considered. On the other hand, if mud tem-perature is lower than that of the formation, the elastic modeloverestimates rock strengths. These temperature change scenarios(heating vs cooling) correspond to the upper and lower segmentsof a well in a high-temperature zone, respectively.

Additional analysis taking into account the influence offilter-cake is necessary to ascertain the full range of the poro-thermoelastic effects on wellbore failure, breakouts and theirutilization to estimate the in situ stress magnitude and rockstrength. Furthermore, consideration of mechanical and thermalanisotropy would be useful in analyses of breakouts for the deter-mination of in situ stress and rock strength, and would enhanceunderstanding of the fundamental mechanisms involved in rockfailure around wellbores.

Acknowledgements

The authors would like to thank Dr. Stephen Hickman and ananonymous reviewer for their helpful suggestions. Additional edi-torial comments by Drs. Sabodh Garg and Marcelo Lippmann areappreciated.

Appendix A. Induced pore pressure and stress around aborehole

The following solution was first derived by Li et al. (1998), andis listed below for completeness. The solution assumes that therock is instantly removed and the wellbore wall subjected to a dif-ferent pressure and temperature and that the following boundaryconditions are applicable (Fig. A1):

at r = a��rr = pm − �∞

rr��r� = −�∞

r��p = pm − psh

�T = Tm − Tsh

(A.1)

at r → ∞��rr = 0��r� = 0 (A.2)

�p = 0�T = 0

The induced stresses (Fig. A2) and pore pressure in Laplace spaceare given by:

258 Q. Tao, A. Ghassemi / Geotherm

Fo

s

s

Ft

ig. A1. Cross section showing the stress, pore pressure and temperature loadingf the borehole.

Pore pressure:

p̃ = pf + cfT

1 − c/cT(Tm − Tf )

[K0(�)K0(ˇ)

− K0(�T )K0(ˇT )

]

+ (pm − pf )K0(�)K0(ˇ)

+ C1B2(1 − v)(1 + vu)2S0 cos 2�

9(1 − vu)(vu − v)K2(�)

+ C2B(1 + vu)S0 cos 2�

3(1 − vu)a2

r2(A.3)

Radial stress:

�̃rr = P0

(a

r

)2− 2�

[(pm − pf ) + cfT

1 − c/cT(Tm − Tf )

]

×[

a2

r2

K1(ˇ)ˇK (ˇ)

− a

r

K1(�)ˇK (ˇ)

]+ 2�(Tm − Tf )

0 0

×[ cfT

1 − c/cT+ K˛m

˛

][a2

r2

K1(ˇT )ˇT K0(ˇT )

− a

r

K1(�T )ˇT K0(ˇT )

]

ig. A2. Diagram showing the stress components acting in the plane perpendicularo the borehole in cylindrical coordinate system.

ics 39 (2010) 250–259

+ C1B(1 + vu)S0 cos 2�

3(1 − vu)

[1�

K1(�) + 6�2

K2(�)

]

− C2S0 cos 2�

1 − vu

(a

r

)2− 3C3S0 cos 2�

(a

r

)4(A.4)

Tangential stress:

s�̃�� = −P0

(a

r

)2+ 2�

[ cfT

1 − c/cT(Tm − Tf ) + (pm − pf )

]

×[

K1(ˇ)ˇK0(ˇ)

(a

r

)2− K1(�)

ˇK0(ˇ)a

r− K0(�)

K0(ˇ)

]

+ 2�(Tm − Tf )( cfT

1 − c/cT+ K˛m

˛

)[− K1(ˇT )

ˇT K0(ˇT )

(a

r

)2

+ K1(�T )ˇT K0(ˇT )

a

r+ K0(�T )

K0(ˇT )

]− C1B(1 + vu)S0 cos 2�

3(1 − vu)

×[

K1(�)�

+(

1 + 6�2

)K2(�)

]+ 3C3S0 cos 2�

(a

r

)4(A.5)

Shear stress:

s�̃r� = 2C1B(1 + vu)S0 cos 2�

3(1 − vu)

[K1(�)

�+ 3K2(�)

�2

]

− C2S0 cos 2�

2(1 − vu)

(a

r

)2− 3C3S0 cos 2�

(a

r

)4(A.6)

where

� = ˛(1 − 2v)2(1 − v)

(A.7)

P0 = �xx + �yy

2(A.8)

S0 = 12

√(�xx − �yy)2 + 4�2

xy (A.9)

� = 12

tan−1

(2�xy

�xx − �yy

)(A.10)

� = r

√s

c(A.11)

ˇ = a

√s

c(A.12)

�T = r

√s

cT(A.13)

ˇT = r

√s

cT(A.14)

C1 = −12ˇ(1 − vu)(vu − v)B(1 + vu)(D2 − D1)

(A.15)

C2 = 4(1 − vu)D2

(D2 − D1)(A.16)

C3 = −ˇ(D2 + D1) + 8(vu − v)K2(ˇ)ˇ(D2 − D1)

(A.17)

D1 = 2(vu − v)K1(ˇ) (A.18)

D = ˇ(1 − v)K (ˇ) (A.19)

2 2

Here K0, K1, K2 are the second kind of Bessel functions with zero,one and two orders, respectively.

The solutions are applicable to wells with any inclination. Foran inclined well, the in situ stresses (�v, �H, �h) are converted to

Q. Tao, A. Ghassemi / Geotherm

Fig. A3. Diagram showing the borehole trajectory in 3D space. Also shown are thegstω

n(ttu

R

B

B

B

B

B

C

C

lobal in situ stress coordinate system and the local borehole coordinate system. Thetress components in the borehole coordinate system are obtained through stressransformation equations using the inclination and rotation (azimuth) angles � and.

ew coordinates with one axis parallel to the axis of the boreholeFig. A3) according to Eq. (A.20). Then, using plane strain assump-ion (i.e., the axial length of borehole is much larger than its radius),he stress, pore pressure and temperature in the plane perpendic-lar to the borehole axis can be obtained as:

�xx = (cos � cos ω)2�H + (cos � sin ω)2�h + (sin �)2�v

�yy = (sin ω)2�H + (cos ω)2�h

�zz = (sin � cos ω)2�H + (sin � sin ω)2�h + (cos �)2�v�xy = (− cos � cos ω sin ω)�H + (cos � cos ω sin ω)�h

�xz = cos � sin �(cos ω)2�H + cos � sin �(sin ω)2�h

− (cos � sin �)�v�yz = (− sin � cos ω sin ω)�H + (sin � cos ω sin ω)�h

(A.20)

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