Mapping stress trajectories and width of the stress-perturbation zonenear a cylindrical wellbore

Ruud Weijermars a,b,n

a Department of Geoscience & Engineering, Delft University of Technology, Stevinweg 1, Delft 2628CN, The Netherlandsb Alboran Energy Strategy Consultants, Delft, The Netherlands

a r t i c l e i n f o

Article history:Received 21 August 2012Received in revised form13 March 2013Accepted 12 August 2013Available online 21 September 2013

Keywords:Wellbore stressStress trajectoriesMud pressureStress-perturbation zoneBalanced boreholes

a b s t r a c t

This study reviews the analytical descriptions for stress characterization around balanced and unba-lanced drill holes. The stress-trajectory patterns around cylindrical wellbores are visualized for a range oftypical physical conditions. The effects of variations in far-field stress, boundary conditions, wellborefluid pressure, and formation pressure are systematically outlined. Axially symmetric and asymmetricfar-field stresses and their interaction with various wellbore pressures are quantified in diagrams scaledfor universal use. The stress-perturbation zone, the region around the wellbore that is affected by a stressperturbation due to the presence of the wellbore, is delineated. Rules are formulated for practicalapplication in wellbore-balancing studies and wellbore-stability analysis. These rules are useful forapplication in drilling activities aimed at the safe and effective extraction of energy resources(geothermal heat, oil, wet gas, dry gas).

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1. Introduction

Stresses around wellbores in rocks cannot be seen directly, butcan be monitored during drilling operations and in experimentalfield-studies, as concisely reviewed in Ref. [1]. Present-day well-bore-stress monitoring-systems rely heavily on automated alarmsfor managing drilling pressure and maintaining wellbore stability[2–4]. The implementation of real-time data-flow-monitoringin boreholes improves process safety and enables the identifica-tion of challenges before they become bigger problems.

Wellbore-stability-control optimization benefits from a betterinsight resulting from wellbore-stress analysis. The classical expres-sions of Kirsch [5] have been used in many engineering applicationsof stressed plates with holes (e.g., [6–11]). The Kirsch equations alsohave been widely used as a basis to analytically model the stateof stress around boreholes, assuming a homogeneous, isotropic andlinear elastic rock continuum. Summaries and historic context of theadapted use of Kirsch equations in borehole problems have beenoutlined in several landmark studies (e.g., [1,12–14]).

In currently available borehole-stress monitoring-tools, littleemphasis has been laid on the visualization of the stress trajec-tories around the wellbore. This is a gap in contemporary wellborestress analysis: although principal stress trajectories and isobar

patterns transparently and comprehensively characterize some ofthe most critical aspect of wellbore stresses, classical studies showonly a standard image of qualitatively sketched stress trajectoriesfor one special case—namely stress trajectories around an emptyborehole subjected to an uniaxial far-field boundary stress(e.g., [10]). Zoback [13] reproduced such a trajectory pattern from[10], and quotes Kirsch [5] as the source of his Fig. 6.1, but thelatter did not include any stress trajectory plot—as Fjaer et al. [14]noted, the original German language paper of Kirsch [5] is anexample of a classic paper that is probably more often citedthan read.

Recent studies [15–18] have shown a wide range of near-wellborestress-trajectory patterns may occur depending on the boundaryconditions for the far-field stress and the mud pressure applied towellbores, even when the wellbore is kept aligned with one of theprincipal stress axes to maintain maximum symmetry in theanalytical solutions. Stress algorithms are applied in software fordrilling control, automated monitoring-systems, acoustic logging-devices, and hydraulic fracturing-tools. The merits of stress-analysistechniques are: (1) risk mitigation by anticipating adverse effectsduring drilling operations (blowout, stuck drill-string, wellboredamage, lost mud-circulation, mud-barrier failure and pressure kicksthat activate blowout preventers), and (2) optimization of wellperformance after drilling (accurate hydraulic fracture placement,proper identification of far-field stresses on the basis of break-outobservations, and the basis of inversion of acoustic signals).

The present study systematically reviews the existing array ofanalytical expressions for themost relevant borehole-stress conditions,

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International Journal ofRock Mechanics & Mining Sciences

1365-1609/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijrmms.2013.08.019

n Corresponding author at: Department of Geoscience & Engineering, DelftUniversity of Technology, Stevinweg 1, Delft 2628CN, The Netherlands.

E-mail address: R.Weijermars@TUDelft.nl

International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159

visualizes the pertinent stress-trajectories, and determines thewidth of the stress-perturbation zone due to the wellbore's pre-sence. Section 2 first considers the simple cases of pressuredwellbores in radial symmetric far-field stress-regimes, illustratedwith new graphs for practical use. Section 3 revisits the stressedhole subject to a uniaxial far-field stress originally considered byKirsch [5], and its expansion to the distinct cases of wellborespenetrating rocks with a bi-axial far-field stress-state. Our syntheticwellbore simulates the effect of progressively higher wellborepressures and shows that the classical stress-trajectory pattern forthe uniaxial far-field tension described by the Kirsch equations isbut one end-member in a continuous range of stress states in theplane of view normal to the wellbore axis. The isotropic or neutralpoints may move away from the wellbore when increasingly higherpressures are applied in so-called overbalanced boreholes. Section 4provides practical rules for wellbore-stress analyses based on ouranalysis and highlights how to avoid common pitfalls. Section 5outlines our conclusions.

2. Stress fields with axially symmetric solutions

2.1. Notation used in this study

Mechanical engineers traditionally use a positive sign for tensilestress in classical material science, and geotechnical engineers oftentake compressive stress as positive. The sign convention adoptedhere takes compressional stresses as positive. Irrespective of signconvention used, the physical meaning of wellbore pressures andstresses is adequately and consistently described by our symbols,subscripts and sign convention.

Assume a wellbore is planned aligned with the intermediateprincipal stress axis s2. The boundary conditions are, using cylindricalcoordinates (R, θ) in a section normal to the future wellbore of radius‘a': sR¼s1 at R¼a and 1 for θ¼0˚; sR¼s3 at R¼a and 1 for θ¼90˚;sθ¼s3 at R¼a and1 for θ¼0˚; and sθ¼s1 at R¼a and1 for θ¼90˚.

We assume an incompressible elastic rock (constant densityassumption for validity of the continuum assumption), but thecontinuum may possess some permeability to account for thepresence of pore pressure within the formation. The unperturbedradial and tangential stresses on surfaces along the trajectory of aplanned wellbore follow from the equations for the Mohr circle ofstress (cf. [19]):

sR ¼ ½ðs1þs3Þ=2�þ½ðs1�s3Þ=2� cos 2θ ð1aÞ

sθ ¼ ½ðs1þs3Þ=2��½ðs1�s3Þ=2� cos 2θ ð1bÞ

sRθ ¼ ½ðs1�s3Þ=2� sin 2θ ð1cÞ

We have chosen the notation to depart from the traditionalapproach in borehole stress analysis, which denotes a maximumtotal stress in the horizontal plane by sH (or sHMAX or SH) and aminimum by sh (or sHMIN or Sh), and a vertical stress by sV (or SV).The reason is that in many natural situations sH and sh are notprincipal stresses at all, while Eqs. (1a)–(1c) are only valid forprincipal stresses. Our notation, using s1, s2, and s3, is arguablymore practical when matching solutions to the natural range oftectonic regimes [16]. Table 1 gives a conversion table for relatingour subscript choice to Andersonian stress regimes, amplified withtranspressional and trans-tensional cases.

In our paper, total stress is denoted by s and deviatoric stressby τ. Deviatoric stresses differ from the total stress in that isotropicstresses (pressure components), which do not contribute todeformation gradients, are taken out. The deviatoric stress canbe represented in both a tensor and principal form. Our notation isin compliance with common practice in structural geology litera-ture [20,21]. There are three principal deviatoric stresses τ1, τ2, andτ3, of which—in our analysis —one of them is assumed to bealigned with the drilling direction; this fixes the two otherprincipal stresses in the transverse direction. We use the physicaleffects of principal stresses, i.e., compression and tension, to fix thesubscripts. Compressional stresses are in our study consistentlydenoted by τ1 and tensional stresses by τ3 in compliance with theprevailing annotation in structural geology and tectonic models.In axially symmetric far-field stresses, with bi-axial stresses τ1 andτ3, τ1 may be equal to τ3, in which case they are both eithertensional or compressional stresses.

2.2. Wellbore stresses around an empty hole with far-field axiallysymmetric tension (no pressure on wellbore)

Kirsch [5] was in his days mostly interested in investigatinghow a far-field stress gives rise to stress concentrations aroundcylindrical holes in elastic thick plates (which justifies a 2Danalysis). He first assumed an axially symmetric far-field tensionalstress, τ0 (Fig. 1a). The boundary conditions are: τR¼τ0 at R¼1;τR¼0 at R¼a; τθ¼τ0 at R¼1; and τθ¼2τ0 at R¼a. These conditionsimpose a radial tension, τR, and a tangentional tension, τθ (Fig. 1b),with the magnitudes at radial distance, R, affected by the presenceof the hole with radius, a, as follows:

τR ¼ τ0½1�ða=RÞ2� ð2aÞ

τθ ¼ τ0½1þða=RÞ2� ð2bÞFor strain compatibility and stress balance, τRþτθ¼�τzz.

Eqs. (2a) and (2b) are plotted in Fig. 2a and b, which shows thatthe radial stresses, τR, completely vanish at the boundary of theempty hole (henceforth τR¼0 at R¼a). At a distance of three timesthe borehole radius a (R¼3a), τR still retains 90% of its far-fieldvalue τ0. In contrast, the tangentional stress, τθ, increases towardthe hole and reaches a maximum value of twice the far-field stress(τθ¼2τ0), which mean the so-called stress-concentration factoris 2 at R¼a. Both the tangentional and radial stresses resumenearly equal magnitudes (resp. 103% and 97%) at six times the holeradius distance (R¼6a, see Fig. 2a and b), which approachesthe outer limit of the near-wellbore zone with an unperturbedfar-field stress state. The principal stress trajectories are given bythe cobweb pattern of Fig. 1b, and stress magnitudes are scaled inFig. 2b.

2.3. Wellbore stresses due to mud load only (no far-field stress onwellbore)

The assumption of an empty borehole with zero-pressure onthe wellbore is not applicable to most practical drilling situations.

Table 1Corresponding principal total stresses and sH, sh and sV, depending on tectonicstress regime (*).

Tectonic Setting of Basin Orientation Invariant PrincipalStresses

Principal stress subscripts valid for all cases s1 s2 s3

Tectonic Setting of Basin Conversion to Spatially FixedSubscripts

Collision sH sh sVExtension sV sh sHStrike Slip sH sV shTrans-tension sV sh sHTrans-pression sH sh sV

n For deviatoric stress conversions, use τ instead of s for all symbols used inthis table.

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159 149

Instead we consider a mud-filled borehole (Fig. 3a) where thetraction exerted on the wellbore is a radial compression P

!M , equal

to the wellbore mud-pressure PM¼ρM.g.h (Fig. 3b). Pressure itself isa scalar quantity, but induces a force vector on the wellbore. Theboundary conditions are: τR¼0 at R¼1; τR ¼ P

!M at R¼a; τθ¼0

at R¼1; and τθ ¼� P!

M at R¼a. For this case, the stress equili-brium Eqs. (2a) and (2b) modify to (see [15]):

τR ¼ P!

Mða=RÞ2 ð3aÞ

τθ ¼� P!

Mða=RÞ2 ð3bÞ

The decay of stress away from such overbalanced (over-pressured)boreholes is plotted in Fig. 4a and b. The principal stress trajec-tories are given by the cobweb pattern of Fig. 3b, and stressmagnitudes are scaled in Fig. 4b.

2.4. Wellbore stress around boreholes with mud pressure andfar-field tension

The stresses around a wellbore with mud load (case of Sections2.3) as well a far-field axially symmetric tension stress (case ofSections 2.2) can be combined to give rise to the followingboundary conditions: τR¼τ0 at R¼1; τR ¼ P

!M at R¼a; τθ¼τ0

at R¼1; and τθ ¼ 2τ0� P!

M at R¼a. The stress state around thewellbore can be modeled by summing Eqs. (2a) and (2b) and Eqs.(3a) and (3b):

τR ¼ τ0½1�ða=RÞ2�þ P!

Mða=RÞ2 ¼ τ0þð P!M�τ0Þða=RÞ2 ð4aÞ

τθ ¼ τ0½1þða=RÞ2�� P!

Mða=RÞ2 ¼ τ0�ð P!M�τ0Þða=RÞ2 ð4bÞEvaluation of Eqs. (4a) and (4b) shows that when the far-field stressτ0 and the normal compression from the mud-balancing pressure

P!

M are equal in strength (but with opposite sign, P!

M ¼�τ0), the

Fig. 1. (a) Bi-axial tensional far-field stress with an empty, pressure-free hole. (b) Principal stress trajectories for the case of bi-axial far-field tension with empty pressure-free hole.

Fig. 2. (a) Stress concentration and dissipation around an uniformly stretched open hole with boundary conditions as in Fig. 1a. The far-field radial stress is nearly 100% at adistance of 10 times the hole's radius, but becomes 0 at the hole's margin. In contrast, the far-field tangential stress increases toward the hole and reaches a maximum valueof twice the far-field stress at the hole's margin. (b) Cross-sectional view of the borehole (blue core) with isobars for the radial tension stress (blue) and tangential tensionstress-isobars (red)—both are principal stresses. The radial stress dissipates toward the hole and has lost 3% of its value at 6 radii away from the hole's, and loses another 25%when only 2 radii away from the center, until it becomes 0 at the wellbore. The isobars for the tangentional stress concentrate toward the hole and the increasing stress-concentration factors, when nearing the hole, are indicated. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159150

radial stress τR becomes a compressive stress at the wellbore equalto 100% the absolute magnitude of the far-field stress |�τ0| (Fig. 5a,as compared to 0% in the absence of a wellbore pressure for the caseof Sections 2.2, Fig. 2a). Due to the mud pressure, the tangentialstress becomes an enhanced tension equal to 300% the magnitudeof the far-field tension τ0 (Fig. 5a, as compared to 200% in theabsence of a wellbore pressure, see Fig. 2a). The stress-perturbation

zone near the wellbore for the case P!

M ¼ τ0 is plotted in Fig. 5a andthe corresponding stress trajectories are plotted in Fig. 5b.

2.5. What is a balanced wellbore?

It is sometimes (wrongly) assumed in drilling operations that anyresidual stress on the wellbore resulting from a far-field stress willvanish if P

!M is appropriately dimensioned. The stress levels at the

wellbore are given by Eqs. (4a) and (4b), which for R¼a simplify to:

τR ¼ P!

M ð5aÞ

τθ ¼ 2τ0� P!

M ð5bÞ

and at R¼1 we have the unperturbed far field stress τ0¼τR¼τθ. Thisshows that for P

!M ¼�2τ0, the radial and tangentional stresses at

the wellbore will both increase (Fig. 6a) and the situationworsens forP!

M ¼�3τ0 (Fig. 6b). The conclusion is that a far-field axiallysymmetric tension-stress cannot be balanced by applying any mudpressure to the wellbore. Axially symmetric tension stress occursabove expanding salt domes. The high tangential well stress cannotbe balanced, unless a suction is applied such that P

!M ¼ 2τ0 or 3τ0.

Although suction pumps are not employed for balancing purposes inregular drilling operations, the extreme environment of a far-fieldradial tension stress may need some experimenting in this direction.Applying a mud-pressure certainly has an adverse effect on wellborestability and open-hole completion would be a better alternative.

2.6. Wellbore stress around boreholes with mud pressure and far-field compression

If we now consider a far field compression on the wellbore,such as due to lithostatic loading, we have at R¼1 a uniformcompression τR¼τθ¼ρgzν/(1�ν) with Poisson ratio v, gravityacceleration g, overburden depth z, and overburden density ρ.

Fig. 4. (a) Stress dissipation around pressured hole with boundary conditions as in Fig. 3a. A far-field stress is absent, which means both the radial and tangential stresses aresolely due to the mud pressure on the wellbore. The pressure-induced radial and tangential stresses have fully dissipated at a distance of about 10 times the hole's radius.(b) Cross-sectional view of the borehole with stress dissipation values for the radial stress isobars (compression, red) and tangential stress isobars (tension, blue)—bothprincipal stresses. The radial and tangentional stresses decrease rapidly in the direction away from the wellbore. (For interpretation of the references to color in this figurelegend, the reader is referred to the web version of this article.)

Fig. 3. (a) Pressurized wellbore with zero far-field stress. (b) Principal stress trajectories for overbalanced wellbore with zero far-field stress.

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159 151

The boundary conditions at the wellbore R¼a are τR ¼ P!

M and

τθ ¼ 2ρgzν=ð1�νÞ� P!

M . The full equations are obtained by adaptingEqs. (4a) and (4b)):

τR ¼ ½ρgzν=ð1�νÞ�½1�ða=RÞ2�þ P!

Mða=RÞ2 ¼ ½ρgzν=ð1�νÞ�þf P!M�½ρgzν=ð1�νÞ�gða=RÞ2

ð6aÞ

τθ ¼ ½ρgzν=ð1�νÞ�½1þða=RÞ2�� P!

Mða=RÞ2 ¼ ½ρgzν=ð1�νÞ�–f P!M�½ρgzν=ð1�νÞ�gða=RÞ2

ð6bÞ

If we now take P!

M ¼ ρgzν=ð1�νÞ at the wellbore R¼a, the radialand tangential stresses τR¼τθ according to Eqs. (6a) and (6b) arerendered equal to ρgzν/(1�ν ), which is the unperturbed far-fieldcompression due to the lithostatic pressure (Fig. 7).

A different situation arises when drilling into permeable rockwhere the formation fluid exerts a resultant pressure PF due tothe lithostatic load and can directly transfer this fluid pressure tothe wellbore and impact the mud pressure P

!M . This situation

seems classical, but a careful discussion is merited. Two basic casescan be distinguished, with a static wellbore net-pressure (Fig. 8a)

and a dynamic net-pressure (Fig. 8b). A static net-pressure requiresa casing or a mud cake that effectively keeps the mud fluid and theformation fluid apart (Fig. 8a). In the case of a dynamic net-pressure, the two fluids communicate and the mud pressure in thewellbore will attain a value close to the formation pressure aftersome time (Fig. 8b). If there is permeable contact between thefluid in the drillhole (due to the mud weight, PM), and theformation fluid (PF), any pressure increase from the overburdenthickness and/or tectonic forces will be communicated to thewellbore pressure.

For all cases, the wellbore net-pressure, PNET, is the unbalancedpart of the wellbore tractions induced by the pressure differential,given by:

P!

NET ¼ PM�!þ PF

�!¼ PM�PF ð7ÞPM�!

is the hydraulic traction on the wellbore due the mud-loadpressure on the wellbore, and PF

�!is the traction (with opposite

sign) due to the formation pressure. The stress-balance equationsresult in the following expressions:

τR ¼ ½ρgzν=ð1�νÞ�þfðPNET�½ρgzν=ð1�νÞ�gða=RÞ2 ð8aÞ

Fig. 6. (a) Stress dissipation around a borehole with mud pressure �2 times the far-field tension. The pressure-induced radial and tangential stresses have fully dissipated ata distance of about 10 times the hole's radius. (b) Same, but with mud pressure equal to �3 times the far-field tension.

Fig. 5. (a) Stress dissipation around borehole with mud pressure PM40 and far-field tension stress τ0 of equal strength but with opposite sign. The pressure-induced radialand tangential stresses have fully dissipated at a distance of about 10 times the hole's radius. (b) Cross-sectional view of the borehole with stress dissipation values for theradial stress isobars (tension, blue) and tangential stress isobars (compression, red), which are both principal stresses. The radial and tangentional stresses decrease rapidly inthe direction away from the wellbore. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159152

τθ ¼ ½ρgzν=ð1�νÞ�–fðPNET�½ρgzν=ð1�νÞ�gða=RÞ2 ð8bÞ

Underbalanced open-hole completions have |PM|o |PF|, which iswhy PNET will then always be negative (Fig. 8a and b, bottom row).

Consequently, the situation of Fig. 7 cannot be achieved inunderbalanced wellbores and instead the net pressure PNETbecomes a traction that contracts the wellbore surface inward,and induces a tensional normal stress at the wellbore wall(at R¼a) with the resulting radial stress τR¼PNET and τθ¼2ρgzν/(1�ν)�PNET (with PNETo0). This corresponds to the situation inFig. 5a, but with a sign reversal: radial stresses are tensional andtangential stresses are compressional. Remember that in allcommercial gas reservoirs, the average reservoir pressure mustbe greater than the wellbore pressure—this is a prerequisite for agiven amount of gas in place to be recovered. During the process ofcompletion, there may be also brief intervals of time when thewell remains underbalanced at the reservoir depth. This is whyunderbalanced well completions require a significant amount ofspecific equipment at the well site to ensure safe operations.

For balanced well completions, PM is engineered with mudloads to aim for PNET close to zero at most depths (Fig. 8a and b,middle cases), which would results again in the stress state plottedin Fig. 7. However, PF is highly variable in natural rocks and maybe difficult to balance at each specific level. Wellbores thereforeare commonly engineered to be over-balanced during the con-struction process by selecting the mud weight such as to provide apositive wellbore net-pressure in the wellbore at all depths.The minimum mud-weight is therefore limited by the highestpressure (in gradient terms) and all of the rest of the exposedformations will then experience a significant overbalance. Thisis primarily a well-control (and safety) issue. The hydrostatic

Fig. 8. (a) and (b): Wellbore pressure profiles. (a): Static cases: overbalanced (top), balanced (middle) and underbalanced (bottom). (b) Dynamic cases: overbalanced (top),balanced (middle) and underbalanced (bottom). The red lines show the pressure distribution. The borehole-fluid pressure is denoted by PM and formation pressure by PF.Static situations apply to the first instant of formation penetration by the drill and will be maintained when a filter-cake screens out the pore space or. cased wells where theformation fluid can not effectively communicate pore pressure with wellbore fluid. (For interpretation of the references to color in this figure legend, the reader is referred tothe web version of this article.)

Fig. 7. Stress dissipation around pressured borehole with mud-pressure equal tothe far-field compression stress due to the lithostatic pressure. The radial andtangential stresses induced by a well chosen mud weight maintain the lithostaticpressure and the hole is perfectly balanced.

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159 153

pressure of the wellbore fluid is the “primary barrier” between theformations fluids (potentially hydrocarbons) and the surface.

For the overbalanced sections of the wellbore, PNET will pushthe wellbore surface outward and thereby exerts a compressionstress normal to the wellbore wall. This is similar to the situationportrayed in Fig. 6a and b. The response of the host rock is elasticdistortion, until failure is initiated when the critical tensilestrength, τc, is reached by the tangential stress τθ and hydraulictensile fractures will appear in the radial direction.

3. Asymmetric stress-field solutions

3.1. Uni-axial far-field stress around an empty borehole

Kirsch [5] derived (using the contemporary textbooks by Bach[22] and Föppl [23]) a concise set of analytical expressions for theprincipal deviatoric radial stress (τR) and principal tangential stress(τθ) and shear stress (τRθ) around a hole in a thick plate subjectedto a uni-axial far-field tension stress (Fig. 9a). The boundaryconditions are: τR¼0 at R¼a and 1 for θ¼0˚; τR¼τ0 at R¼1 forθ¼90˚; τR¼0 at R¼a for θ¼90˚; τθ¼2τ0 at R¼a for θ¼0˚; τθ¼0at R¼a and 1 for θ¼90˚; τθ¼0 at R¼a for θ¼90˚. The Kirsch [5]equations for uni-axially stretched thick perforated plates havebeen widely used as a basis to analytically model the state of stressaround boreholes:

τR ¼ ðτ0=2Þf½1�ða=RÞ2�þ½1�ða=RÞ2�½1�3ða=RÞ2� cos 2θg ð9aÞ

τθ ¼ ðτ0=2Þf½1þða=RÞ2��½1þ3ða=RÞ4� cos 2θg ð9bÞ

τRθ ¼�ðτ0=2Þ½1�ða=RÞ2�½1þ3ða=RÞ2� sin 2θ ð9cÞThese are his classical expressions subsequently used in manyengineering applications (e.g., [6–11]) and adapted for use in

borehole problems (e.g., [1,12–14]. It can also be readily seen thatfor θ¼0, Eqs. (9a) and (9b) become equal to Eqs. (2a) and (2b).

The far-field stress assumption by Kirsch [5], for the thickelastic plate containing a traction-free hole under uni-axial ten-sion, τ0, implies the boundary condition at r-1 is given by τR¼τ0for θ¼01 and τθ¼0 for θ¼901 (Fig. 9a). Essentially this means thatthe far-field direction normal to the uni-axial tension is a stress-free (rollers or free-hanging) boundary.

Kirsch [5] did not plot the principal stress trajectories. For theuni-axial tension, the stress trajectories are no longer axi-symmetric and these therefore depart from the spiderwebpatterns in Section 3 made up by radial and tangential stresses.The trajectories of the principal stresses may be mapped in (R,θ)-space, using Eqs. (9a)–(9c) to determine the inclination β of anyspatial stress trajectory with respect to the R-axes from thefollowing expression (see [15]):

tan 2β¼ 2τrθ=ðτr�τθÞ ð10Þ

with two solution for β separated by π/2. A continuous solution ofEq. (10) in a particular space outlines the τ1 and τ3 stresstrajectories. Fig. 9b shows the classical case of stress trajectoriesaround an empty hole with a uni-axial far-field tension stress τ0.The stress-concentration factor is two times the far-field stress.

The magnitudes of the principal deviatoric stresses, τR and τθ,follow from Eqs. (9a)–(9c) based on knowledge of the far-fieldstress magnitude, τ0, in the location of each point determined bypolar coordinates (R, θ), with radial distance, R, drill hole withradius, a, and angular position θ. Fig. 10a shows the stress isobarmap for the uni-axial stress boundary condition. The isobars forthe reactive stress are outlined in Fig. 10b. It is clear that tensionalfailure will occur in the direction perpendicular to the tension atthe locations where the tensional stress reaches maxima of up to2 times the far field tension, at the wellbore margin along thehorizontal surface at the bottom of Fig. 10a. Kirsch [5] included

Fig. 9. (a) Boundary conditions for uni-axial far-field stress, τ0, as assumed by Kirsch [5]. The empty hole is pressure-free and the far-field long edges are free boundaries(rollers). (b) Principal stress trajectories for uni-axial far-field stress as per boundary conditions in (a). Direction of the tensional principal stress trajectories are outlinedby the blue curvi-linears. Compressional stress trajectories are aligned with the red curves. Isotropic points (black dots) on the wellbore wall are not 901 apart, unlike thebi-axial plane-stress. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159154

experimental data of plates uni-axially stretched until failure,which persistently showed the plates developed a tension fractureat either side of the holes in the predicted stress concentrationregion.

3.2. Uni-axial far-field stress with overbalanced mud pressure

The magnitude of the elements of the stress tensor for a uni-axial stress field with a borehole net pressure in polar (R, θ)—spaceis (for details see [15]):

τR ¼ PNET ða=RÞ2þðτ0=2Þf½1�ða=RÞ2�½1�4ða=RÞ2þ3ða=RÞ4� cos 2θgð11aÞ

τθ ¼�PNET ða=RÞ2þðτ0=2Þf½1þða=RÞ2��½1þ3ða=RÞ4� cos 2θg ð11bÞ

τRθ ¼�ðτ0=2Þ½1�3ða=RÞ4þ2ða=RÞ2� sin 2θ ð11cÞThe boundary conditions are: τR¼PNET at R¼a for θ¼0˚ and 90˚;τR¼0 at R¼1 for θ¼90˚; τR¼τ0 at R¼1 for θ¼0˚; τθ¼2τ0�PNET atR¼a for θ¼0˚; τθ¼�PNET at R¼a for θ¼90˚; τθ¼0 at R¼1 forθ¼90˚; τθ¼τ0 at R¼1 for θ¼0˚. Introduction the non-dimensionalstresses τnR ¼ τR=PNET , τnθ ¼ τθ=PNET , τnRθ ¼ τRθ=PNET and the non-dimensional scaling parameter F¼PNET /τ0 in polar coordinatespace (Rn, θ) with radial coordinates normalized according to Rn¼R/a transforms Eqs. (11a)–(11c) to their non-dimensional form [15]:

τnR ¼ ð1=Rn2Þ�ð1=2FÞf½1�ð1=Rn2Þ�þ½1�ð4=Rn2Þþð3=Rn4Þ� cos 2θgð12aÞ

τnθ ¼�ð1=Rn2Þ�ð1=2FÞf½1þð1=Rn2Þ��½1þð3=Rn4Þ� cos 2θg ð12bÞ

τnRθ ¼�ð1=2FÞ½1�ð3=Rn4Þþð2=Rn2Þ� sin 2θ ð12cÞThe principal stress trajectory pattern in (Rn, θ)—space is fullydetermined by the inclination β of a stress trajectory with respect tothe R-axes [15]:

tan 2β¼ 2τnRθ=ðτnR�τnθ Þ ð13Þwith the two conjugate solutions for β separated by π/2. A contin-uous solution of Eq. (13) outlines the stress trajectories in a spatialplane perpendicular to the wellbore. The principal stress trajectoriesare mapped for the continuous range of all possible ratios ofhydraulic pressure PNET and regional far field stress τ0 [and sub-stituted in the scaling factor, F, in Eqs. (12a)–(12c) and (13)].

The principal stress trajectory patterns for overbalanced well-bores—which occur when the F number becomes progressivelypositive—are mapped in Fig. 11a–d. The result is that for larger Fnumbers, all radial stresses are compressional—even in the direc-tion of the far field tension stress. The two aligned principalstresses τ1 (compression near the wellbore) and τ3 (equal to the farfield tension τ0) are separated by the isotropic or neutral points(red dots). Within the region outlined by the τ3-elliptical stresstrajectory between the two neutral points, tension fractures canform only in the radial planes and curved surfaces that followτ3-trajectories. Due to increasing positive PNET in overbalancedwellbores, the directions of τ1 and τ3 (largest and smallestprincipal stresses) interchange near the wellbore (Fig. 11b).This effect may have been overlooked in wellbore breakout studies.

These results have important implications for hydraulic fractur-ing operations: as the hydraulic pressure increases PNET increasesand therefore the F number will rapidly shoot up. The resultingstress trajectories for heavily overbalanced wellbores imply thattension fractures will open in radial directions without a clearpreferential direction.

3.3. General bi-axial far-field stress around an empty borehole

A generic solution of the force balance equation for the state ofstress in wellbores subjected to a bi-axial tectonic backgroundstress (Fig. 12a and b) can be obtained by the superposition of twostress functions (Φ1 for the far-field uni-axial stress in theX-direction), and Φ2 for the far-field uni-axial stress in theY-direction). These stress functions are simple summations of theAiry stress function Φ(R,θ) for a non-hydraulically pressurized,circular hole in a plate subjected to a uni-axial far-field stress, τ0 [7]:

ΦðR; θÞ ¼ ðτ0=4ÞR2�ðτ0=2Þa2 ln ðR=aÞþ½ðτ0=4ÞR2�ðτ0=2Þa2þðτ0=4Þða4=R2Þ� cos 2θ

ð14ÞThe stress functions Φ1(R,θ) and Φ2(R,θ) for the two perpendicularuni-axial far-field stresses τ1 and τ3 (Fig. 11a) may be summed toyield a generic bi-axial stress function:

ΦTotalðR; θÞ ¼Φ1ðR; θÞþΦ2ðR; θÞ ¼ ðτ1=4ÞR2�ðτ1=2Þa2 ln ðR=aÞþ½ðτ1=4ÞR2�ðτ1=2Þa2þðτ1=4Þða4=R2Þ� cos 2θ

þðτ3=4ÞR2�ðτ3=2Þa2 lnðR=aÞ�½ðτ3=4ÞR2�ðτ3=2Þa2þðτ3=4Þða4R2Þ�cos2θ ð15Þ

The boundary condition at r-1 is given by τR¼τ1 for θ¼01 andτR¼τ3 for θ¼901. The radial and tangential normal stresses andsymmetric shear stresses follow from differentiation of the stressfunction ΦTotal(R,θ) according to Eqs. (16a)–(16c):

τR ¼ ð1=RÞð∂Φ=∂RÞþð1=R2Þð∂2Φ=∂θ2Þ ð16aÞ

τθ ¼ ∂2Φ=∂R2 ð16bÞ

τRθ ¼�ð∂=∂RÞð1=RÞð∂Φ=∂θÞ ð16cÞ

Fig. 10. (a) Isobars for principal tension stress expressed as a percentage of the uni-axial far-field tension stress, τ0. (b) Isobars for the principal tangential stress,expressed as a percentage of the far-field tension stress, τ0. Boundary conditions areas per Fig. 9a.(according to [5]).

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159 155

Application to Eq. (15) yields the generic expressions for stressesaround a borehole subject to the conjugate far field stresses τ1 and τ3:

τR ¼ ½ðτ1þτ3Þ=2�½1�ða=RÞ2�þ½ðτ1�τ3Þ=2�½1�ða=RÞ2�½1�3ða=RÞ2Þ� cos 2θ

ð17aÞ

τθ ¼ ½ðτ1þτ3Þ=2�½1þða=RÞ2��½ðτ1�τ3Þ=2�½1þ3ða=RÞ4� cos 2θ ð17bÞ

τRθ ¼�½ðτ1�τ3Þ=2�1�ða=RÞ2½1þ3ða=RÞ2� sin 2θ ð17cÞThe magnitudes of the principal deviatoric stresses, τR and τθ, followfrom Eqs. (17a)–(17c) based on knowledge of the far-field stressmagnitude, τ1 and τ3, in the location of each point determined bypolar coordinates (R, θ), with radial distance, R, and drill hole withradius, a, and angular position θ. One can see that at the boreholeradius the radial stress vanishes, i.e., τR¼0 at (a/R)¼1. The tangentialstress-concentration factor at the wellbore at R¼a for θ¼0 isτθ¼3τ1�τ3.

Adopting a plane stress assumption, τ3¼�τ1 (at R-1), sim-plifies Eqs. (17a)–(17c) to (valid only for RZa):

τR ¼ τ1½1�4ða=RÞ2þ3ða=RÞ4� cos 2θ ð18aÞ

τθ ¼�τ1½1þ3ða=RÞ4� cos 2θ ð18bÞ

τRθ ¼�τ1½1þ2ða=RÞ2�3ða=RÞ4� sin 2θ ð18cÞThis solution differs from Eqs. (9a)–(9c), which is for a uni-axialfar-field stress and Eqs. (18a)–(18c) are for a bi-axial (plane)far-field stress. The principal stress trajectories around thepressure-free hole due to a far-field plane stress have been visualizedin Fig. 12b. The tangential stress-concentration factor at the wellboreis 4 times τ1 at R¼a for θ¼0, i.e., τθ¼4τ1. Radial tension fracturesform when the tangential (hoop) stress is tensional and exceeds therock's tensile strength, τc, according to τθZτc.

3.4. Bi-axial far-field stress with mud pressure

The balancing effect of any fluid pressure on the wellbore canbe accounted for by including in Eqs. (17a)–(17c)—analogous toexpressions (11a–11c)—the net pressure:

τR ¼ PNET ða=RÞ2þ½ðτ1þτ3Þ=2�½1�ða=RÞ2�þ½ðτ1�τ3Þ=2�½1�ða=RÞ2�½1�3ða=RÞ2� cos 2θ

ð19aÞ

T3

T1

T1

T1

T1

T1

T1

T1

F =

+1

F =

+5

T3

T3T3

T3

T3

T3

T3

T3

Isotropic Point

T1

F =

+20

F =

+10

Fig. 11. (a–d) Stress trajectory patterns around overbalanced wellbores. Formation pressure is lower than the wellbore pressure (PNET40). The ratio of PNET and far fieldstress is positive (F40) for all cases shown. Color fringes are for τ1 normalized by PNET and contour interval is 0.25 for (a) and 0.1 for (b–d). Red dots are isotropic points. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)(after [16]).

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159156

τθ ¼�PNET ða=RÞ2þ½ðτ1þτ3Þ=2�½1þða=RÞ2��½ðτ1�τ3Þ=2�½1þ3ða=RÞ4� cos 2θ

ð19bÞ

τRθ ¼�½ðτ1�τ3Þ=2�½1�ða=RÞ2�½1þ3ða=RÞ2� sin 2θ ð19cÞThe implied boundary conditions for the stresses at the boreholeare: τR¼PNET at R¼a; τR¼τ1 at R¼1 for θ¼0˚; τR¼τ3 at R¼1 forθ¼90˚; τθ¼3τ1�τ3�PNET at R¼a for θ¼0˚; τθ¼-PNET at R¼a forθ¼90˚; τθ¼τ1 at R¼1 for θ¼90˚; τθ¼τ3 at R¼1 for θ¼0˚.

Someone with ordinary skill in geo-mechanics can prove that thecase for net pressured wellbores described by Eqs. (19a)–(19c) becomeidentical to the generally accepted summary format of Zang andStephansson [1], which is based on the work of Scheidegger [24]:

sRR ¼ ð1�ρ2Þ½ðSHþShÞ=2�þ½1�4ρ2þ3ρ4�½ðSH�ShÞ=2� cos 2φþΔpρ2

ð20aÞ

sφφ ¼ ð1þρ2Þ½ðSHþShÞ=2��½1þ3ρ4�½ðSH�ShÞ=2� cos 2φ�Δpρ2 ð20bÞ

sRφ ¼�½1þ2ρ2�3ρ4� ½ðSH�ShÞ=2� sin 2φ ð20cÞwith ρ¼a/R and Δp¼the net-pressure in the borehole. The referenceframe uses polar coordinates (R,φ), and two stresses SH and Sh actingat infinity correspond to τ1 and τ3 used in our approach. One canreconcile the expression of Zang and Stephansson [1] with the originalKirsch equations (Eqs. (9a)–(9c) in our study) by taking Sh¼0 andΔp¼0 (Kirsch [5] did not consider over- or under-pressured holes).

4. Discussion and practical rules

4.1. State-of-the-art gaps

Wellbore integrity is compromised when the safe mud-weightwindow is overstepped. Wellbore pressures are “safe” when thenet-pressure in the wellbore does not exceed the pore pressure or(formation pressure) as this would lead to permanent loss of mud-circulation fluid—which is costly. If well sections are drilledunderbalanced, rather than overbalanced, mud pressures that aretoo low for the pressure in certain formations increase the risk ofmud-barrier failure due to pressure kicks and may lead to

blowouts. High residual-stress and high stress concentration-factors, due to incomplete balancing of the wellbore stresses bythe mud-weight, in either under- or over-balanced drilling, maylead to tensile failure and shear failure (breakout).

With drilling operations moving to HP/HT conditions in everdeeper wells [25–27], the theoretical models of drilling risk anduncertainty envelopes must become more sophisticated. Reser-voirs with riskier pore pressure and fracture-gradient windowsrequire sophisticated stress analysis to ensure the well remainswithin the margins of the safe-drilling window. Pre-drillingdecisions about drill-hole-path-trajectory planning and the asso-ciated wellbore-stability analysis will benefit from the inclusionsof stress-trajectory visualization around the full length of thewellbore. Our present analysis aims to contribute to the develop-ment of improved stress-analysis tools for inclusion in contem-porary monitoring tools an automated dash-boards or wellbore-stability analysis.

Basic analytical equations for wellbore-stress quantificationhave been available and adapted for use in borehole stabilityanalysis since the ground-breaking work of Kirsch [5]. However,this has not translated into a comprehensive visualizationof wellbore-stress trajectories for a range of far-field and inner-wellbore boundary-conditions. Our study intends to fill part of thisgap and a systematic analysis revealed a number of worthwhilepractical implications articulated below.

4.2. Practical rules

Numerous practitioners use the following set of “generic”expressions to estimate the radial and tangential stresses atpressured vertical wellbores with an assumed far-field stress sHand sh (e.g., [28]; correcting for a sign reversal in sR):

sR ¼ P ð21aÞ

sθ ¼ 3sH�sh�P ð21bÞ

sRθ ¼ 0 ð21cÞ

Fig. 12. (a) Boundary conditions for a general bi-axial far-field stress comprised of a principal tension stress, τ3, and perpendicular far-field compression stress, τ1. The hole isstress free. (b) Principal stress trajectories for bi-axial far-field plane-stress as per boundary conditions in (a). Direction of the tensional principal stresses are outlined by theblue curvi-linears. Compressional stresses are aligned with the red trajectories. Isotropic points emanating from the wellbore wall are 901 apart. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article.)

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159 157

The adoption of such a generic set of equations introduces severalsources of potential inaccuracy in wellbore-stress analysis:

(A1) Total stresses sH and sh may at depth no longer be alignedwith principal stresses. Use of s1, s2, and s3, (and correspond-ing principal deviatoric stresses τ1, τ2, and τ3) is arguably morepractical. In any case, when sH and sh are used assurances areneeded on how these relate to the principal stress directions. Ifstresses are no longer Andersonian (see Section 2.1) at depth(as in most natural deformations, especially when drilling nearsalt bodies [29–34]), the tensor transformation rules should beused to relate sH and sh to the principal stresses acting on thewellbore.(A2) The stress-concentration factor of 3 times the maximumfar-field stress implied by Eq. (21b) is only valid for a bi-axialfar-field stress [described in our study by Eqs. (17a)–(17c)]. Thestress-concentration factor for the special case of a bi-axial far-field plane-stress is 4 times the maximum far-field stress[described in our study by Eqs. (18a)–(18c)]. If a far-field stressis axially symmetric [such as outlined in Eqs. (2a)–(2c)] thestress-concentration factor is 2 times the far-field stress. Thisdistinction is of practical importance as such far-field stressesoccur above expanding salt domes. The same stress concentra-tion factor of 2 applies to wellbores penetrating salt-weldcontrolled depo-centers, which are situated in radial symmetricfar-field compression stress (assuming these are tectonicallystill active).(A3) For all cases considered, the effective shear stress τRθ willgenerally not assume 0 (counter to Eq. (21c) as demonstratedearlier in our analysis.

The visualization of stress trajectories and stress intensity around wellbores further reveals the following practical implications:

(B1) The zone of stress perturbation around a wellbore, wherethe far-field stress magnitude and orientation are affected bythe presence and the mud-pressure in the wellbore, does notextend more than 10–15 wellbore radii. The radial stressbeyond five wellbore radii distance from the wellbore is equalto 95% the magnitude of the far-field stress in (a) axiallysymmetric, (b) uni-axial and (c) bi-axial far-field stress regimes.(B2) Loading of the wellbore by a mud-weight or hydraulicpressure does not enlarge the stress perturbation zone aroundthe wellbore, the effect is constrained to the region outlined inrule (B1).(B3) So-called balancing of wellbores is impossible when far-field stresses are radial axi-symmetric tensions (such as aboveactive salt domes), as has been illustrated in Section 2.5.(B4) A reversal of principal stress directions may occur insidethe stress perturbation zone for uni-axial and bi-axial far-fieldstresses (illustrated here in Fig. 11a–d). The isotropic pointsmark a region around the wellbore where the principal stressreversals occur. This is particularly relevant for fracking opera-tions [15–18].(B5) Breakout-stress measurements need a correction for the90 degree rotation of principal stresses (Fig. 11a–d) thattypically occur in the stress-perturbation zones near over-balanced wellbores.

5. Recommendations and conclusions

Risk management of drilling and completion operations demandcontinuous improvements to wellbore-stability models [35]. Thisstudy aims to contribute to the general learning-curve which aims tomake drilling operations safer, faster and cheaper [36,37]. The state

of stress around mud-load-pressured boreholes affected by a far-field stress can be modeled analytically, commonly based onequations first derived by Gustav Kirsch over a century ago. Theprincipal stress trajectories around wellbores, as they interact withthe ambient far-field principal stresses and in response to manip-ulation of the mud-pressure in the wellbore, have not been mappedsystematically before. Our study highlights the importance ofattempts to map such trajectories in order to improve wellbore-stability analyses as well as the pre-drilling planning of wellbore-trajectory paths.

Safe-drilling windows in wellbore-stress monitoring-systemsneed to take into account the effects of the far-field stresses.A universal expression valid for all far-field boundary conditionsdoes not exist: using a generic expression for estimating the stresscomponents on a wellbore [cf. Eqs. (21a)–(21c)] can easily lead tomis-interpretations of the stress-concentration factor—too low ifthe actual far-field stress is bi-axial plane-stress and too high ofthe stress is axially symmetric.

The present study systematically reviewed the analyticalexpressions valid for specific settings and demonstrated thatmeticulous attention must be paid to the far-field boundary-conditions in order to reveal the appropriate stress-state nearthe wellbore. Principal stress trajectories around wellbores canassume an unlimited range of patterns, depending upon theboundary conditions (including far-field stresses and wellborenet-pressures), geo-mechanical properties of the rock (incompres-sible, elastic, or poro-elastic), and thermal conditions. Any addi-tional thermal stresses have been excluded from our analysis, butcan be relatively simply accounted for analytically (by a thermalstress function superposition, as these generally are symmetricaround the wellbore).

In our study, wellbores were assumed to be aligned with one ofthe principal stress axes to bring out the salient details of the stressequations. Principal stresses along real wellbores will generally onlybe aligned with principal stresses at shallow depths due to the freesurface condition at zero depth. The stress trajectories around deeperwellbores inclined to the principal stress axes can be derived byincorporating stress-tensor transformation-formulae in our analysis.The widespread practice of using the horizontal total stresses sH(or sHMAX or SH) and sh (or sHMIN or Sh) as inputs for the principalstresses featuring in the wellbore stress equations (as common innearly all wellbore-stress analysis-tools) is itself a potential source offalse conclusions for wellbore stability measures when the principalstresses are no longer aligned with sH and sh. Practical rules forimproved wellbore-stress analyses, based on our review of theanalytical expressions currently used, have been formulated inSection 4 (rules A1–A3 and B1–B5).

Acknowledgments

Alboran Energy Strategy Consultants has generally sponsoredpart of this study.

References

[1] Zang A, Stephansson O. Stress field of the Earth's crust. London: Springer;2009.

[2] Van Oort E, Griffith J Schneider A. How to accelerate drilling learning curves.Paper SE/IADC 140333; 2011.

[3] Sadlier A, Wolfe C, Reese M. Automated alarms for Managing drilling pressureand maintaining wellbore stability—new concepts in while-drilling decisionmaking. Paper PE146298; 2011.

[4] Li S, George J, Purdy C. Pore-pressure and wellbore stability prediction toincrease drilling efficiency. Journal of Petroleum Technology 2012;63(2):98–101.

[5] Kirsch G. Die Theorie der Elastizität und die Bedürfnisse der Festigkeitslehre.Zeitschrift des Vereines deutscher Ingenieure 1898;42(28):797–807.

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159158

[6] Muskhelishvili NI. Some basic problems of the mathematical theory ofelasticity. Groningen: Noordhoff; 1954.

[7] Savin GN. Stress concentrations around holes. New York: Pergamon; 1961.[8] Malvern LE. Introduction to the mechanics of a continuous medium. Engle-

wood Cliffs, NJ: Prentice-Hall; 1969.[9] Timoshenko SP, Goodier JN. Theory of elasticity. 3rd ed.. New York: McGraw-

Hill; 1970.[10] Jaeger JC, N.G.W. COOK. Fundamentals of rock mechanics. 3rd ed.. London:

Chapman & Hall; 1979.[11] Delale F. Stress analysis of multilayered plates around circular holes. Interna-

tional Journal of Engineering Science 1984;22(1):57–75.[12] Jaeger JC, Cook NGW, Zimmerman RW. Fundamentals of rock mechanics.

Oxford: Blackwell; 2007.[13] Zoback M. Reservoir geomechanics. Cambridge: Cambridge University Press;

2007.[14] Fjær E, Holt RM, Horsrud P, Raaen AM, Risnes R. Petroleum related rock

mechanics. 2nd ed.. Amsterdam: Elsevier; 2008.[15] Weijermars R. Analytical stress functions applied to hydraulic fracturing:

scaling the interaction of tectonic stress and unbalanced borehole pressures.In: Proceedings of the 45th US rock mechanics/geomechanics symposium heldin San Francisco, CA, 26–29 June 2011, Paper ARMA 11-598.

[16] Weijermars R, Schultz-Ela D. Visualizing stress trajectories around pressurizedwellbores. In: Proceedings SPE/EAGE European unconventional resourcesconference and exhibition, held in Vienna, Austria, 20–22 March 2012, PaperSPE 152559.

[17] Weijermars R, Zhang X, Schultz-Ela D. Unrecognized ‘fracture caging’ couldmake shale gas drilling safer and more profitable. First Break 2012;30(2):35–6.

[18] Weijermars R., Zhang X., Schultz-Ela D. Geomechanics of fracture caging inwellbores, Geophysical Journal International 2013;193:1119-1132.

[19] Means WD. Stress and strain. basic concepts of continuum mechanics forgeologists. New York: Springer-Verlag; 1976.

[20] Fossen H. Structural geology. Cambridge: Cambridge University Press; 2010.[21] Weijermars R. Principles of Rock Mechanics. Alboran Science Publishing; 1997.

Open Course ware January 2011. Available from: ⟨http://ocw.tudelft.nl/courses/applied-earth-sciences/principles-of-rock-mechanics/course-home/⟩⟨http://ocw.tudelft.nl/courses/applied-earth-sciences/principles-of-rock-mechanics/readings/⟩.

[22] Bach C. Elastizität und festigkeit. Die für die technik wichtigsten sätze undderen erfahrungsmäßige Grundlage. Berlin: Springer; 1898.

[23] Föppl A. Festigkeitslehre. Leipzig; 1897.[24] Scheidegger AE. Stresses in the Earth's crust as determined from hydraulic

fracturing data. Geologie und Bauwesen 1962;27:45–53.[25] Kernche F, Hannegan D, Sammat E, Arnone M. Managed-pressure drilling

enables drilling beyond the conventional limit in a deepwater HP/HAT well,SPE142312; 2011.

[26] Ridley K, Portas B, Robbie J, Pinkstone H. Continuous circulation reduces NPT,World Oil 2011; March: p. 37–40.

[27] Salguero A, Almanze E, Haddad J. Challenging well-testing operations in HighT-Temperature environments—worldwide experience and best practiceslearned. Paper SPE 148575; 2011.

[28] Carcione JM, Helle HB, Gangi AF. Theory of borehole stability when drillingthrough salt formations. Geophysics 2006;71(3):F31–47.

[29] Sanz P. F., Dasari G.R., 2010. Controls on in-situ stresses around salt bodies. In:Proceedings of the 44th U.S. rock mechanics symposium and 5th U.S.—Canadarock mechanics symposium, Salt Lake City, Utah, 27–30 June 2010; paperARMA p. 10–169.

[30] Luo G, Nikolinakou M, Flemings P, Hudec M. Near-salt stress and wellborestability: a finite-element study and its application, Paper ARMA 12-309.

[31] Luo G, Nikolinakou M, Flemings P, Hudec M. Geomechanical modeling ofstresses adjacent to salt bodies: 1. Uncoupled models. AAPG Bulletin2012;96:43–64.

[32] Nikolinakou M, Luo G, Hudec M, Flemings P. Geomechanical modeling ofstresses and pore pressures in mudstones adjacent to salt bodies, Paper ARMA11-271.

[33] Nikolinakou M, Luo G, Hudec M, Flemings P. Geomechanical modeling ofstresses adjacent to salt bodies: 2. Poro-elasto-plasticity and coupled over-pressures. AAPG Bulletin 2012;96:65–85.

[34] Schutjens PMTM, Snippe JR, Mahani H, Turner J, Ita J, Mossop AP. Production-induced stress change in and above a reservoir pierced by two salt domes: ageomechanical model and its applications. SPE Journal 2012;17(1):80–97.

[35] Aldred W, Plumb D, Bradford I, Cook J, Gholkar V, Cousins L, et al. ManagingDrilling Risk. Oilfield Review 1999;11(2):2–19.

[36] Pritchard D, York PL, Beattie S, Hannegan D Drilling hazard management:Integrating mitigation methods, World Oil 2010; December: p. 49–54.

[37] Pritchard D, Cunha JC. Trends in Monitoring: How to use real-time dataeffectively. Journal of Petroleum Technology 2012;64(1):48–52.

R. Weijermars / International Journal of Rock Mechanics & Mining Sciences 64 (2013) 148–159 159