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Journal of Natural Gas Science and Engineering 36 (2016) 986e1003

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Journal of Natural Gas Science and Engineering

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Stress cages and fracture cages in stress trajectory models ofwellbores: Implications for pressure management during drillingand hydraulic fracturing

Ruud WeijermarsHarold Vance Department of Petroleum Engineering, Texas A&M University, 3116 TAMU College Station, TX 77843-3116, USA

a r t i c l e i n f o

Article history:Received 22 June 2016Received in revised form22 October 2016Accepted 24 October 2016Available online 27 October 2016

Keywords:Wellbore stressesStress cagesFracture cagesFrac number

E-mail address: R.Weijermars@TAMU.edu.

http://dx.doi.org/10.1016/j.jngse.2016.10.0541875-5100/© 2016 Elsevier B.V. All rights reserved.

a b s t r a c t

This study visualizes in high resolution the stress trajectory patterns around a wellbore resulting fromthe interaction between the induced, effective borehole pressures with and without far-field tectonicstresses. Near-wellbore stress trajectory maps are produced using non-dimensional solutions of thestress function for balanced, overbalanced and underbalanced borehole sections. Particular attention ispaid to the isotropic points, where no deviatoric stress occurs, as these mark the locations whereprincipal stress reversals occur, an aspect that has not been highlighted before. When wellbores areoverbalanced, stress cages develop, and the region occupied by a stress cage grows when the net pressure(positive) on the wellbore is much larger than the far-field principal stress. Fracture cages are a com-plementary phenomenon occurring in underbalanced wellbores. Fracture cages occupy larger areaswhen the absolute value of the (negative) net pressure grows larger relative to the far-field (native) stressstate. A novel nomogram visualizes the continuous range of stress trajectories around overbalanced aswell as underbalanced borehole sections, and is scalable by a single dimensionless number, 1/z, coinedthe Frac number. A proper understanding of the principal stress trajectories is extremely useful forpredicting the planes of least resistance for fracture initiation. The Frac number can thus contribute to theoptimization of well design and hydraulic fracture placement.

© 2016 Elsevier B.V. All rights reserved.

1. Introduction

Detailed analysis of principal stress trajectories and stressmagnitudes around wellbores in the presence of fluid pressure isimportant for at least two major applications in petroleum engi-neering: (1) Prevention of wellbore damage during drilling towarda target reservoir, and (2) controlling the initiation and propagationdirection of hydraulic fractures for stimulation of the reservoirzone. Once the principal stress orientations and magnitudes areknown, the respective regions where maximum tensile stress andmaximum shear stress occur can be identified to next determinewhere the critical stress for tensile failure and shear slip arereached. Tensile fractures will follow the maximum principal stresstrajectories, and shear fractures follow the sliplines of maximumshear stress determined by the failure criterion (e.g. angle of in-ternal friction).

Current tool suites used by industry for borehole-stress

monitoring do not map the principal stress trajectories in anydetail. Likewise, classical studies of wellbore stresses commonlyshow stress trajectories for one special case only e namely stresstrajectories around an empty borehole subjected to an uni-axial far-field boundary stress (e.g., Jaeger and Cook, 1979; Jaeger et al.,2007; Zoback, 2007, Fig. 6-1). However, a wide range of near-wellbore stress-trajectory patterns may occur depending on suchfactors as the presence or absence of any far-field stress and therelative magnitudes of the formation pressure and mud weightapplied to the wellbore (Weijermars, 2011; Weijermars et al.,2013a). There is considerable merit in characterization of thestress state around the wellbore by visualizing the stress trajec-tories as a powerful visual image.

The present study expands the analysis of stress trajectories byhighlighting the stress reversals that may occur near the wellboredue to the interaction of the net pressure on the wellbore and thetectonic background stress. Such stress reversals do not occur in theabsence of a far-field stress, which is why we first review thatsimpler case. After first laying out basic assumptions (Section 2),typical orientations of tensile and compression stresses in the

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 987

wellbore during overbalanced and underbalanced drilling withoutany far-field tectonic stress are mapped (Section 3). We then pro-ceed to visualize the range of stress patterns induced aroundwellbores due to various types of Andersonian tectonic (far-field)stress, again accounting for whether drilling is underbalanced oroverbalanced (Section 4). The Frac number nomogram is explained(Section 5), followed by a brief discussion and conclusions (Sections6 and 7).

2. Basic assumptions

This study presents the practical application of analytical stressfunctions that model stress patterns around overbalanced,balanced and underbalanced sections of a drillhole. We assume thehost rock behaves as an elastic continuum. Poro-elastic effects canbe included but are neglected in our approach for simplicity. Tensornotation can also be included, but we want to focus on the physicalaspects of unbalanced borehole stresses, and therefore seekmaximum computational transparency by aligning the borehole inthe direction of one of the principal stresses (assuming verticaldrillholes or horizontal wellbore sections aligned with one of theprincipal stresses). Our stress pattern solutions can be translated toany drillhole section (vertical, horizontal) that remains alignedwith one of the Andersonian stresses (Fig. 1).

The traditional approach in borehole stress analysis also avoidstensor notation, denoting a maximum total stress in the horizontalplane by sH (or sH MAX or SH) and a minimum by sh (or sHMIN or Sh)

Fig. 1. Principal stress orientations for three basic configurations of crustal faulting (shallo(deeper crust) based upon depth of brittle-ductile transition (see Weijermars, 1998): a) northickening in collision belts, and c) strike-slip faulting and deep-seated ductile shear zoneshallow as 5 km, may affect drilling and wellbore integrity for deep offshore boreholes inangular relationship of principal stress with strike-slip movement involving transpression

and a vertical stress by sV (or SV). When sH, sh, and sV are used asinputs, assurances are needed on how these relate to the principalstress directions s1, s2, and s3. The conversion of sH, sh, and sV toprincipal stresses s1, s2, and s3 is given in Table 1. For application inbasins with regional far-field stress due to active tectonics, Table 1relates sH, sh, and sV to the Andersonian principal stresses (c.f.,Anderson, 1951) for common regional tectonic regimes.

The basic equations for the stress concentration factors inboreholes are only valid for principal stresses, which are invariantsof the stress tensor independent of the coordinate system. Cautionis required when applying the following set of ‘’generic’’ expres-sions to estimate the radial, tangential and shear stresses at pres-sured vertical wellbores with an assumed far-field stress sH and sh(e.g., Carcione et al., 2006; correcting for a sign reversal in sR):

sR ¼ P (1a)

sq ¼ 3sH � sh � P (1b)

sRq ¼ 0 (1c)

The adoption of such a generic set of equations introducesseveral potential sources of inaccuracy in wellbore-stress analysis,especially when drilling near salt bodies where these stresseschange spatial orientation rapidly (Fig. 2). A first assumption forexpressions (1a-c) to be valid is that either sH∕∕s1 and sh∕∕s3; orsH ¼ s2, and sh ¼ s3 . Second, the stress-concentration factor of 3

w crust) according to Anderson (1951), augmented with ductile deformation modesmal faulting and crustal thinning in extensional basins, b) reverse faulting and crustals. Brittle-ductile transition, which occurs in areas of high heat flow and dry rocks asthe Gulf of Mexico currently reaching such depths. Velocity vectors are accurate andand transtension are detailed in Weijermars (1993).

Table 1Cartesian stress components sH, sh and sV and corresponding principal total stresses and deviatoric stresses, depending on the tectonic stress regime (see Fig. 1). This set ofassumptions also applies to the stress invariants for the deeper wellbore position in Fig. 2.

Tectonic Setting of Basin Orientation Invariant Principal Stresses Relative magnitude of invarants

Principal Total Stress s1 s2 s3 s1 � s2 � s3Principal Deviatoric Stress t1 t2 t3 t1 � t2 � t3

Tectonic Setting of Basin Conversion to Spatially Fixed Subscripts Relative Magnitude of fixed subscripts

Extension sV sH sh sV � sH � shCollision sH sh sV sH � sh � sVStrike Slip sH sV sh sH � sV � sh

Fig. 2. a: Vertical section across salt dome showing principal stress trajectories and zones of stress reversal along the wellbore (adapted fromWeijermars, 1998). b: Stress trajectoryorientations in two horizontal sections A-B and C-D, as outlined in (a). c: At depths Z1, drillhole axis is aligned with principal stresses on the wellbore: sv ¼ s1, sH ¼ s2, and sh ¼ s3. Atdepths Z2, the drillhole is no longer aligned with the stress invariants and orthogonal stresses on the wellbore (sv, sH, and sh) will no longer correspond to any of the stress invariantss1, s2 and s3. At depths Z3, the drillhole axis is again aligned with principal stresses on the wellbore, but now: sv ¼ s2, sH ¼ s1, and sh ¼ s3.

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003988

times the maximum far-field stress, s1, implied by Eq. (1b) is onlyvalid for bi-axial far-field stress with sH ¼ s1, sh ¼ s3 and sH s sh(Weijermars, 2013). If instead the far-field stress is axially sym-metric sH ¼ sh ¼ s3 (for example, above a radial symmetricallyexpanding salt dome) the stress-concentration factor is 2 times thefar-field stress, s3 (Weijermars, 2013). The upper section of Fig. 2 hassV∕∕s1, sH ¼ s2, and sh ¼ s3; the deeper section has sV∕∕s2,sH ¼ > s1, and sh ¼ s3. The intermediate section of the wellbore isoblique to all principal stresses, and tensor transformation rulesshould be used to relate sH and sh to the principal stresses acting onsuch sections of the wellbore (not elaborated here).

The stress-concentration factor for the special case of anaxisymmetric constriction sH ¼ sh ¼ s1 (such as may occur deeperin mini-basins sinking into a salt canopy) is 4 times the maximumfar-field stress, s1 (Weijermars, 2013). Making careful distinctionsbetween different types of far-field stress regimes therefore is ofpractical importance when establishing stress concentration fac-tors and must be taken into account in any wellbore stabilityanalysis.

In this paper, total stress is denoted by s and deviatoric stress byt. Compressional stresses are marked positive and consistentlyreferred to as the first principal axis, denoted by s1 (total stress) and

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 989

t1 (deviatoric stress) in compliance with the prevailing annotationin structural geology and tectonic models (Fossen, 2010;Weijermars, 1998). There is no difference in determining theprincipal stress trajectories from far-field deviatoric stresses t1 andt3, and from total stresses with symbols s1 and s3, except for anisotropic term. The in-plane ‘far field’ stresses t1 and t3 willdetermine the stress trajectory exactly the same as that determinedby total stress. However, deviatoric stresses differ from the totalstress in that the pressure component (isotropic stress) causesvolumetric deformation (only when the rock is compressible, i.e.,Poisson ratio<0.5) and deviatoric stresses cause shape deformation.The deviatoric stress can be represented in both a tensor andprincipal form and stress arrow representation of the smallestinvariant, t3, typically points opposite to s3 (due to removal of thepressure component).

In summary, using sH, sh, and sV [Eqs. (1a-c) and Table 1] insteadof s1, s2, and s3 will be correct only when stress concentrationfactors are adapted (ranging between 2 and 4, see Weijermars,2013). However, sH, sh, and sV near vertical wellbores proximal tosalt domes often are no longer aligned with any Andersonianprincipal stresses s1, s2, and s3 (such as at depth Z2 in Fig. 2a). Insuch cases use of Eqs. (1a-c) gives incorrect results.

3. Wellbore stresses without regional background stress

This section considers the case of preventing wellbore damageduring drilling toward a target reservoir. The importance of drillingoptimization by continuous feedback has been highlighted innumerous studies (e.g., Hougaz et al., 2012; McManus et al., 2012).Present-day wellbore-stress monitoring-systems rely heavily onautomated alarms for managing drilling pressure and maintainingwellbore stability (Sadlier et al., 2011; Van Oort et al., 2011; Li et al.,2012). The implementation of real-time data-flow-monitoring inboreholes improves process safety and helps to maintain theintegrity of wellbores during drilling operations (e.g., Azar andSamuel, 2007; Watson et al., 2003). Balancing wellbore stresses isa main challenge, which is complicated by two sets of highly var-iable conditions: (1) formation pressures vary with depth andinnate pore pressure, and (2) formations may be differentiallystressed by tectonic loading and deformation.

This section reviews the impact of varying formation porepressure with depth, introducing a proper definition for PNET todistinguish pure cases of underbalanced, balanced and over-balanced drilling without asymmetry of stress trajectories aroundthe wellbore due to any far-field stress. Section 4 includes the effectof a far-field stress.

3.1. Wellbore pressure differentials

Twodistinctmethods can be distinguished forwellbore pressurecontrol: overbalanced and underbalanced drilling, each of whichrequire their own well-control devices. For overbalanced drilling(OBD),which is the traditional drillingmode, the focus is on pressurecontrol and devices are in place to ensure the mud weight remainsbelow the fracture pressure and above the pore pressure of theformation (Rehm et al., 2012). Any undue rise in well pressure isrelieved by valves that are built into the blow-out preventer (BOP).For underbalanceddrilling (UBD), the focus is onflowcontrol and theprincipal device for well-control is the choke which is dynamicallyresponding to any change in pressure; slowing circulation of drillingfluid via the choke counters increases in the wellbore pressure,while faster circulation increases the pressure (Lyons et al., 2001).The BOP is only a secondary device for well-control when the chokecontrol is insufficient to prevent build-up of excess well pressure.

UBD is selectively applied by industry to produce low pressure

reservoirs when the extra cost for equipment to control flow ismerited. When pressing reasons for UBD are not present, industrysimply uses OBD for well control. Wellbores are commonly engi-neered to be overbalanced during the well construction process byselecting the mud weight such as to avoid the occurrence of largepressure differentials between the mud and the rim of rock aroundthe wellbore. The overbalanced drilling must ensure wellboreintegrity at every depth.

Even when the intention is to drill balanced, both overbalancedand underbalanced well pressures may occur. In the US offshorewells one usually must drill overbalanced due to governmentalconstraints on maintaining well barriers (apart from dedicatedUBD). But with pore pressure varying when drilling throughdifferent formations, a mudweight and casing program designed tostay overbalanced in all sections of the wellbore may be under-balanced in specific sections of the same wellbore. That is whyblowouts occur. For example, the blow out frequency for explorationand development drilling in the US Gulf of Mexico's outer conti-nental shelf is 2.5 per mille (1 blowout per 400 wells drilled, using31,574 wells drilled between 1981 and 2011; BOEM, 2014). TheInternational Association of Oil and Gas Producers suggests a globalfrequency for full blow outs of 1 per 2000 wells drilled (OGP, 2010).Precisely that is why it is so important to monitor and correctlycharacterize the stress pattern and stress magnitudes at each depthin the wellbore, as advocated in this study.

In what follows, we employ two pressure balance expressions,an essential distinction to separate pressure scalar effects fromphysical stress tensor effects. One pressure balance term is the fluidpressure differential, DPFLUID, which is always a scalar quantity:

DPFLUID ¼ PMUD � PPORE (2a)

The minimum mud weight for an OBD approach is generallylimited by the formation with the highest fluid pressure and all ofthe rest of the intersected formations may experience a significantoverbalance. This is primarily a well control (safety) issue thatstrives to balance the fluid pressure in the wellbore at all depths.The hydrostatic pressure of the wellbore fluid is the “primary bar-rier” between the formations fluids (potentially hydrocarbons) andthe surface (e.g., Ahmed and Meehan, 2012). Any over-pressuredreservoir fluid that has PPORE > PMUD may rise via the annulus tothe surface. Reversely, when PMUD > PPORE, drilling fluid may invadethe reservoir and could lead to reservoir damage and screen out.Dynamic pressure equilibrium will develop any depth where dril-ling and reservoir fluids meet and the local fluid pressure in theannulus (e.g., bottom hole pressure at the drill bit) is:

PFLUID/ðPMUDVANNULUS þ PPOREVRESERVOIRÞ=ðVANNULUS

þ VRESERVOIRÞ (2b)

with respective volumes of annulus and reservoir indicated in thedenominator. This expression neglects any (P,T) effects for whichsuitable expressions can be found in textbooks when applicableunder certain reservoir conditions. The pressure communicationwith the formation fluid may be restricted by low permeability(e.g., shale) or by building of filter cake (permeable formations)which cases would retard or preclude the achievement of a dy-namic pressure equilibrium described in Eq. (2b).

The second pressure balance term that should be consideredmonitors the balance of elastic stresses on the wall rock of the drillhole due to the lithostatic load, POVERBURDEN, and any counterpressure from PFLUID. The lithostatic load is relatively well con-strained POVERBURDEN¼rgzy/(1- y), with rock density, r, gravity, g,depth, z, and Poisson ratio, y. After drilling, the net radial pressuredue to elastic expansion or compression of the wellbore rim,

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003990

defined here as:

P!

NET ¼ P!

OVERBURDEN � PFLUID (2c)

In most regular drilling operations, well control aimed at well-bore integrity preservation PFLUID will be close to PMUD (as per Eq.(1b)) because the mud pressure will be kept close to PPORE. How-ever, during hydraulic fracturing, PFLUIDwill by far exceed both PPOREand subsequently P

!OVERBURDEN to make sure the tensile strength is

reached by P!

NET for fracture creation and propagation.We re-examine the stress patterns induced by P

!NET on the rim

rock of the wellbore, with and without a tectonic backgroundstress. In all cases, P

!NET represents the effective pressure exerted

on the wellbore surface with an elastic stress response of the wallrock. While PFLUID considers fluid pressure which a scalar quantity,the elastic response to P

!NET causes deviatoric stress which is a

tensor quantity. When P!

OVERBURDEN > PFLUID, rock in the rim of thewellbore dilates into the borehole and tensional radial stress de-velops (which is why P

!NET can be negative). Reversely, when PFLUID

> P!

OVERBURDEN, drilling fluid exerts a radial net pressure on theelastic wellbore resulting in compressional radial stress. Note thatthere is a sign reversal involved: when P

!NET>0, the deviatoric

stress on the wellbore is compressional (positive in this study), andfor P

!NET<0, the deviatoric stress on the wellbore is tensional

(negative in this study).The terms under and overbalanced in the remainder of this

study exclusively refer to pressure differentials in the wellbore ofeither PFLUID or P

!NET which may have only a loose connection with

applied or intended drilling methods (see Section 3.4 for furtherdiscussion). For example, underbalanced overburden pressure

Fig. 3. Central sketch shows the three principal cases of mud pressure gradient effects information fluid pressure PPORE, and lithostatic rock pressure POVERBURDEN. Top row left: Outjectory pattern around over-pressured (overbalanced) wellbore with PNET>0. The direction ot1 trajectories, and normal to the least principal stress t3. Breakout can not occur, due to unfabullet passage pressurizing (bullet proof) glass plate. Bottom row left: Inward directed netaround underbalanced wellbore with PNET<0. The direction of preferred tensional failure (around under-pressured (underbalanced) wellbores. Breakouts may occur in all radial direcfractures are superposed on radial fractures when a bullet draws vacuum upon exit of the hdirection [Photos are shareware domain].

( P!

NET<0) is the norm rather than an exception even in over-balanced drilling, whereas underbalanced pore pressure (DPFLUID <0) is the norm for underbalanced drilling.

Fig. 3 shows the definition of three basic net pressure statesstates in the wellbore adopted in this study: overbalanced,balanced and underbalanced. The fluid pressure of a formation,PPORE (also known as formation pressure or reservoir pressure) thatis sealed at the topmay correspond to the lithostatic pressure of theoverburden. The wellbore is stable and flow of any reservoir fluidinto the annulus is prevented by the counter pressure of the drillingfluid, PMUD, which in this example also matches the lithostaticpressure, P

!OVERBURDEN. In many cases, the relationship between

PFLUID and P!

OVERBURDEN may be different from the equality assumedin Fig. 3 (see later). The lower limit for fluid pressure commonlyassumed is due to a hydrostatic gradient of ~10 MPa/km burial. Theupper limit of the overburden pressure is commonly assumed dueto the lithostatic gradient of ~25 MPa/km burial (e.g., Weijermars,1998).

3.2. Stress around non-balanced wellbore

The state of stress around a circular vertical borehole of radius, a,solely subject to a net pressure P

!NET is radial symmetric (for con-

venience P!

NET and P!

OVERBURDEN are simply denoted as PNET andPOVERBURDEN in what follows). The stress field can be described by asimple stress function, f1, in a 2-D cylindrical reference frame (r,q),oriented normal to the wellbore axis (Weijermars, 2011):

f1ðr; qÞ ¼ PNETa2 lnr (3)

wellbores: (1) overbalanced, (2) balanced, and (3) underbalanced. Mud pressure PMUD,ward directed net pressure corresponds to “explosion mode”. Top Middle: Stress tra-f preferred tensional failure (incipient fracs) are indicated by black spikes parallel to thevorable shear stress orientations. Top Right:Multiple radial tension fractures formed bypressure corresponds to “implosion mode”. Bottom Middle: Stress trajectory patternssolid black circles) follows concentric rings, due to a reversal of the principal stressestions (red cusps), due to favorable shear stress orientations. Bottom Right: Concentricole in an ordinary glass plate, aided by elastic rebound of the earlier compressed radial

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 991

The two normal and symmetric shear stresses follow from dif-ferentiation of the stress function according to Eq. (4a to c):

tr ¼ ð1=rÞðvf=vrÞ þ�1.r2��

v2f=vq2�

(4a)

tq ¼ �PNET ða=rÞ2 (4b)

trq ¼ �ðv=vrÞð1=rÞðvf=vqÞ (4c)

Application to Eq. (3) yields (valid only for r�a):

tr ¼ PNET ða=rÞ2 (5a)

tq ¼ �PNET ða=rÞ2 (5b)

trq ¼ 0 (5c)

Radial tension fractures form when the tangential (hoop) stressis tensional and reaches the rock's tensile strength, tCT, according totq ¼ tCT. Typically, tCT ranges between 4 and 19MPa (Molenda et al.,2013). The stress pattern and diagnostic failure mode of suchoverbalanced wellbores is included in Fig. 3 (top row). In over-balanced wellbores, radial tension fractures may form with arbi-trary geographical orientation (determined by perturbations of therock continuum, after which the stress cage/stress shadow effectcommonly results in bi-wing tension cracks, but not necessarily sowhen the stress pattern is perfectly axial symmetric). Fig. 4b il-lustrates a natural case of magmatic dike intrusion due to over-pressure in 3 radial directions different from the bi-wing failuremode commonly assumed.

Consequent to principal stress orientations in underbalancedwellbores without a tectonic background stress (Fig. 3, centralimage bottom row), breakouts can form, but no radial tensionfractures. Tension failure may occur along the tangential direction,in concentric rings instead. Such concentric tension fracturesdevelop when the radial stress, tr, is tensional and reaches therock's critical tensile strength, tCT, according to tr ¼ tCT. The con-dition for shear failure is given by trq ¼ tCS, with critical shearstrength tCS. Unlike the tensile strength, tCT, the shear strength, tCS,varies greatly with the fluid pressure (seeWeijermars,1998). Fig. 4dillustrates multiple rings of concentric tension failure due to under-pressure in a central 107 m deep shaft by nuclear implosion.

3.3. Recommendations for this subsection

Two different modes of tension failure may occur around ver-tical wellbores with radial symmetric stress fields. When farefieldstress is absent, net pressure on the wellbore induces radial sym-metric stress patterns (Figs. 3 and 4). When fluid pressure in thewellbore is smaller than the lithostatic pressure, concentric rings oftension failure may develop when the wellbore dilates inward atdepths where radial tension reaches the critical tensile strength ofrocks (Fig. 3, bottom row and Fig. 4d). In contrast, radial tensionfractures may develop at depths where the fluid pressure is higherthan the elastic tension due to the overburden pressure [Fig. 3, topand Fig. 4b; Eq. (2c)].

The possibility of concentric spalling, breakouts and wellborecollapse due to underbalancing the overburden pressure in thewellbore is well known, which is why slightly overbalanced drillingis the norm. OBDmay lead to PNET reaching the tensional strength ofthe rock, which creates radial tension fractures in arbitrary di-rections. For example, such fractures may be 120�apart as is thecase in the magmatic dike intrusion fed by a central magma pipepreserved after exhumation by uplift and now known as Ship Rock

(Fig. 4b). In the case of a shattering glass plate some 18 radialfractures developed, each about 20� apart (Fig. 3, top row). When abiaxial tectonic stress field is superposed on thewellbore, the stresstrajectory pattern becomes anisotropic, which is when so-called bi-winged tension fractures (180� apart) are most likely to develop(see Section 4).

One should realize that along a single wellbore, in theory, theradial tension failure of Fig. 4b and concentric failure of Fig. 4d mayeach occur and alternate in subsequent layers, due to variations inoverburden pressure associated with different Poisson ratios. Theradial tension in an empty borehole increases with the lithostaticload t3 ¼ P

!OVERBURDEN ¼ rgzy/(1-y), and for v ¼ 0.45, t3 ¼ 0.82rgz.

However, Poisson ratios vary significantly per rock type and there-fore POVERBURDENwill vary accordingly, whichmay render PNET eithersmaller or larger than zero along individual sections of thewellbore[applying Eq. (2c)] with fluid pressure balanced at some sections(Fig. 5). For the cases illustrated in Figs. 3 and 4 the “regional back-ground stress” is negligible. For such cases the pre-drilling estab-lishment of the safe mud weight window suffices with only onevertical profile which will be valid for any radial direction from thewellbore axis outward. A practical example is shown in Fig. 6,highlighting that the tensile fracture gradient is steeper in forma-tions with lower Poisson ratios (assuming equal tensile strength,which in practice also varies with rock type and must be accountedfor where possible when balancing the wellbore fluid). Tensionfractures formwhen PNET > 0, breakouts formwhen PNET < 0.

Notably, it has been argued that inweak formations the collapselimit may differ from the pore pressure, for example, by addingwellbore strenghtening materials that stimulate casing smearduring drilling-with-casing and casing-while-drillling (Van Oortet al., 2009; Razavi et al., 2016).

3.4. Use of underbalanced and overbalanced terminology

Industry commonly relates the terms underbalanced and over-balanced to mud weight versus pore pressure. This is arguably anoversimplified usage, because the mud weight needs to control thewell integrity which is not only affected by pore pressure but alsoby the far field-stres (if any). The degree of overbalance required tomaintain (time-dependent) borehole stability depends upon (1)the native stress and pore pressure state, (2) mechanical propertiesof the formation (here assumed isotropic elastic), and (3) theorientation of the wellbore (here assumed parallel to one of theprincipal stresses). Generally, the casing program is designed tomaintain overbalance in each wellbore section (Fig. 6). However,the analysis of Section 3.1 shows that the parameters relevant forstability in the wellbore relate to the net pressure on the wellbore(Eq. (2c)). This is well understood in industry, but may not alwaysprevent underbalancing from occurring at a particular level in awellbore designed to remain overbalanced at all sections. This iswhy in this article the terms overbalanced and underbalanced, arereferred to as design terms when used in conjunction with “dril-ling” as in underbalanced drilling (UBD) and overbalanced drilling(OBD). Separately, the terms underbalanced and overbalanced areemployed to describe the pressure balance at specific depths in thewellbore. For example, in practical drilling operations an OBDprogram may still encounter, due to incomplete information,anomalous formation pressures that can render a specific section ofthe wellbore underbalanced.

4. Wellbore stresses with regional background stress

The classical expressions of Kirsch [1898; based on thecontemporary textbooks by Bach (1898) and F€oppl (1897)] havebeen used in many engineering applications of stressed plates with

Fig. 4. a: 3D view of overbalanced wellbore with radial planes of preferred tension failure normal to the least principal stress t3 . Breakouts cannot occur, due to unfavorable shearstress orientation. b: Corresponding natural fractures due to magma pressure emanating radial from a vertical magma pipe intruded through Tertiary sediments at Shiprock Peak(Rubin, 1995) situated near the remote Four Corners area where Arizona, New Mexico, Utah and Colorado join in a single point (image courtesy Google Earth). c: Underbalancedwellbore with cylindrical surfaces of preferred tension. Due to a reversal of the principal stresses, breakouts may occur in all radial directions, as stress orientations are favorable toshear failure. d: Corresponding circumferential tension fractures accompanied by ground subsidence after implosion triggered by plutonium fission in India's 1974 nuclear test(trees for scale). Feature is 200 m across and located 25 km NW of Pokhran in Rajastan, India.

Fig. 5. Mud weight is used to balance both the fluid and overburden pressures on thewellbore. Due to different pressures at different depths, the average mud density maylead to imbalances at specific depths. The aim is to maintain a stable wellbore, but theoccurrence of overbalanced and underbalanced sections cannot be excluded to occur.

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003992

holes (e.g., Muskhelishvili, 1954; Savin, 1961; Malvern, 1969;Timoshenko and Goodier, 1970; Jaeger and Cook, 1979; Delale,1984). The Kirsch (1898) equations also have been widely used toanalytically model the state of stress around boreholes, consideringa vertical hole in a homogeneous, isotropic and linear elastic rockcontinuum. Summaries and historic context of the adapted use ofKirsch equations in borehole problems have been outlined inseveral landmark studies (e.g., Jaeger et al., 2007; Zoback, 2007;Fjær et al., 2008; Zang and Stephansson, 2009). What is new hereis the introduction of non-dimensional expressions which showthat each stress function pattern can be characterized by a Fracnumber.

4.1. Equations

When a drill hole perforates a rock mass subject to a regional,tectonic stress field, the far field stress trajectories will be deflectedin the vicinity of the penetrating borehole (Fig. 7). The Airy stressfunction, f2, for a non-hydraulically pressurized, circular hole in aplate subjected to a far field stress corresponding to a horizontalcompression (t1), is (Savin, 1961, p. 106, eq. 2.150):

Fig. 6. Gray shaded area is potentially safe drilling window. Risk of wellbore instability increases when the fluid pressure nears either the lower or higher limits of the safe drillingenvelope. Toward the lower limit, risk of underbalancing (PNET < 0) increases; a condition which may trigger kicks, spalling, breakout and washout. Toward the upper limit, risk ofoverpressure rises (PNET > 0), which is favorable to tensile failure and lost circulation when critical strength of rock is exceeded.

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 993

f2ðr; qÞ ¼ þðt1=4Þr2 � ðt1=2Þa2 lnðrÞ �hðt1=4Þr2 � ðt1=2Þa2

þ ðt1=4Þ�a4

.r2� i

cos 2q

(6)

This assumes zero hydrostatic pressure on the walls of the holeand assumes deviatoric stresses are solely due to regional tectonics.Differentiation of Eq. (6) yields stress components similar to theKirsch (1898) equations (c.f., Timoshenko and Goodier, 1970, eqs.61, p. 91) commonly applied to describe the stress pattern nearboreholes in connection with breakout studies (Gough and Bell,1982; Zheng et al., 1989). In an alternative approach, the stressaround a free hole subjected to far field stress has been discussed byJaeger and Cook (1979) in terms of a complex variable function dueto Muskhelishvili (1954). General discussions of borehole stressesare found in Malvern (1969) and recent textbooks (Zoback, 2007;Zang and Stephansson, 2009).

Inwellbores with both a net pressure and a tectonic backgroundstress present, the superposition of two stress functions [f1 for thenet pressure induced stress field (Eq. (3)), and f2 for the tectonicstress (Eq. (6))] can be used to reveal the total stress state:

fTOTALðr; qÞ ¼ f1ðr; qÞ þ f2ðr; qÞ ¼ PNETa2 lnðrÞ þ ðt1=4Þr2

� ðt1=2Þa2 lnðrÞ �hðt1=4Þr2 � ðt1=2Þa2

þ ðt1=4Þ�a4

.r2� i

cos 2q

(7)

The magnitude of elements of the stress tensor is obtained by

differentiation in cylindrical coordinates (valid only for r � a):

tr ¼ PNET ða=rÞ2 þ ðt1=2Þhh1� ða=rÞ2

iþh1� 4ða=rÞ2

þ 3ða=rÞ4icos 2q

i(8a)

tq ¼ �PNET ða=rÞ2 þ ðt1=2Þhh1þ ða=rÞ2

i�h1þ 3ða=rÞ4

icos 2q

i

(8b)

trq ¼ �ðt1=2Þ½1� 3ða=rÞ4 þ 2ða=rÞ2 �sin 2q (8c)

Eqs. (8a-c) may be non-dimensionalized for general applicationby defining the non-dimensional stress t* (t normalized by PNET),according to:

t*r ¼ tr=PNET ¼ ða=rÞ2 þ ðt1=2PNET Þ½½1� ða=rÞ2 �þ ½1� 4ða=rÞ2 þ 3ða=rÞ4 �cos 2q �

(9a)

t*q ¼ tq=PNET ¼ �ða=rÞ2 þ ðt1=2PNET Þ½½1þ ða=rÞ2 �� ½1þ 3ða=rÞ4 �cos 2q �

(9b)

t*rq ¼ trq=PNET ¼� ðt1=2ÞPNETh1� 3ða=rÞ4 þ 2ða=rÞ2

isin 2q

(9c)

Introduction of the scaling parameter z¼t1/ PNET in cylindricalspace (r*,q) with radial coordinates normalized according to r* ¼ r/a

Fig. 7. Stress trajectory pattern around overbalanced borehole, with PNET positive and 10times the far field deviatoric stress jt1j, which means F ¼ þ10 (see later).

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003994

transforms Eqs. (9aec) to:

t�r ¼�1.r*2

�þ ð2=2Þ

hh1�

�1.r*2

� iþh1�

�4.r*2

�

þ�3.r*4

� icos 2q

i(10a)

t�q ¼ ��1.r*2

�þ ð2=2Þ

hh1þ

�1.r*2

�i�h1þ

�3.r*4

�icos 2q

i

(10b)

t�rq ¼ �ð2=2Þh1�

�3.r*4

�þ�2.r*2

� isin 2q (10c)

The scaling parameter z is the non-dimensional ratio t1/PNET,using the tectonic background stress t1 in numerator and theeffective hydraulic pressure, PNET, on the walls of the borehole, inthe denominator. The inverse of 1/z is the dimensionless Fracnumber (F; introduced in Weijermars et al., 2013a):

F ¼ 1=2 ¼ PNET=jt1j (11)

The stress trajectories pattern may now be mapped in (r*,q)-space using various ratios of hydraulic pressure PNET and regionalfar field stress t1 as expressed in the scaling factor z (Fig. 7). Theinclination b of a stress trajectory with respect to the r*-axesmay bedetermined from:

tan 2b ¼ 2trq���tr� � tq

�� (12)

with two solutions for b separated by p/2. A continuous solution ofEq. (12) outlines the stress trajectories in a spatial plane perpen-dicular to the wellbore.

4.2. Overbalanced wellbores with background stress

The transition from the Kirsch state of isotropic stress (F ¼ 0) tothe extreme case of a highly overbalanced (over-pressured) bore-hole (F¼þ50) is mapped out in Fig. 8. A range of cases for positive Fnumber conditions (0 < F <þ∞) are included with pressure isobarscolor-coded. These solutions show the variety of stress trajectorypatterns possible in overbalanced wellbores, depending upon the Fvalue. Stress trajectory patterns for positive F values occur whenPNET > 0.

When the F number becomes progressively positive, whichcorresponds to increasing positive PNET in overbalanced wellbores,then the directions of t1 and t3 (largest and smallest principalstresses) interchange near the wellbore. The effect has not beenrecognized in any earlier analysis, except for earlier companionstudies (Weijermars et al., 2013a) which first drew attention to theoccurrence of such near-wellbore stress reversals. The result of thestress reversal is that, near the wellbore, all radial stresses arecompressional, even in the direction of the far field tension stress.The isotropic points delineate the precise location at the ellipticalregion around the wellbore where principal stress reversals occurwith respect to the far-field stresses. The two aligned principalstresses t1 (compression near the wellbore) and t3 (far field ten-sion) are separated by such isotropic or neutral points (red dots).Within the region outlined by the t3-elliptical stress trajectorybetween the two neutral points, tension fractures can form only inthe radial planes. Breakout is unlikely as long as t3-trajectoriesfollow fully closed curved surfaces in the immediate vicinity of thewellbore with potential slip direction solutions precluding move-ment of such rock breakout (Fig. 9).

The formation of stress cages in overbalanced wellbores is amechanismwhere the opening of initial tension fractures (and anyproppant emplacement; Fig. 10a and b) leads to a change in thehoop stress. Prior authors stated hoop stress after stress cagingwould be an increased compression (Tovar and Bhat, 2011), which isnot supported by our results. The change of the stress state due tostress caging actually is a drop in tensional hoop stress (Fig. 10c). Thedrop in hoop stress (less tension)makes the formation of additionaltensional fractures inside the ‘stress cage’ more difficult. Conse-quent to the stress cage concept (Ito et al., 1999; Alberty andMcLean, 2004; Wang et al., 2008; Tovar and Bhat, 2011), and theearlier work by Haimson and Fairhurst (1969), tension fracturescommonly form as two diametrically opposed fracture branches,so-called “bi-wings”.

We believe bi-winged tension fractures typically form when atectonic far-field stress superimposes a bi-axial anisotropy ofstresses around the wellbore (Fig. 10a and b). However, we arguethat deviations are well possible for higher Frac numbers. Forexample, the tangential tension stresses due to the shock wave (F/ þ∞) in a previously stress free plexi-glass plate hit by a bulletmay create a starburst pattern of tension fractures (Fig. 3, top row).Likewise, tension fractures formed around vertical volcanic pipeintrusions are know to emanate outward in multiple radial di-rections (Muller, 1986; Muller and Pollard, 1977; Koenig andPollard, 1998) that are not necessarily bi-winged (Fig. 4b).Episodic injection offers a possible explanation, but we believe thatintermediate Frac numbers (0 <<F <þ50) may, in fact, generatemulti-branched radial tension fractures. Speed of injection mayplay a role as well in favoring either bi-winged or multi-branched‘starburst’ tension failure.

4.3. Underbalanced wellbores with background stress

Underbalanced drilling, open-hole completions and regularOBD operations hitting an unexpected, high-pressure formation all

Fig. 8. Stress trajectory patterns around overbalanced wellbores (PNET > 0). The ratio of PNET and far field stress t1 is positive (F > 0) for all cases shown. Color fringes are for t1normalized by PNET. Red dots are neutral points. Stress pattern solutions based on Eqs. (9a-c) and (11). (For interpretation of the references to colour in this figure legend, the readeris referred to the web version of this article.)

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 995

are examples of situations where negative Frac numbers may arise.These cases are all counter-intuitive, as we found many practicallyoriented drilling engineers primarily think in terms of over-balanced wellbore stresses. Our solutions for the transition fromthe Kirsch state of stress (F ¼ 0) to the extreme case of a highlyunderbalanced or under-pressured hole (F ¼ �50) are mapped outin Fig. 11. These solutions show the range of stress trajectory pat-terns possible in underbalancedwellbores, which occur for PNET< 0.

In underbalanced wellbore sections, fracture cages occur, whichpreclude any tension fracture initiated at the wellbore to escapeoutward from the elliptical area. The fracture cage, outlined by theregion occupied by the elliptical t1 trajectories, is an ellipse thatintersects the two neutral points (Fig. 12). The horizontal linethrough the two neutral points connects zones of principal stressreversal, with a change in the deviatoric stress sign. The compres-sional stress t1 due to the far field stress will be deflected inside thefracture cage zone. A tensional stress t3 on the horizontal line in-side the fracture cage is aligned with a compressional stress t1outside the fracture cage zone. These two principal stresses areseparated by the neutral points, where their respective magnitudesare equal.

The slip line solutions related to the principal stresses in thefracture cage (Fig. 12) are compatible with common breakout ob-servations (Prensky, 1992). The occurrence of circular tension fail-ure around underbalanced holes has been observed in laboratoryexperiments (Roest et al., 1989; Vardoukalis et al., 1988) and isknown from implosions around cavities in nuclear tests (Fig. 4d).These examples are for confining pressure and do not involve anydeviatoric background stress, which therefore corresponds to drillholes with a negative net pressure (F/ -∞). The fracture cage isoutlined by an ellipse of t1 trajectories that intersects the twoneutral points. An equation for determining the fracture cage el-lipse dimensions have been given in an earlier study (Weijermars

et al., 2013a).It is important to realize that for the fracture cage state, radial

tension fractures cannot initiate at the wellbore because tensionfractures may only propagate along the t1-trajectories. Such tensilefractures always form perpendicular to the tensile stress (t3) anddilate in the direction of t3. Underbalanced wellbores (F < 0) haveprincipal stresses t1 and t3 in the fracture cage (Fig. 12) rotated 90�

compared to those in the stress cage of overbalanced wells (F > 0;Fig. 9). Tension fractures may grow after hydraulic pressure isapplied only when an initial crack is already present normal to thewellbore (see Weijermars et al., 2013a, Fig. 10). However, suchfractures will not grow radial outward. Instead such fractures willdeflect into the direction of t1 trajectories, which from closed el-lipses as long as F < 0. Fig. 13a shows the analytical solution toexactly match a numerical simulation for a case where an incipientfracture at the wellbore is respectively shorter and longer than thedistance to the fracture cage rim. The short fracture stays trapped inthe fracture cage when hydraulically loaded (Fracture 1, Fig. 13a).The short fracture of Fig. 13a may only propagate further by curlingalong the t1-trajectories, because tensile fractures always formperpendicular to the tensile stress (t3) and open in the direction of t3. The longer fracture extends long enough to have its fracture tipoutside the fracture cage and can propagate further outward intothe rock matrix to align with the far field t1 trajectories (Fracture 2,Fig. 13a). In fact, outside the fracture cage region a stress cage effectmay again occur when the hydraulic fracture escapes the fracturecage (Fig.13b). However, without hydraulic loading of any such pre-existing cracks, wellbore sections that are underbalanced (F < 0)will typically develop breakouts when the critical shear strength isreached (Figs. 12 and 13c,d). Concentric spalling is likely to occur insuch underbalanced wellbore sections when the critical tensilestrength is reached by the radial tension stress (e.g., Fig. 3 bottom;Fig. 4d).

Fig. 9. Analytical stress trajectory solution around overbalanced wellbore sections(F ¼ þ10). Neutral points (red dots) act as points of principal stress reversal: far-field t3becomes t1 when continuing straight along the path of the external t3 trajectory to theinternal t1 trajectory that meet in each of the two neutral points. The neutral pointsalso define the outline of stress cage region (yellow shade) with the long axis coin-cident with a t1 trajectory. The outline of the elliptical stress cage itself is entirely madeup of an elliptical t3 trajectory that intersects the two neutral points. Inside theelliptical stress cage all radial stresses are compression and largely coincident with t1;all hoop stresses are tensional and always close to t3 trajectories. The stress cageconditions imply only radial tension fractures can initiate and will propagate along t1trajectories, normal to t3 trajectories. Hypothetical breakout zone (pink shade) cannotexist, because the slip direction along the potential shear failure lines [dashed bluelines; Eq. (11)] are incompatible with such breakout. (For interpretation of the refer-ences to colour in this figure legend, the reader is referred to the web version of thisarticle.)

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003996

4.4. Key conclusions and practical use of F-number

The Fracture number is diagnostic for, and can be used to,concisely summarize, key features of the stress trajectory patternsaround vertical wellbores aligned with the stress invariants (i.e., theprincipal stresses). The following key features are concluded basedupon our analytical models.

1. Stress induced near overbalanced wellbores will be 2D radialsymmetric (3D axi-symmetric) around the wellbore when a far-field stress is absent F / þ∞ (Section 3).

2. When a far-field stress is present, the stress pattern becomesprogressively anisotropic, reflecting the relative influence of thewellbore fluid pressure and the far-field stress normal to thewellbore on the near wellbore stress. Overbalanced wellboreswith far field stress are characterized by þ∞ > F > 0 (Section 4).

3. Stress cages may develop at the wellbore proper for all caseswhere þ∞ > F > 0. Radial tension fractures are likely to be 180�

apart only when F / 0, and multiple radial fractures may formwhen F >>0.

4. Stress trajectory patterns for underbalanced wellbores appearidentical to those for overbalanced wellbores but are different inthat t1 and t3-trajectories are exchanged. Negative net pressureon the wellbore will build radial tensile stress and hoop stresson the wellbore rim is compressive. Underbalanced wellboreswith far-field stress can be scaled by �∞ < F < 0. When a far-field stress is absent F / �∞ (Section 3).

5. Radial tension fractures cannot initiate from the wellbore rimwhen F < 0. Instead concentric tension fractures may initiate(spalling) when the radial extension reaches the critical tensionstrength and shear failure is imminent too. Fracture cages mayoccur and any high angle fracture at the wellbore will propagatestaying trapped inside the fracture cage unless its initial lengthis long enough to reach into the stress cage zone beyond thefracture cage zone (Fig. 13a and b). The occurrence of breakoutsis typical for underbalanced sections of a wellbore, because thedirection of the shear stress on potential slip lines allows frag-ments to slide out into the wellbore (Fig. 12).

For determination of the F-number at any level in a particularborehole, two parameters need to be constrained: the far-fieldstress (fixed by the native tectonic state of the rock), which canbe estimated by a variety of practical methods (not discussed here,but see summary in Zang and Stephansson, 2009). For optimalwellbore stability, the second parameter, fluid pressure is estimatedvia the mud pressure in the wellbore to control the F-number suchas to avoid the occurrence of both huge positive and large values(i.e. �100 < F < þ100). For hydraulic fracking operations, the F-number will progressively approach F / þ∞, and radial tensionfractures will develop, not all necessarily bi-winged, as arguedabove.

5. Frac number nomogram

A new plot introduced by Weijermars et al. (2013a) succinctlycharacterizes the stress trajectory patterns resulting from theinteraction between induced, effective borehole pressures andnatural tectonic stresses in the host rock near a wellbore (Fig. 14).This plot is based on the non-dimensional solutions of the stressfunction in Eqs. (9aec), scaled by the Frac number as a singledimensionless number [F ¼ 1/z, Eq. (10)]. The nomogram of Fig. 14visualizes the continuous range of stress trajectories around over-balanced as well as underbalanced boreholes. Stress trajectorypatterns remain identical along slope lines of constant F-value.Overbalanced wellbores are characterized by positive Fracnumbers. Underbalanced wellbores are characterized by negativeFrac numbers. All F-lines converge on the origin, where PNET¼ 0 andt1¼0, for which the borehole is completely stress free and no stresstrajectories exist. Moving outward from the origin along an F-line ofconstant slope, a stress trajectory pattern immediately will appearas characterized by the particular F-value. The absolute magnitudeof the principal stresses increases when moving away from theorigin, but the stress trajectory pattern for that F-value remainsunchanged.

The following five characteristic cases can be distinguished,based on the value of the Frac number (Fig. 14):

1. F is equal to or close to zero (F / 0): This situation with PNETclose to zero occurs in shallow open holes, and in deeper drillholes that are perfectly balanced. A tectonic background stressmay or may not be present. If the borehole stress is dominatedby a far field stress and PNET/ 0, the classical Kirsch situationoccurs, and tension failure can occur (Fig. 10).

2. F is very large and positive ( þ 100 < F � þ ∞): This impliesnear wellbore stresses due to wellbore pressure act largely

Fig. 10. a: Preferential tension failure in overbalanced wellbore sections and the relation with the stress cage orientation (see Figs. 8 and 9 for details). b: Opening of a radial tensionfracture in the stress cage will lower the tension of the hoop stress. c: Tangential an radial stress magnitudes prior to (light shaded) and after (dark shaded) tension fractureformation. The hoop tension is lowered by the opening of the tension fracture. All curves in light-gray-shaded region shift into the dark-gray-shaded region near the wellbore.

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 997

unperturbed by the tectonic far field stress on the wellbore. Tokeep F positive and very large, the net pressure on the innerwellbore needs to counteract and exceed the compounded ef-fects of the lithostatic pressure and any far-field stress. If thecritical tensile strength is reached, stress cage effects favor bi-winged fractures at lower positive Frac numbers, but mayinclude multiple radial cracks at higher Frac numbers. This couldbe especially the case during hydraulic fracturing with plug-and-perf fracture initiation at multiple angles to thewellbore. We cannot excludemore than one set of bi-winged tension fractures willdevelop, particularly when the F-value becomes larger. Forexample, multiple radial tension fractures formed in a pressurewave around a bullet hitting a plexiglass plate (Fig. 3, top row),and around overpressured volcanic pipes (Fig. 4b).

3. F is positive (ranging from 0< F < þ 100): The stress trajectorypatterns arising around overbalanced wellbores with Frac

numbers in this range are increasingly affected by an anisotropicfar-field stress superposed on the radial compression due to thefluid pressure. Two practical situations can be distinguished:those where wellbore stability is aimed for, in which case F willbe kept close to 0; and hydraulic fracturing of production sec-tions, in which case F may reach any value in the range 0 < F <þ100. The stress cage effect is significant: all radial tensionfractures tend to curve toward the direction of largest principalstress and normal to the minimum principal stress (Fig. 15a).This has been independently confirmed in a fully coupled fluidflow and fracture numerical model (Olson and Wu, 2012).Breakout by shear failure is unlikely for positive Frac numbers,because the direction of the shear stress is unfavorable formovement along the slip lines (Fig. 9).

4. F is negative (ranging from ¡100 < F < 0): These conditionsmay arise when the wellbore becomes vastly underbalanced

Fig. 11. Stress trajectory patterns around underbalanced wellbores (PNET < 0). The F number (ratio of PNET and far field stressjt1j) is negative for all wellbores shown. Color fringes arefor t3 normalized by PNET. Red dots are neutral or isotropic points and dotted ellipse gives the fracture cage outline. Stress pattern solutions based on Eqs. (9aec) and (11). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 12. Analytical stress trajectory solution for underbalanced wellbore sections (Eqs.9aec; F ¼ �10). Fracture cage (blue shade) outlines region where radial fracturescannot form (for details see Fig. 13) and is outlined by the largest elliptical t1 trajectoryrunning through the two neutral points. Breakouts (pink zone) can occur because theslip lines and relative sense of shear are compatible with such motion (Eq. (11)). (Forinterpretation of the references to colour in this figure legend, the reader is referred tothe web version of this article.)

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003998

due to the drill bit entering a highly over-pressured formationand takes a kick. The stress pattern for underbalanced wellboresincludes a ’fracture cage’, an elliptical region where the hoopstress coincides with the largest principal stress axis (Fig. 15b).Tension fractures cannot escape from this region, unless initialcracks are long enough to reach beyond the zone of the fracturecage. Alternatively, fluid pressure rings may develop outwardand thus transfer hydro-pressure to the rim of the ‘fracture cage’,

fromwhere tension joints can form normal to the least principalstress axis (Weijermars et al., 2013a). Although the realm ofstress cages developing near the wellbore generally occurs forF > 0, it can be seen from the icons of stress trajectory solutionsincluded in Fig. 12 that stress caging may still occur for F<0, butthe stress caging region lies outside the central region wherefracture caging prevails (Fig. 13b).

5. F is very large and negative (-∞≤ F <¡100): The net pressureon the wellbore acts inward and may cause cavitation of thewellbore (Fig. 15b). This case appears in highly underbalancedboreholes, commonly caused by unexpectedly high formationpressures that are not adequately compensated for by theweight of the drill-mud column in the borehole. The influence ofthe far-field stress is relatively minor because either PFLUID >> t1or t1 / 0. The occurrence of concentric spalling (or exfoliation)by tension failure near over-pressured circular holes has beenreported before in laboratory experiments (Roest et al., 1989;Vardoukalis et al., 1988). Slabbing is a similar process reportedfrom deep tunnels and shafts (Durrheim et al., 1995; Ortlepp,2001). For F / ¡∞, stress caging is impossible, because thefull region of rock near and further away from the wellbore isthen occupied by the potential fracture caging domain. Thefracture cage dominates and precludes the radial propagation ofany fractures (Fig. 13aec). The concentric tension fracturesformed in the depressurized wake of the bullet passing througha glass plate exemplifies the swing from a positive F number to anegative F number (Fig. 3, bottom). Fig. 4d shows anotherexample of concentric tension fractures formed by implosion ofa man-made central cavity.

6. Discussion

6.1. Real-time frac number

Detailed stress trajectory analysis and concise characterization

Fig. 13. a: Tension fracture 1 (red) is too short to escape from the fracture cage formedby elliptical t1 - stress trajectories around the wellbore and can only propagate byopening up along these trajectories. Fracture 2 is long enough to escape the fracturecage and grows further outward in the direction of the maximum far-field stress.Borehole is underbalanced with negative net pressure 1.66 times the compressionaldeviatoric stress (t1), so that F ¼ �1.66. Isobar contour interval is 0.3 (t3/PNET). Detailsof the numerical simulation are given in Weijermars et al. (2013a) and Zhang et al.(2011). b: Principal stresses inside the fracture cage have appropriate orientation toform breakouts (red cusps) by shear failure. c: Example of breakout from borehole indeep South African gold mine. Surface is to the left (after Maury, 2006). d: Fragments oftypical breakout splints coming to the surface with drilling fluid (after Osisanya, 2012).(For interpretation of the references to colour in this figure legend, the reader isreferred to the web version of this article.)

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 999

of such patterns by the Frac number are useful concepts forimproved understanding of the physical conditions required inwellbores for the occurrence of tension cracks and breakouts.Traditional representation of breakouts and tension cracks with

respect to principal stresses (Fig. 16) is consistent with solutions inthis study (Fig. 15a and b). However, the present analysis demon-strates that breakouts and tension cracks are unlikely to formsimultaneous at the same wellbore level because tension cracksrequire overbalancing and breakouts require underbalancing. Thekey parameter is the Frac number and its sign is determined by thenet pressure on the wellbore: when negative (F < 0) breakouts andconcentric spalling are likely, but radial tension failure cannotoccur; when positive (F > 0) no breakouts can form, but radialtension failure is likely. Fig. 16 shows the coeval occurrence oftension failure and breakouts, which may occur when F ¼ 0, whichis the Kirsch standard solution (only a far field stress but no netpressure in the borehole). In practice, the dominant state of awellbore is determined by OBD or UBD well designs. Coevalbreakouts and tension fractures are therefore less likely to apply toan entire drill hole because much of the wellbore section is morelikely than not either overbalanced or underbalanced most of thetime.

Assuming a situation with incomplete fluid pressure informa-tion, tension fracks and breakouts may occur in the same wellbore,albeit each at different levels. Fig. 17 shows a synthetic cross-section with formation pore pressures varying such that a largerange of fluid pressures, PFLUID, may exist at different levels in thewellbore, which implies the full range of Frac numbers may beencountered within a single borehole. The range of subsurfaceconditions encountered when drilling in extreme subsurface en-vironments makes wellbore balancing a challenging process. Dril-ling and well control would merit from the inclusion of diagnosticreal-time Frac numbers and stress trajectory visualizations in pre-drilling planning and during drilling. Because the full theoreticalrange of Frac numbers may occur in any drilled wellbore, using Fracnumbers to characterize the state of stress around wellbores iscompact, informative and suitable for taking real time decisions onwell control interventions. Visualization of the associated stresstrajectories is extremely useful for predicting the likely location ofhydraulic fracture initiation and likely directions of their propaga-tion. The Frac number can therefore also contribute to the opti-mization of well design for fracture stimulation, which is muchneeded to enhance worldwide production.volumes from uncon-ventional oil and gas fields.

6.2. Model limitations

Four native states of rock petrophysics and geomechanicalproperties may further modify the mode of fracture developmentaround wellbores. First, poro-elasticity could dampen stress con-centrations, which are generally lowered by up to 50% in stresspeak regions of such poro-elastic media, as compared to elasticmedia (cf. Nikolinakou et al., 2011). Second, many rock formationstargeted for unconventional hydrocarbon production are jointedand faulted (Engelder and Lash, 2008). Such discrete surfaces willattract stress concentrations when drilled by a wellbore. For highlyfractured formations the modeling of hydraulic fracture develop-ment is advanced in discrete finite network models (Dershowitzet al., 2011) and discrete element models (Pettitt et al., 2011).Third, many rocks with a diagenetic fabric like shales, platy silts andlaminated limestones may have anisotropic elastic moduli analo-gous to their viscous flow anisotropy behavior for rock below thebrittle-ductile transition (Weijermars, 1998). Such anisotropy canbe incorporated in the closed form stress function solutions pre-sented here (possible future work). Fourth, special care must betaken when drilling through salt formations, which unlike othersedimentary rock types behave visco-elastic rather than poro-elastic (Jackson and Hudec, 2016). On the instant of penetrationby the drill bit, the stress patterns analyzed here will apply to

Fig. 14. Stress trajectory pattern solutions for the full range of Frac numbers (-∞ < F < þ∞). Tension joints (black lines) in a vertical borehole are formed perpendicular to the leastprincipal stress axis of total tectonic stress, t3, and parallel to the t1 and t2 (out of plane). Vertical and horizontal axes are scaled differently to highlight sensitivity of stress patternsfor F values in the ranges �100 > F >þ100 (after Weijermars et al., 2013a).

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e10031000

wellbores traversing vertical salt sections. However, elastic stressesdissipate by crystalloplastic creep after passage of the Maxwellrelaxation time (quantified in an earlier study; Weijermars et al.2013b). The elastic stresses induced by drilling are then replacedby drag on the cemented drillhole due to regional creep of the saltor ductile closure of the wellbore (when still uncased). Methods forthe analysis of such stresses and displacement of wellbores in salthave been outlined systematically in a series of comprehensivestudies dedicated to wellbore stability analysis in salt sections (seereferences in Weijermars et al., 2013b).

Fig. 15. a: Stress pattern solution for F ¼ þ10. The likely directions of tension failure are mdirections of tension failure are marked by the blackened, concentric ellipses, which outlisections are considered horizontal map view, the cases shown are for a vertical wellbore inwellbore, then the cases apply to a collision belt (Fig. 1b). (For interpretation of the reference

7. Conclusions

This study visualizes the stress trajectories around cylindricalwellbores aligned with one of the principal stress directions suchthat the two other principal stresses act perpendicular to thewellbore. When the wellbore is also loaded by a drilling fluid, atdepth develops a certain net pressure (PNET) on the welllbore due tothe balance of mud pressure with that of native fluids in the rockpenetrated by the drillhole. Although stress trajectories are notroutinely mapped by any commercial package available for

arked by the blackened curvi-linears. b: Stress pattern solution for F ¼ �10. The likelyne a ‘fracture cage’. Possible breakouts by shear failure are marked with red cusps. Ifa strike-slip basin (Fig. 1c). If sections are considered vertical planes across a horizontals to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 16. Traditional image of borehole breakout and tension fracture due to hydraulicfracturing. Breakouts (red cusps) are always normal to the maximum stress directionand highlight the locations where maximum shear stresses occur. Tension fractures(black fissures) form normal to the direction of the least principal stress. Shown hereare the deviatoric stress arrows. (For interpretation of the references to colour in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 17. (a) Geological cross-section of hypothetical drilling operation in sedimentary sequensolutions of the stress field for the subsurface conditions in layer 3 models cases with F ¼ 0. (-∞ < F < þ∞. PNET is the effective hydraulic pressure on the wellbore, and t1, t2, and t3 arefractures (black lines and curves) and breakouts (red cusps) are highlighted. Stresses are vlegend, the reader is referred to the web version of this article.)

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e1003 1001

monitoring borehole stresses, such patterns are useful for under-standing and visualizing common borehole failure phenomenasuch as washout, splintering, breakout, ballooning and hydraulicfracturing, including associated features such as lost circulation andborehole collapse. A set of analytical equations derived from asuitable pair of stress functions accurately models the initiation ofhydraulically induced stress patterns on a wellbore e with andwithout a regional background stress. Our results are generalizedby introducing the dimensionless Frac number (F). The Frac numberis a concise quantitative measure to characterize stress patternsaround wellbores. Frac numbers can be diagnostic for the occur-rence of spalling, radial failure and shear breakouts. The typical F-cases outlined may be helpful in well design and hydraulic fractureengineering in the worldwide emerging development of uncon-ventional oil and gas fields. Improving the well productivity andlowering the cost of fracture stimulation can help to postpone thedepletion of fossil fuels.

Patent

A U.S. provisional patent (EFS-ID13032284; Application no.61660622; conf. no. 3115 Attorney docket number PA6078PRV) wasfiled 15 June 2012 covering “Algorithms for supreme control ofwellbore pressures and stresses to optimize oil and gas extraction”.

ce in strike-slip basin (Fig. 1c) with highly variable formation pressures. (b) The Kirschc) The stress function solutions of this study models the full range of stress patterns forthe principal deviatoric stresses in cylindrical coordinates (r, q). The potential tensionalid for uncased wellbore. (For interpretation of the references to colour in this figure

R. Weijermars / Journal of Natural Gas Science and Engineering 36 (2016) 986e10031002

Acknowledgements

Dan Schultz-Ela is thanked for early discussions and coding toproduce reliable trajectory plots. Responsibility for the contentresides entirely with the author.

Nomenclature

Notation Used

Physical Quantity Symbol Justification & Meaning

Total Stress s Includes pressure components (when absent, s ¼ t).Deviatoric Stress t Effective stress, excludes the confining pressure component.Principal Total Stress s1 s2 s3 Invariants for coordinate system used, these are the real

physical total stresses independent of tensor orientation.Principal Deviatoric Stress t1 t2 t3 Invariants for coordinate system used, these are the real

physical deviatoric stresses independent of tensororientation.

Far Field StressTotalDeviatoric s1 s2 s3 t1 t2 t3 ‘Far field’ principal stress values for both total and deviatoricforms.

Net Pressure on wellbore P!

NET , PNET Scalar pressure on the wall rock surface of the wellbore PNET

induces a normal stress equal to P!

NET .Compressive Stress Sign Positive Compressive stress and confining pressure is positive.

Extensional stress and expansive pressure is negative.Largest Absolute Deviatoric Stress jt1j Concurs with applied geoscientist use of maximum

compressive stress as s1, even when compressive stress istaken negative as in our study.

Frac Number P!

NET /jt1j Uniquely scales the stress trajectory pattern around thewellbore based on non-dimensional stress function

solutions and deviatoric stress contributions of P!

NET and t1.

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