Research ArticleProbabilistic Approach in Wellbore StabilityAnalysis during Drilling

Mahmood R Al-Khayari Adel M Al-Ajmi and Yahya Al-Wahaibi

Department of Petroleum and Chemical Engineering Sultan Qaboos University Muscat Oman

Correspondence should be addressed to Mahmood R Al-Khayari mahmwoodgmailcom

Received 20 April 2016 Accepted 21 July 2016

Academic Editor Alireza Bahadori

Copyright copy 2016 Mahmood R Al-Khayari et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

In oil industry wellbore instability is the most costly problem that a well drilling operation may encounter One reason for wellborefailure can be related to ignoring rock mechanics effects A solution to overcome this problem is to adopt in situ stresses inconjunction with a failure criterion to end up with a deterministic model that calculates collapse pressure However the uncertaintyin input parameters can make this model misleading and useless In this paper a new probabilistic wellbore stability modelis presented to predict the critical drilling fluid pressure before the onset of a wellbore collapse The model runs Monte Carlosimulation to capture the effects of uncertainty in in situ stresses drilling trajectories and rock properties The developed modelwas applied to different in situ stress regimes normal faulting strike slip and reverse faulting Sensitivity analysis was applied toall carried out simulations and found that well trajectories have the biggest impact factor in wellbore instability followed by rockproperties The developed model improves risk management of wellbore stability It helps petroleum engineers and field plannersto make right decisions during drilling and fieldsrsquo development

1 Introduction

Drilling is one of the most important and costly operationsin oil industry due to the fact that if the wellbore stabilitycannot be accomplished drilling will lead to higher thannecessary costs So maintaining a stable wellbore is one ofthe main challenges encountered during drilling whethera well is being drilled with overbalanced or underbalancedtechniques [1]

The possible mechanical wellbore failures (rock failures)can be grouped in two main types tensile failure and shearfailure The tensile failure represents a hydraulic fracturewithin the formation which normally leads to a loss in systemcirculation during drilling whereas the shear failure can besimply described as the collapse of the borehole wall whichmay result in stuck pipe and in severe cases a loss of theopen hole section [2] The upper limit of the mud weight isthe maximum mud weight before the onset of tensile failure(fracturing) It can be obtained by conducting a leak off testOn the other hand the minimum allowable mud pressure isthe critical pressure before a wellbore collapse

Themost common well failure during drilling is boreholecollapse The method of analyzing this case involves the useof wellbore stability modeling consisting of a constitutivemodel coupledwith a failure criterionThe constitutivemodeldescribes the deformation properties of the rock and thefailure criterion determines the limits of the deformation [3]

This paper presents a model that accounts for uncertain-ties of rock properties and in situ stresses as well as wellboretrajectories in order to estimate the critical mud pressureduring drilling Unlike conventional stability models thatdeal with static values this probabilistic model uses MonteCarlo simulation method in order to study the impact ofuncertainties on wellbore instability [4] It was developed byutilizing a linear elastic and isotropic constitutive model inconjunction with the Mogi-Coulomb rock failure law

Al-Ajmi andAl-Harthy [4] have developed a probabilisticborehole model to study the influence of uncertainty of inputvariables in vertical borehole stability This paper discussesa probabilistic model to predict the collapse pressure fornonvertical wellbores (horizontal and deviated wells) as wellas vertical wellbores during drilling That will be done by

Hindawi Publishing CorporationJournal of Petroleum EngineeringVolume 2016 Article ID 3472158 13 pageshttpdxdoiorg10115520163472158

2 Journal of Petroleum Engineering

applying Monte Carlo simulation to the stability modeldeveloped by Al-Ajmi and Zimmerman [5 6]

For nonvertical wellbores there is no closed-form analyt-ical solution to calculate collapse pressure in order to preventborehole instability To solve this problem an excel programwas developed to conduct stability analysis in vertical andnonvertical boreholes and perform risk analysis in wellboreinstability

11 Stresses around Boreholes A constitutive model is a set ofequations needed to determine the stress state of a boreholeand describe its behavior Different models have been devel-oped andwidely studied however linear elastic analysis is themost common approach due to its ease of application [7] andits requirements of fewer input parameters comparedwith therest of complicated models

The stress components in cylindrical coordinates can berepresented by

120590119903 = 1205901199091015840cos2120579 + 1205901199101015840sin2120601 + 212059011990910158401199101015840 sin 120579 cos 120579120590120579 = 1205901199091015840sin2120579 + 1205901199101015840cos2120601 minus 212059011990910158401199101015840 sin 120579 cos 120579120590119911 = 1205901199111015840 120590119903120579 = (1205901199101015840 minus 1205901199091015840) sin 120579 cos 120579 + 12059011990910158401199101015840 (cos2120579 minussin2120579) 120590119903119911 = 12059011990910158401199111015840 cos 120579 + 12059011991010158401199111015840 sin 120579120590120579119911 = 12059011991010158401199111015840 cos 120579 minus 12059011990910158401199111015840 sin 120579

(1)

where 120590119903 120590120579 and 120590119911 are called radial tangential andaxial stress components in a cylindrical coordinate systemrespectively while the virgin formation stresses in (119909 119910 119911)coordinate system before excavation can be expressed as

120590119900119909 = (120590119867 cos2120572 + 120590ℎ sin2120572) cos2119894 + 120590V sin2sin2119894120590119900119910 = 120590119867 sin2120572 + 120590ℎ cos2120572120590119900119911 = (120590119867 cos2120572 + 120590ℎ sin120572) sin2119894 + 120590V cos2119894120590119900119909119910 = 05 (120590ℎ minus 120590119867) sin 2120572 cos 119894120590119900119910119911 = 05 (120590ℎ minus 120590119867) sin 2120572 sin 119894120590119900119909119911 = 05 (120590119867 cos2120572 + 120590ℎ sin2120572 minus 120590V) sin 2119894

(2)

Equation (2) represents the virgin formation stresses whichare denoted by the superscript ldquo119900rdquo on the stresses The virginin situ stresses will be altered by excavation In cylindricalcoordinates the complete stress solution around an arbitraryoriented wellbore can be represented by the following

120590119903 = (120590119900119909 + 1205901199001199102 )(1 minus 11988621199032 )

+ (120590119900119909 + 1205901199001199102 )(1 + 311988641199034 minus 41198862

1199032 ) cos 2120579+ 120590119900119909119910 (1 + 311988641199034 minus 4119886

2

1199032 ) sin 2120579 + 119875119908 11988621199032

120590120579 = (120590119900119909 + 1205901199001199102 )(1 + 11988621199032 )

minus (120590119900119909 minus 1205901199001199102 )(1 + 311988641199034 ) cos 2120579minus 120590119900119909119910 (1 + 311988641199034 minus 4119886

2

1199032 ) sin 2120579 minus 119875119908 11988621199032 120590119911 = 120590119900119911

minus V[2 (120590119900119909 minus 120590119900119910) 11988621199032 cos 2120579 + 4120590119900119909119910 1198862

1199032 sin 2120579]

120590119903120579 = [minus(120590119900119911 minus 1205901199001199102 )(1 minus 311988641199034 + 2119886

2

1199032 ) sin 2120579]+ 120590119900119909119910 (1 minus 311988641199034 + 2119886

2

1199032 ) cos 2120579120590120579119911 = (minus120590119900119909119911 sin 120579 + 120590119900119910119911 cos 120579)(1 + 11988621199032 ) 120590119903119911 = (120590119900119909119911 cos 120579 + 120590119900119910119911 sin 120579)(1 minus 11988621199032 )

(3)

where ldquo119886rdquo is the radius of the wellbore 119875119908 is the internalwellbore pressure and V is Poissonrsquos ratio The angle 120579 ismeasured clockwise from 119909-axis as shown in Figure 1

In a linear elasticmaterial the largest stress concentrationoccurs at the borehole wall Therefore borehole failure isexpected to initiate there [8ndash10] For wellbore instabilityanalysis consequently stresses at the borehole wall are theones that should be compared against a failure criterion

Estimating the stresses at borehole wall in the case ofdeviated wellbore is done by setting 119903 = 119886 in (2) the resultswill be as follows

120590119903 = 119875119908120590120579 = 120590119900119909 + 1205900119910 minus 2 (120590119900119909 minus 120590119900119910) cos 2120579 minus 4120590119900119909119910 sin 2120579 minus 119875119908120590119911 = 120590119900119911 minus V [2 (120590119900119909 minus 120590119900119910) cos 2120579 + 4120590119900119909119910 sin 2120579] 120590120579119911 = 2 (minus120590119900119909119911 sin 120579 + 120590119900119910119911 cos 120579) 120590119903120579 = 0120590119903119911 = 0

(4)

For vertical wellbore (Figure 2) the stresses at borehole wallare determined by setting the inclination angle 119894 = 0 in (2)and orienting the horizontal axes so that the direction 120579 = 0is parallel to 120590119867 (ie 120572 = 0) The stresses become

120590119903 = 119875119908120590120579 = 120590119867 + 120590ℎ minus 2 (120590119867 minus 120590ℎ) cos 2120579 minus 119875119908120590119911 = 120590V minus 2V (120590119867 minus 120590ℎ) cos 2120579

Journal of Petroleum Engineering 3

120590

120590ℎ

y998400

y

120572

i

120590H

z z998400

x

x998400

120579

Figure 1 Stress transformation system for deviated borehole

120590120579119911 = 0120590119903120579 = 0120590119903119911 = 0

(5)

The stresses around a horizontal wellbore wall after excava-tion (Figure 3) can be represented by setting 119894 equal to 1205872and plugging the resulted equation into (2) which gives

120590119903 = 119875119908120590120579 = (120590V + 120590119867 sin2120572 + 120590ℎ cos2120572)

minus (120590V minus 120590119867 sin2120572 minus 120590ℎ cos2120572) cos 2120579 minus 119875119908120590119911 = 120590119867 cos2120572 + 120590ℎ sin2120572

minus 2V (120590V minus 120590119867 sin2120572 minus 120590ℎ cos2120572) cos 2120579120590120579119911 = (120590ℎ minus 120590119867) sin 2120572 cos 120579120590119903120579 = 0120590119903119911 = 0

(6)

In this case where the wellbore axis lies along the maximumhorizontal principal stress (ie 120572 = 0) the stress at boreholewall is described as follows

120590119903 = 119875119908120590120579 = 120590V + 120590ℎ minus 2 (120590V minus 120590ℎ) cos 2120579 minus 119875119908120590119911 = 120590119867 minus 2V (120590V minus 120590ℎ) cos 2120579120590120579119911 = 0120590119903120579 = 0120590119903119911 = 0

(7)

12 Rock Failure Criteria (Mogi-Coulomb Criterion) Thesecond important element in wellbore stability is rock failure

120590

120590ℎ

y998400

120590H

z998400

x998400120579

Figure 2 Stress transformation system for vertical borehole [12]

120590

120590ℎy998400

y

120572

120590H

z z998400

x

x998400120579

Figure 3 Stress transformation system for a vertical borehole

criteria These criteria are sets of functions in stress or strainspaces that describe or predict the condition at which rockfails under external load In literature many of failure criteriahave been developed by scientists who suggested differentapproaches trying to simulate actual rock failures In thispaper the concentration will be onMogi-Coulomb criterion

In 2005 Al-Ajmi and Zimmerman presented a linearform of Mogi criterion [11] Al Ajmi found that the plottedpolyaxial data of some rocks using Mogi failure envelope in120591oct 1205901198982 space can be fit to a linear model as follows

120591oct = 119886 + 1198871205901198982 (8)

where 1205901198982 is the effective normal stress defined by

1205901198982 = 1205901 + 12059032 (9)

120591oct is the octahedral shear stress defined by

120591oct = 13radic(1205901 minus 1205902)2 + (1205902 minus 1205903)2 + (1205903 minus 1205901)2 (10)

And 119886 and 119887 are the intersection of the line on 120591oct-axis and its inclination respectively (Figure 4) Al Ajmi [12]found and verified experimentally that failure under polyaxialstresses can be predicted from triaxial test data using thelinear Mogi criterion Accordingly he developed a new true-triaxial failure criterion called Mogi-Coulomb criterion Fortriaxial stress states (1205902 = 1205903) the linear Mogi criterion isexactly equivalent to Mohr-Coulomb criterion [13]

4 Journal of Petroleum Engineering

Triaxial

Triaxial

Uniaxial

Uniaxial

tension

compression

compression

120591oct

c cot120601 T02 C02 120590m

l

a

b

Figure 4 Mogi-Coulomb failure envelope [12]

Combining (8) and (9) and (7) and rearranging theequation the failure function 119865 will be

119865 = 119886 + 119887 (1205901 + 12059032 )minus 13radic(1205901 minus 1205902)2 + (1205902 minus 1205903)2 + (1205903 minus 1205901)2

(11)

Failure occurs if 119865 le 0The Mogi-Coulomb parameters (119886 119887) can be related to

Coulomb failure parameters (119888 120601) as follows119886 = 2radic23 119888 cos120601119887 = 2radic23 sin120601

(12)

where 119888 is the shear strength or cohesion of the material and120601 is the internal friction angle [13]

2 Modeling Wellbore Stability

21 Vertical Wellbore Failure To ensure mechanical wellborestability in drilling vertical boreholes the lower limit of themud pressure (119875119908) must be considered to be the collapsepressure To find the collapse pressure Kirsch solution isapplied to find stresses around a borehole radial (120590119903) axial(120590119911) and tangential (120590120579) This solution then must be coupledwith a failure criterion (in this research the Mogi-Coulombcriterion will be used) to end up with collapsed pressuremodel

UsingKirsch solution the stresses at the boreholewall canbe defined by the following

120590119903 = 119875119908120590120579 = 119860 minus 119875119908120590119911 = 119861

(13)

119860 and 119861 are constants and can be defined by

119860 = 3120590119867 minus 120590ℎ119861 = 120590V + 2V (120590119867 minus 120590ℎ) (14)

where 120590V is vertical in situ stress 120590119867 and 120590ℎ aremaximum andminimum horizontal in situ stresses respectively And V is amaterial constant called Poissonrsquos ratio

A closed-form solution for collapsed pressure can beachieved by applying (13) and (14) to the Mogi-Coulombcriterion This solution can be used as a borehole failuremodel However in this model there are three differentpermutations of the three principle stresses (120590119903 120590119911 and 120590120579)which need to be studied in order to determine the collapsepressureThese permutations are as follows (1) 120590119911 ge 120590120579 ge 120590119903(2) 120590120579 ge 120590119911 ge 120590119903 and (3) 120590120579 ge 120590119903 ge 120590119911 (Figure 5)

Equations (15) illustrate closed-form solutions to calcu-late collapse pressures in vertical wellbore for in situ stressregimes normal faulting (120590119911 gt 120590120579 gt 120590119903) reverse faulting(120590120579 gt 120590119911 gt 120590119903) and strike slip (120590120579 gt 120590119903 gt 120590119911) respectively1198751199081198871= 16 minus 211988710158402 [(3119860 + 21198871015840119896) minus radic119867 + 12 (1198702 + 1198871015840119860119870)] 1198751199081198872= 12119860 minus 16radic12 [1198861015840 + 1198871015840 (119860 minus 21198750)]2 minus 3 (119860 minus 2119861)21198751199081198873= 16 minus 211988710158402 [(3119860 + 21198871015840119866) minus radic119867 + 12 (1198662 + 1198871015840119860119866)]

(15)

In these equations failure occurs if 119875119908 (internal wellborepressure) le 119875119908119887 (the lower limit of the mud pressure) where

119867 = 1198602 (411988710158402 minus 3) + (1198612 minus 119860119861) (411988710158402 minus 12) (16)

119870 = 1198861015840 + 1198871015840 (119861 minus 21198750) (17)

119866 = 119870 + 1198871015840119860 (18)

It has been proven in literature that the most commonstress distribution is the second case when 120590120579 ge 120590119911 ge 120590119903 andit is called reverse faulting stress regime (eg [4ndash6 8ndash10])

22 Probabilistic Approach Risk Analysis in Vertical WellboreStability In 2010Al-Harthi andAl-Ajmi applied risk analysisto wellbore stability They utilized the Mogi-Coulomb stabil-ity model (see (15) (19) and (20)) to conduct a sensitivityanalysis in vertical wellbores They carried out Monte Carlosimulation using Risk program They also conducted asensitivity analysis using Mohr-Coulomb stability modelThe comparison between the two analyses showed that theMohr-Coulomb criterion tends to overestimate minimumoverbalance mud pressure that is required to keep boreholestable during drilling (as illustrated in Figure 6) They alsoconcluded that risk analysis is a good tool to estimate thecritical mud pressure of drilling fluid with existence ofuncertainties in input parameters The sensitivity analysis ofthe Mogi-Coulomb criterion (which is taking in account theeffect of the intermediate principal stress) showed that themaximum horizontal stress (120590119867) is the most critical factor

Journal of Petroleum Engineering 5

1205901

1205902

1205903

(a)

1205901

1205902

1205903

(b)

1205901

1205902

1205903

(c)

Figure 5 In situ stress regimes (a) normal fault (b) reverse fault and (c) strike slip [12]

times10minus4

Mohr-CMogi-C

0

05

1

15

2

25

3

35

4

45

Prob

abili

ty d

istrib

utio

n fu

nctio

n

12 13 14 15 16 17 18 19 20 2111

Overbalance pressure (times103 kPa)

Figure 6 Overbalance pressure in the strike slip regimeTheMogi-Coulomb criterion on the left side of the graph and the Mohr-Coulomb criterion on the right side [4]

1800

0

1700

0

1600

0

17500

16500

18500

15500

Mean of overbalance (kPa) (Mogi-C)

Vertical stressPoissonrsquos ratio

Minimum horizontal stressPore pressure

CohesionFriction angle

Maximum horizontal stress

Figure 7 Sensitivity analysis in the strike slip regime (tornado chartfor Mogi-Coulomb) [4]

in vertical wellbore stability analysis under any in situ stressregime (as illustrated in Figures 7 and 8)

23 Nonvertical Wellbore Failure In order to determine thecritical mud pressure (the minimum mud pressure required

2100

0

2200

0

2000

0

2300

0

22500

20500

23500

21500

Mean of overbalance (kPa) (Mohr-C)

Poissonrsquos ratioVertical stress

Minimum horizontal stressPore pressure

CohesionFriction

Maximum horizontal stress

Figure 8 Sensitivity analysis in the strike slip regime (tornado chartfor Mohr-Coulomb) [4]

to prevent borehole collapse) in deviated boreholes thestresses at the borehole wall should be calculated using (4)and then compared with Mogi-Coulomb failure criterionHowever the resulting tangential and axial stresses are notprincipal stresses as the shear stress 120590120579119911 may not be equal tozero As a result the tangential and the axial stresses shouldbe determined using the following

1205901 = 12 (120590119909 + 120590119910) + radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

1205902 = 12 (120590119909 + 120590119910) minus radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

(19)

where 1205901 and 1205902 are tangential and axial stresses respectively120590119909 and 120590119910 are normal stresses in 119909 and 119910 directions respec-tively and 120591119909119910 is the shear stress in 119909-119910 direction

Introducing the resulting principal stresses to the Mogi-Coulomb failure criterion a 4th-degree equation needs tobe solved in order to find the collapse pressure This meansthere is no closed-form analytical solution that can be derivedfor collapse pressure in deviated boreholes utilizing Mogi-Coulomb failure criterion similar to vertical boreholes or ifMohr-Coulomb failure criterion had been utilized [12]

6 Journal of Petroleum Engineering

Figure 9 Interface of the deterministic instability model in Excelfor nonvertical boreholes

Table 1 Vertical wellbore failure for different stress regimes

Input Value120590vg (psift) 11120590Hg (psift) 1120590hg (psift) 091198750 (psift) 046Cohesion (psi) 870Friction angle (deg) 313Poissonrsquos ratio 033

3 Probabilistic Numerical Models forNonvertical Wellbores

The previous section illustrates that there is no closed-formsolution to model deviated wellbore stability However Al-Ajmi [12] has applied iterative loops in a computer programto calculate the collapse pressure for nonvertical boreholesMathCAD program was used to perform these iterated loopsto find the critical mud pressure (see the Appendix)

This model was utilized to build a stability model in theExcel program In the developed model equations of stressesand Mogi-Coulomb failure criterion have been convertedfrom MathCAD language to the language of the Excelprogram Also the macro tool (which is using Visual BasicProgramming Language) in the Excel has been utilized to dothe iterated loops to calculate the collapse pressure Figure 9illustrates the interface of the developed model

The input parameters of the developed model are depth(ℎ) vertical stress gradient (120590vg) maximum horizontal stressgradient (120590Hg) minimum horizontal stress gradient (120590hg)pore pressure gradient (1198750119892) cohesion (119888) friction angle (120601)and Poissonrsquos ratio (v)

The purpose of this model is to calculate the collapsepressure for a given set of input parameters that are con-stant (having static values) for 100 different well trajectoryscenarios In other words themodel will find out the collapsepressure for 10 values of inclination angle (119894) ranging fromzero to 90 degrees multiplied by 10 values of azimuth angle(119896) ranging from zero to 90 degrees too

To ensure the validity of this model it was run for acertain case (inputs are listed inTable 1) and compared againstAl-Ajmirsquos MathCADmodel The output result was absolutelyidentical as illustrated in Figures 10 and 11

Figure 10 Resulting collapse pressure from the developed Excelmodel

Figure 11 Resulting collapse pressure from the MathCAD model

The model is consistent with Al-Ajmirsquos MathCAD modelin calculating collapse pressure for static input parametershowever the target is to end up with a probabilistic modelin order to conduct instability risk analysis for nonvertical(deviated) wellbores As a result the Risk program wasutilized with the new Excel based instability model to give aprobabilistic model (see Figure 12)

31 An Overview of Risk Program Risk is an add-in pro-gram to Microsoft Excel It is using Monte Carlo simulationthat allows user to test variety of scenarios in a given modeland determine the probability distribution of each outcomeWhen combined with a well-built model it can give you agood feel for how likely a given outcome is for amodeled planor business venture In this model the sixth version of Riskprogram is used to run Monte Carlo simulation It allowsusers to make decisions using the sensitivity risk analysis Inthis program parameters can be set fixed or can be given adesired range and distribution Due to the limitation of in situstresses and rock properties data uniform distribution wasused mostly in data simulations

4 Probabilistic Collapse Pressure Analysis

Thedeveloped probabilistic wellbore collapse pressuremodelwas run to calculate the critical mud pressure for different

Journal of Petroleum Engineering 7

Figure 12 Interface of the probabilistic instability model in Excelfor deviated boreholes

casesThemodel utilizes theRisk program and (as has beenmentioned in the previous chapter) the model conductedMonte Carlo simulation for each case to capture the effectof uncertainty in input variables (in situ stresses and rockproperties) in collapse mud pressure In situ stresses androck properties data of a sandstone formation provided inliterature are used in this analysis Uniform distributionwas applied due to limitation of data Then for each casea sensitivity analysis was carried out to figure out whichparameter has more impact on collapse pressures

41 Collapse Pressure Risk Analysis for Normal Faulting StressRegime In this case Monte Carlo simulation was run tocalculate the collapse mud pressure at normal faulting in situstress regime The borehole was assumed to be drilled in asandstone formation with a depth of 3048 meters The usedfield stresses condition and rock properties are described inTables 2 and 3 respectively

In the first simulation all input values were given rangesof values with uniform distributions including well trajecto-ries (inclination angle 119894 and azimuth angle 119896) To calculate thecollapse pressures the developed probabilisticmodel was runapplying Monte Carlo simulation with 10000 iterations usingRisk program [14]

The histogram in Figure 13 illustrates the resulting valuesof possible overbalance pressures (collapse mud pressure)and their occurrence frequencies within the given rangesof input parameters The result shows that the minimumpossible value of collapse pressure is 36780 kPa whereas themaximum possible value is 44790 kPa Table 4 summarizesthe statistics of the simulation

To study the relations between input parameters and theresulting collapse pressures scattered plots of collapse pres-sure versus input parameters were constructed (Figures 14ndash16) The scattered plots of 119875119908 versus input parameters showthat 119875119908 increases with deviation angle (119894) in a relation similar

Table 2 Normal faulting in situ stress regime [4]

Field stresses (kPam) Minimum Maximum120590vg 2194 2262120590Hg 1934 2036120590hg 1876 19341198750 1011 1052

Table 3 Rock properties for a sandstone formation [4]

Rock properties Minimum MaximumCohesion (kPa) 4500 5500Friction angle (deg) 31 33Poissonrsquos ratio 02 03

Table 4 Normal faulting Monte Carlo simulation statistics (chang-ing all input parameters)

Statistics 119875119908 (kPa)Minimum 36780Maximum 44792Mean 40598Standard deviation 168790 confidence interval 38310ndash43250

Table 5 Normal faulting Monte Carlo simulation statistics (con-stant input parameters except 119894)Statistics 119875119908 (kPa)Minimum 38360Maximum 40866Mean 39480Standard deviation 9287190 confidence interval 38363ndash40852 thousand

to (S) shape However there are a lot of noises associated withthis relation due to the effect of other parameters For the restof the parameters nowell-defined relations can be figured outfrom plots even though the overall trends of plotted valuesmay give indications of general relations between 119875119908 andinput parameters

Another type of simulation was conducted while settingall input parameters constant except one in order to studythe effect of uncertainty of unfixed parameter on the criticalmudpressureThen the simulationwas repeated several timesand in each time a different parameter was set variably Forthe first case deviation angle (119894) was the only parameterthat is given a range (from 0 to 90 degrees) whereas therest of parameters were kept constant at the mid values ofthe given ranges The occurrence frequencies of possiblecollapse pressures are illustrated in Figure 17 And Table 5summarizes the statistics of the simulation In order to studythe relation between deviation angle (119894) and collapse pressure119875119908 a scattered plot has been constructed The analysis ofthe resulting collapse pressure shows that deviation angle(119894) has an (S) shape relation with the critical mud pressure(see Figure 18) ) Figures 19 and 20 illustrate the relation of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

2 Journal of Petroleum Engineering

applying Monte Carlo simulation to the stability modeldeveloped by Al-Ajmi and Zimmerman [5 6]

For nonvertical wellbores there is no closed-form analyt-ical solution to calculate collapse pressure in order to preventborehole instability To solve this problem an excel programwas developed to conduct stability analysis in vertical andnonvertical boreholes and perform risk analysis in wellboreinstability

11 Stresses around Boreholes A constitutive model is a set ofequations needed to determine the stress state of a boreholeand describe its behavior Different models have been devel-oped andwidely studied however linear elastic analysis is themost common approach due to its ease of application [7] andits requirements of fewer input parameters comparedwith therest of complicated models

The stress components in cylindrical coordinates can berepresented by

120590119903 = 1205901199091015840cos2120579 + 1205901199101015840sin2120601 + 212059011990910158401199101015840 sin 120579 cos 120579120590120579 = 1205901199091015840sin2120579 + 1205901199101015840cos2120601 minus 212059011990910158401199101015840 sin 120579 cos 120579120590119911 = 1205901199111015840 120590119903120579 = (1205901199101015840 minus 1205901199091015840) sin 120579 cos 120579 + 12059011990910158401199101015840 (cos2120579 minussin2120579) 120590119903119911 = 12059011990910158401199111015840 cos 120579 + 12059011991010158401199111015840 sin 120579120590120579119911 = 12059011991010158401199111015840 cos 120579 minus 12059011990910158401199111015840 sin 120579

(1)

where 120590119903 120590120579 and 120590119911 are called radial tangential andaxial stress components in a cylindrical coordinate systemrespectively while the virgin formation stresses in (119909 119910 119911)coordinate system before excavation can be expressed as

120590119900119909 = (120590119867 cos2120572 + 120590ℎ sin2120572) cos2119894 + 120590V sin2sin2119894120590119900119910 = 120590119867 sin2120572 + 120590ℎ cos2120572120590119900119911 = (120590119867 cos2120572 + 120590ℎ sin120572) sin2119894 + 120590V cos2119894120590119900119909119910 = 05 (120590ℎ minus 120590119867) sin 2120572 cos 119894120590119900119910119911 = 05 (120590ℎ minus 120590119867) sin 2120572 sin 119894120590119900119909119911 = 05 (120590119867 cos2120572 + 120590ℎ sin2120572 minus 120590V) sin 2119894

(2)

Equation (2) represents the virgin formation stresses whichare denoted by the superscript ldquo119900rdquo on the stresses The virginin situ stresses will be altered by excavation In cylindricalcoordinates the complete stress solution around an arbitraryoriented wellbore can be represented by the following

120590119903 = (120590119900119909 + 1205901199001199102 )(1 minus 11988621199032 )

+ (120590119900119909 + 1205901199001199102 )(1 + 311988641199034 minus 41198862

1199032 ) cos 2120579+ 120590119900119909119910 (1 + 311988641199034 minus 4119886

2

1199032 ) sin 2120579 + 119875119908 11988621199032

120590120579 = (120590119900119909 + 1205901199001199102 )(1 + 11988621199032 )

minus (120590119900119909 minus 1205901199001199102 )(1 + 311988641199034 ) cos 2120579minus 120590119900119909119910 (1 + 311988641199034 minus 4119886

2

1199032 ) sin 2120579 minus 119875119908 11988621199032 120590119911 = 120590119900119911

minus V[2 (120590119900119909 minus 120590119900119910) 11988621199032 cos 2120579 + 4120590119900119909119910 1198862

1199032 sin 2120579]

120590119903120579 = [minus(120590119900119911 minus 1205901199001199102 )(1 minus 311988641199034 + 2119886

2

1199032 ) sin 2120579]+ 120590119900119909119910 (1 minus 311988641199034 + 2119886

2

1199032 ) cos 2120579120590120579119911 = (minus120590119900119909119911 sin 120579 + 120590119900119910119911 cos 120579)(1 + 11988621199032 ) 120590119903119911 = (120590119900119909119911 cos 120579 + 120590119900119910119911 sin 120579)(1 minus 11988621199032 )

(3)

where ldquo119886rdquo is the radius of the wellbore 119875119908 is the internalwellbore pressure and V is Poissonrsquos ratio The angle 120579 ismeasured clockwise from 119909-axis as shown in Figure 1

In a linear elasticmaterial the largest stress concentrationoccurs at the borehole wall Therefore borehole failure isexpected to initiate there [8ndash10] For wellbore instabilityanalysis consequently stresses at the borehole wall are theones that should be compared against a failure criterion

Estimating the stresses at borehole wall in the case ofdeviated wellbore is done by setting 119903 = 119886 in (2) the resultswill be as follows

120590119903 = 119875119908120590120579 = 120590119900119909 + 1205900119910 minus 2 (120590119900119909 minus 120590119900119910) cos 2120579 minus 4120590119900119909119910 sin 2120579 minus 119875119908120590119911 = 120590119900119911 minus V [2 (120590119900119909 minus 120590119900119910) cos 2120579 + 4120590119900119909119910 sin 2120579] 120590120579119911 = 2 (minus120590119900119909119911 sin 120579 + 120590119900119910119911 cos 120579) 120590119903120579 = 0120590119903119911 = 0

(4)

For vertical wellbore (Figure 2) the stresses at borehole wallare determined by setting the inclination angle 119894 = 0 in (2)and orienting the horizontal axes so that the direction 120579 = 0is parallel to 120590119867 (ie 120572 = 0) The stresses become

120590119903 = 119875119908120590120579 = 120590119867 + 120590ℎ minus 2 (120590119867 minus 120590ℎ) cos 2120579 minus 119875119908120590119911 = 120590V minus 2V (120590119867 minus 120590ℎ) cos 2120579

Journal of Petroleum Engineering 3

120590

120590ℎ

y998400

y

120572

i

120590H

z z998400

x

x998400

120579

Figure 1 Stress transformation system for deviated borehole

120590120579119911 = 0120590119903120579 = 0120590119903119911 = 0

(5)

The stresses around a horizontal wellbore wall after excava-tion (Figure 3) can be represented by setting 119894 equal to 1205872and plugging the resulted equation into (2) which gives

120590119903 = 119875119908120590120579 = (120590V + 120590119867 sin2120572 + 120590ℎ cos2120572)

minus (120590V minus 120590119867 sin2120572 minus 120590ℎ cos2120572) cos 2120579 minus 119875119908120590119911 = 120590119867 cos2120572 + 120590ℎ sin2120572

minus 2V (120590V minus 120590119867 sin2120572 minus 120590ℎ cos2120572) cos 2120579120590120579119911 = (120590ℎ minus 120590119867) sin 2120572 cos 120579120590119903120579 = 0120590119903119911 = 0

(6)

In this case where the wellbore axis lies along the maximumhorizontal principal stress (ie 120572 = 0) the stress at boreholewall is described as follows

120590119903 = 119875119908120590120579 = 120590V + 120590ℎ minus 2 (120590V minus 120590ℎ) cos 2120579 minus 119875119908120590119911 = 120590119867 minus 2V (120590V minus 120590ℎ) cos 2120579120590120579119911 = 0120590119903120579 = 0120590119903119911 = 0

(7)

12 Rock Failure Criteria (Mogi-Coulomb Criterion) Thesecond important element in wellbore stability is rock failure

120590

120590ℎ

y998400

120590H

z998400

x998400120579

Figure 2 Stress transformation system for vertical borehole [12]

120590

120590ℎy998400

y

120572

120590H

z z998400

x

x998400120579

Figure 3 Stress transformation system for a vertical borehole

criteria These criteria are sets of functions in stress or strainspaces that describe or predict the condition at which rockfails under external load In literature many of failure criteriahave been developed by scientists who suggested differentapproaches trying to simulate actual rock failures In thispaper the concentration will be onMogi-Coulomb criterion

In 2005 Al-Ajmi and Zimmerman presented a linearform of Mogi criterion [11] Al Ajmi found that the plottedpolyaxial data of some rocks using Mogi failure envelope in120591oct 1205901198982 space can be fit to a linear model as follows

120591oct = 119886 + 1198871205901198982 (8)

where 1205901198982 is the effective normal stress defined by

1205901198982 = 1205901 + 12059032 (9)

120591oct is the octahedral shear stress defined by

120591oct = 13radic(1205901 minus 1205902)2 + (1205902 minus 1205903)2 + (1205903 minus 1205901)2 (10)

And 119886 and 119887 are the intersection of the line on 120591oct-axis and its inclination respectively (Figure 4) Al Ajmi [12]found and verified experimentally that failure under polyaxialstresses can be predicted from triaxial test data using thelinear Mogi criterion Accordingly he developed a new true-triaxial failure criterion called Mogi-Coulomb criterion Fortriaxial stress states (1205902 = 1205903) the linear Mogi criterion isexactly equivalent to Mohr-Coulomb criterion [13]

4 Journal of Petroleum Engineering

Triaxial

Triaxial

Uniaxial

Uniaxial

tension

compression

compression

120591oct

c cot120601 T02 C02 120590m

l

a

b

Figure 4 Mogi-Coulomb failure envelope [12]

Combining (8) and (9) and (7) and rearranging theequation the failure function 119865 will be

119865 = 119886 + 119887 (1205901 + 12059032 )minus 13radic(1205901 minus 1205902)2 + (1205902 minus 1205903)2 + (1205903 minus 1205901)2

(11)

Failure occurs if 119865 le 0The Mogi-Coulomb parameters (119886 119887) can be related to

Coulomb failure parameters (119888 120601) as follows119886 = 2radic23 119888 cos120601119887 = 2radic23 sin120601

(12)

where 119888 is the shear strength or cohesion of the material and120601 is the internal friction angle [13]

2 Modeling Wellbore Stability

21 Vertical Wellbore Failure To ensure mechanical wellborestability in drilling vertical boreholes the lower limit of themud pressure (119875119908) must be considered to be the collapsepressure To find the collapse pressure Kirsch solution isapplied to find stresses around a borehole radial (120590119903) axial(120590119911) and tangential (120590120579) This solution then must be coupledwith a failure criterion (in this research the Mogi-Coulombcriterion will be used) to end up with collapsed pressuremodel

UsingKirsch solution the stresses at the boreholewall canbe defined by the following

120590119903 = 119875119908120590120579 = 119860 minus 119875119908120590119911 = 119861

(13)

119860 and 119861 are constants and can be defined by

119860 = 3120590119867 minus 120590ℎ119861 = 120590V + 2V (120590119867 minus 120590ℎ) (14)

where 120590V is vertical in situ stress 120590119867 and 120590ℎ aremaximum andminimum horizontal in situ stresses respectively And V is amaterial constant called Poissonrsquos ratio

A closed-form solution for collapsed pressure can beachieved by applying (13) and (14) to the Mogi-Coulombcriterion This solution can be used as a borehole failuremodel However in this model there are three differentpermutations of the three principle stresses (120590119903 120590119911 and 120590120579)which need to be studied in order to determine the collapsepressureThese permutations are as follows (1) 120590119911 ge 120590120579 ge 120590119903(2) 120590120579 ge 120590119911 ge 120590119903 and (3) 120590120579 ge 120590119903 ge 120590119911 (Figure 5)

Equations (15) illustrate closed-form solutions to calcu-late collapse pressures in vertical wellbore for in situ stressregimes normal faulting (120590119911 gt 120590120579 gt 120590119903) reverse faulting(120590120579 gt 120590119911 gt 120590119903) and strike slip (120590120579 gt 120590119903 gt 120590119911) respectively1198751199081198871= 16 minus 211988710158402 [(3119860 + 21198871015840119896) minus radic119867 + 12 (1198702 + 1198871015840119860119870)] 1198751199081198872= 12119860 minus 16radic12 [1198861015840 + 1198871015840 (119860 minus 21198750)]2 minus 3 (119860 minus 2119861)21198751199081198873= 16 minus 211988710158402 [(3119860 + 21198871015840119866) minus radic119867 + 12 (1198662 + 1198871015840119860119866)]

(15)

In these equations failure occurs if 119875119908 (internal wellborepressure) le 119875119908119887 (the lower limit of the mud pressure) where

119867 = 1198602 (411988710158402 minus 3) + (1198612 minus 119860119861) (411988710158402 minus 12) (16)

119870 = 1198861015840 + 1198871015840 (119861 minus 21198750) (17)

119866 = 119870 + 1198871015840119860 (18)

It has been proven in literature that the most commonstress distribution is the second case when 120590120579 ge 120590119911 ge 120590119903 andit is called reverse faulting stress regime (eg [4ndash6 8ndash10])

22 Probabilistic Approach Risk Analysis in Vertical WellboreStability In 2010Al-Harthi andAl-Ajmi applied risk analysisto wellbore stability They utilized the Mogi-Coulomb stabil-ity model (see (15) (19) and (20)) to conduct a sensitivityanalysis in vertical wellbores They carried out Monte Carlosimulation using Risk program They also conducted asensitivity analysis using Mohr-Coulomb stability modelThe comparison between the two analyses showed that theMohr-Coulomb criterion tends to overestimate minimumoverbalance mud pressure that is required to keep boreholestable during drilling (as illustrated in Figure 6) They alsoconcluded that risk analysis is a good tool to estimate thecritical mud pressure of drilling fluid with existence ofuncertainties in input parameters The sensitivity analysis ofthe Mogi-Coulomb criterion (which is taking in account theeffect of the intermediate principal stress) showed that themaximum horizontal stress (120590119867) is the most critical factor

Journal of Petroleum Engineering 5

1205901

1205902

1205903

(a)

1205901

1205902

1205903

(b)

1205901

1205902

1205903

(c)

Figure 5 In situ stress regimes (a) normal fault (b) reverse fault and (c) strike slip [12]

times10minus4

Mohr-CMogi-C

0

05

1

15

2

25

3

35

4

45

Prob

abili

ty d

istrib

utio

n fu

nctio

n

12 13 14 15 16 17 18 19 20 2111

Overbalance pressure (times103 kPa)

Figure 6 Overbalance pressure in the strike slip regimeTheMogi-Coulomb criterion on the left side of the graph and the Mohr-Coulomb criterion on the right side [4]

1800

0

1700

0

1600

0

17500

16500

18500

15500

Mean of overbalance (kPa) (Mogi-C)

Vertical stressPoissonrsquos ratio

Minimum horizontal stressPore pressure

CohesionFriction angle

Maximum horizontal stress

Figure 7 Sensitivity analysis in the strike slip regime (tornado chartfor Mogi-Coulomb) [4]

in vertical wellbore stability analysis under any in situ stressregime (as illustrated in Figures 7 and 8)

23 Nonvertical Wellbore Failure In order to determine thecritical mud pressure (the minimum mud pressure required

2100

0

2200

0

2000

0

2300

0

22500

20500

23500

21500

Mean of overbalance (kPa) (Mohr-C)

Poissonrsquos ratioVertical stress

Minimum horizontal stressPore pressure

CohesionFriction

Maximum horizontal stress

Figure 8 Sensitivity analysis in the strike slip regime (tornado chartfor Mohr-Coulomb) [4]

to prevent borehole collapse) in deviated boreholes thestresses at the borehole wall should be calculated using (4)and then compared with Mogi-Coulomb failure criterionHowever the resulting tangential and axial stresses are notprincipal stresses as the shear stress 120590120579119911 may not be equal tozero As a result the tangential and the axial stresses shouldbe determined using the following

1205901 = 12 (120590119909 + 120590119910) + radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

1205902 = 12 (120590119909 + 120590119910) minus radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

(19)

where 1205901 and 1205902 are tangential and axial stresses respectively120590119909 and 120590119910 are normal stresses in 119909 and 119910 directions respec-tively and 120591119909119910 is the shear stress in 119909-119910 direction

Introducing the resulting principal stresses to the Mogi-Coulomb failure criterion a 4th-degree equation needs tobe solved in order to find the collapse pressure This meansthere is no closed-form analytical solution that can be derivedfor collapse pressure in deviated boreholes utilizing Mogi-Coulomb failure criterion similar to vertical boreholes or ifMohr-Coulomb failure criterion had been utilized [12]

6 Journal of Petroleum Engineering

Figure 9 Interface of the deterministic instability model in Excelfor nonvertical boreholes

Table 1 Vertical wellbore failure for different stress regimes

Input Value120590vg (psift) 11120590Hg (psift) 1120590hg (psift) 091198750 (psift) 046Cohesion (psi) 870Friction angle (deg) 313Poissonrsquos ratio 033

3 Probabilistic Numerical Models forNonvertical Wellbores

The previous section illustrates that there is no closed-formsolution to model deviated wellbore stability However Al-Ajmi [12] has applied iterative loops in a computer programto calculate the collapse pressure for nonvertical boreholesMathCAD program was used to perform these iterated loopsto find the critical mud pressure (see the Appendix)

This model was utilized to build a stability model in theExcel program In the developed model equations of stressesand Mogi-Coulomb failure criterion have been convertedfrom MathCAD language to the language of the Excelprogram Also the macro tool (which is using Visual BasicProgramming Language) in the Excel has been utilized to dothe iterated loops to calculate the collapse pressure Figure 9illustrates the interface of the developed model

The input parameters of the developed model are depth(ℎ) vertical stress gradient (120590vg) maximum horizontal stressgradient (120590Hg) minimum horizontal stress gradient (120590hg)pore pressure gradient (1198750119892) cohesion (119888) friction angle (120601)and Poissonrsquos ratio (v)

The purpose of this model is to calculate the collapsepressure for a given set of input parameters that are con-stant (having static values) for 100 different well trajectoryscenarios In other words themodel will find out the collapsepressure for 10 values of inclination angle (119894) ranging fromzero to 90 degrees multiplied by 10 values of azimuth angle(119896) ranging from zero to 90 degrees too

To ensure the validity of this model it was run for acertain case (inputs are listed inTable 1) and compared againstAl-Ajmirsquos MathCADmodel The output result was absolutelyidentical as illustrated in Figures 10 and 11

Figure 10 Resulting collapse pressure from the developed Excelmodel

Figure 11 Resulting collapse pressure from the MathCAD model

The model is consistent with Al-Ajmirsquos MathCAD modelin calculating collapse pressure for static input parametershowever the target is to end up with a probabilistic modelin order to conduct instability risk analysis for nonvertical(deviated) wellbores As a result the Risk program wasutilized with the new Excel based instability model to give aprobabilistic model (see Figure 12)

31 An Overview of Risk Program Risk is an add-in pro-gram to Microsoft Excel It is using Monte Carlo simulationthat allows user to test variety of scenarios in a given modeland determine the probability distribution of each outcomeWhen combined with a well-built model it can give you agood feel for how likely a given outcome is for amodeled planor business venture In this model the sixth version of Riskprogram is used to run Monte Carlo simulation It allowsusers to make decisions using the sensitivity risk analysis Inthis program parameters can be set fixed or can be given adesired range and distribution Due to the limitation of in situstresses and rock properties data uniform distribution wasused mostly in data simulations

4 Probabilistic Collapse Pressure Analysis

Thedeveloped probabilistic wellbore collapse pressuremodelwas run to calculate the critical mud pressure for different

Journal of Petroleum Engineering 7

Figure 12 Interface of the probabilistic instability model in Excelfor deviated boreholes

casesThemodel utilizes theRisk program and (as has beenmentioned in the previous chapter) the model conductedMonte Carlo simulation for each case to capture the effectof uncertainty in input variables (in situ stresses and rockproperties) in collapse mud pressure In situ stresses androck properties data of a sandstone formation provided inliterature are used in this analysis Uniform distributionwas applied due to limitation of data Then for each casea sensitivity analysis was carried out to figure out whichparameter has more impact on collapse pressures

41 Collapse Pressure Risk Analysis for Normal Faulting StressRegime In this case Monte Carlo simulation was run tocalculate the collapse mud pressure at normal faulting in situstress regime The borehole was assumed to be drilled in asandstone formation with a depth of 3048 meters The usedfield stresses condition and rock properties are described inTables 2 and 3 respectively

In the first simulation all input values were given rangesof values with uniform distributions including well trajecto-ries (inclination angle 119894 and azimuth angle 119896) To calculate thecollapse pressures the developed probabilisticmodel was runapplying Monte Carlo simulation with 10000 iterations usingRisk program [14]

The histogram in Figure 13 illustrates the resulting valuesof possible overbalance pressures (collapse mud pressure)and their occurrence frequencies within the given rangesof input parameters The result shows that the minimumpossible value of collapse pressure is 36780 kPa whereas themaximum possible value is 44790 kPa Table 4 summarizesthe statistics of the simulation

To study the relations between input parameters and theresulting collapse pressures scattered plots of collapse pres-sure versus input parameters were constructed (Figures 14ndash16) The scattered plots of 119875119908 versus input parameters showthat 119875119908 increases with deviation angle (119894) in a relation similar

Table 2 Normal faulting in situ stress regime [4]

Field stresses (kPam) Minimum Maximum120590vg 2194 2262120590Hg 1934 2036120590hg 1876 19341198750 1011 1052

Table 3 Rock properties for a sandstone formation [4]

Rock properties Minimum MaximumCohesion (kPa) 4500 5500Friction angle (deg) 31 33Poissonrsquos ratio 02 03

Table 4 Normal faulting Monte Carlo simulation statistics (chang-ing all input parameters)

Statistics 119875119908 (kPa)Minimum 36780Maximum 44792Mean 40598Standard deviation 168790 confidence interval 38310ndash43250

Table 5 Normal faulting Monte Carlo simulation statistics (con-stant input parameters except 119894)Statistics 119875119908 (kPa)Minimum 38360Maximum 40866Mean 39480Standard deviation 9287190 confidence interval 38363ndash40852 thousand

to (S) shape However there are a lot of noises associated withthis relation due to the effect of other parameters For the restof the parameters nowell-defined relations can be figured outfrom plots even though the overall trends of plotted valuesmay give indications of general relations between 119875119908 andinput parameters

Another type of simulation was conducted while settingall input parameters constant except one in order to studythe effect of uncertainty of unfixed parameter on the criticalmudpressureThen the simulationwas repeated several timesand in each time a different parameter was set variably Forthe first case deviation angle (119894) was the only parameterthat is given a range (from 0 to 90 degrees) whereas therest of parameters were kept constant at the mid values ofthe given ranges The occurrence frequencies of possiblecollapse pressures are illustrated in Figure 17 And Table 5summarizes the statistics of the simulation In order to studythe relation between deviation angle (119894) and collapse pressure119875119908 a scattered plot has been constructed The analysis ofthe resulting collapse pressure shows that deviation angle(119894) has an (S) shape relation with the critical mud pressure(see Figure 18) ) Figures 19 and 20 illustrate the relation of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Petroleum Engineering 3

120590

120590ℎ

y998400

y

120572

i

120590H

z z998400

x

x998400

120579

Figure 1 Stress transformation system for deviated borehole

120590120579119911 = 0120590119903120579 = 0120590119903119911 = 0

(5)

The stresses around a horizontal wellbore wall after excava-tion (Figure 3) can be represented by setting 119894 equal to 1205872and plugging the resulted equation into (2) which gives

120590119903 = 119875119908120590120579 = (120590V + 120590119867 sin2120572 + 120590ℎ cos2120572)

minus (120590V minus 120590119867 sin2120572 minus 120590ℎ cos2120572) cos 2120579 minus 119875119908120590119911 = 120590119867 cos2120572 + 120590ℎ sin2120572

minus 2V (120590V minus 120590119867 sin2120572 minus 120590ℎ cos2120572) cos 2120579120590120579119911 = (120590ℎ minus 120590119867) sin 2120572 cos 120579120590119903120579 = 0120590119903119911 = 0

(6)

In this case where the wellbore axis lies along the maximumhorizontal principal stress (ie 120572 = 0) the stress at boreholewall is described as follows

120590119903 = 119875119908120590120579 = 120590V + 120590ℎ minus 2 (120590V minus 120590ℎ) cos 2120579 minus 119875119908120590119911 = 120590119867 minus 2V (120590V minus 120590ℎ) cos 2120579120590120579119911 = 0120590119903120579 = 0120590119903119911 = 0

(7)

12 Rock Failure Criteria (Mogi-Coulomb Criterion) Thesecond important element in wellbore stability is rock failure

120590

120590ℎ

y998400

120590H

z998400

x998400120579

Figure 2 Stress transformation system for vertical borehole [12]

120590

120590ℎy998400

y

120572

120590H

z z998400

x

x998400120579

Figure 3 Stress transformation system for a vertical borehole

criteria These criteria are sets of functions in stress or strainspaces that describe or predict the condition at which rockfails under external load In literature many of failure criteriahave been developed by scientists who suggested differentapproaches trying to simulate actual rock failures In thispaper the concentration will be onMogi-Coulomb criterion

In 2005 Al-Ajmi and Zimmerman presented a linearform of Mogi criterion [11] Al Ajmi found that the plottedpolyaxial data of some rocks using Mogi failure envelope in120591oct 1205901198982 space can be fit to a linear model as follows

120591oct = 119886 + 1198871205901198982 (8)

where 1205901198982 is the effective normal stress defined by

1205901198982 = 1205901 + 12059032 (9)

120591oct is the octahedral shear stress defined by

120591oct = 13radic(1205901 minus 1205902)2 + (1205902 minus 1205903)2 + (1205903 minus 1205901)2 (10)

And 119886 and 119887 are the intersection of the line on 120591oct-axis and its inclination respectively (Figure 4) Al Ajmi [12]found and verified experimentally that failure under polyaxialstresses can be predicted from triaxial test data using thelinear Mogi criterion Accordingly he developed a new true-triaxial failure criterion called Mogi-Coulomb criterion Fortriaxial stress states (1205902 = 1205903) the linear Mogi criterion isexactly equivalent to Mohr-Coulomb criterion [13]

4 Journal of Petroleum Engineering

Triaxial

Triaxial

Uniaxial

Uniaxial

tension

compression

compression

120591oct

c cot120601 T02 C02 120590m

l

a

b

Figure 4 Mogi-Coulomb failure envelope [12]

Combining (8) and (9) and (7) and rearranging theequation the failure function 119865 will be

119865 = 119886 + 119887 (1205901 + 12059032 )minus 13radic(1205901 minus 1205902)2 + (1205902 minus 1205903)2 + (1205903 minus 1205901)2

(11)

Failure occurs if 119865 le 0The Mogi-Coulomb parameters (119886 119887) can be related to

Coulomb failure parameters (119888 120601) as follows119886 = 2radic23 119888 cos120601119887 = 2radic23 sin120601

(12)

where 119888 is the shear strength or cohesion of the material and120601 is the internal friction angle [13]

2 Modeling Wellbore Stability

21 Vertical Wellbore Failure To ensure mechanical wellborestability in drilling vertical boreholes the lower limit of themud pressure (119875119908) must be considered to be the collapsepressure To find the collapse pressure Kirsch solution isapplied to find stresses around a borehole radial (120590119903) axial(120590119911) and tangential (120590120579) This solution then must be coupledwith a failure criterion (in this research the Mogi-Coulombcriterion will be used) to end up with collapsed pressuremodel

UsingKirsch solution the stresses at the boreholewall canbe defined by the following

120590119903 = 119875119908120590120579 = 119860 minus 119875119908120590119911 = 119861

(13)

119860 and 119861 are constants and can be defined by

119860 = 3120590119867 minus 120590ℎ119861 = 120590V + 2V (120590119867 minus 120590ℎ) (14)

where 120590V is vertical in situ stress 120590119867 and 120590ℎ aremaximum andminimum horizontal in situ stresses respectively And V is amaterial constant called Poissonrsquos ratio

A closed-form solution for collapsed pressure can beachieved by applying (13) and (14) to the Mogi-Coulombcriterion This solution can be used as a borehole failuremodel However in this model there are three differentpermutations of the three principle stresses (120590119903 120590119911 and 120590120579)which need to be studied in order to determine the collapsepressureThese permutations are as follows (1) 120590119911 ge 120590120579 ge 120590119903(2) 120590120579 ge 120590119911 ge 120590119903 and (3) 120590120579 ge 120590119903 ge 120590119911 (Figure 5)

Equations (15) illustrate closed-form solutions to calcu-late collapse pressures in vertical wellbore for in situ stressregimes normal faulting (120590119911 gt 120590120579 gt 120590119903) reverse faulting(120590120579 gt 120590119911 gt 120590119903) and strike slip (120590120579 gt 120590119903 gt 120590119911) respectively1198751199081198871= 16 minus 211988710158402 [(3119860 + 21198871015840119896) minus radic119867 + 12 (1198702 + 1198871015840119860119870)] 1198751199081198872= 12119860 minus 16radic12 [1198861015840 + 1198871015840 (119860 minus 21198750)]2 minus 3 (119860 minus 2119861)21198751199081198873= 16 minus 211988710158402 [(3119860 + 21198871015840119866) minus radic119867 + 12 (1198662 + 1198871015840119860119866)]

(15)

In these equations failure occurs if 119875119908 (internal wellborepressure) le 119875119908119887 (the lower limit of the mud pressure) where

119867 = 1198602 (411988710158402 minus 3) + (1198612 minus 119860119861) (411988710158402 minus 12) (16)

119870 = 1198861015840 + 1198871015840 (119861 minus 21198750) (17)

119866 = 119870 + 1198871015840119860 (18)

It has been proven in literature that the most commonstress distribution is the second case when 120590120579 ge 120590119911 ge 120590119903 andit is called reverse faulting stress regime (eg [4ndash6 8ndash10])

22 Probabilistic Approach Risk Analysis in Vertical WellboreStability In 2010Al-Harthi andAl-Ajmi applied risk analysisto wellbore stability They utilized the Mogi-Coulomb stabil-ity model (see (15) (19) and (20)) to conduct a sensitivityanalysis in vertical wellbores They carried out Monte Carlosimulation using Risk program They also conducted asensitivity analysis using Mohr-Coulomb stability modelThe comparison between the two analyses showed that theMohr-Coulomb criterion tends to overestimate minimumoverbalance mud pressure that is required to keep boreholestable during drilling (as illustrated in Figure 6) They alsoconcluded that risk analysis is a good tool to estimate thecritical mud pressure of drilling fluid with existence ofuncertainties in input parameters The sensitivity analysis ofthe Mogi-Coulomb criterion (which is taking in account theeffect of the intermediate principal stress) showed that themaximum horizontal stress (120590119867) is the most critical factor

Journal of Petroleum Engineering 5

1205901

1205902

1205903

(a)

1205901

1205902

1205903

(b)

1205901

1205902

1205903

(c)

Figure 5 In situ stress regimes (a) normal fault (b) reverse fault and (c) strike slip [12]

times10minus4

Mohr-CMogi-C

0

05

1

15

2

25

3

35

4

45

Prob

abili

ty d

istrib

utio

n fu

nctio

n

12 13 14 15 16 17 18 19 20 2111

Overbalance pressure (times103 kPa)

Figure 6 Overbalance pressure in the strike slip regimeTheMogi-Coulomb criterion on the left side of the graph and the Mohr-Coulomb criterion on the right side [4]

1800

0

1700

0

1600

0

17500

16500

18500

15500

Mean of overbalance (kPa) (Mogi-C)

Vertical stressPoissonrsquos ratio

Minimum horizontal stressPore pressure

CohesionFriction angle

Maximum horizontal stress

Figure 7 Sensitivity analysis in the strike slip regime (tornado chartfor Mogi-Coulomb) [4]

in vertical wellbore stability analysis under any in situ stressregime (as illustrated in Figures 7 and 8)

23 Nonvertical Wellbore Failure In order to determine thecritical mud pressure (the minimum mud pressure required

2100

0

2200

0

2000

0

2300

0

22500

20500

23500

21500

Mean of overbalance (kPa) (Mohr-C)

Poissonrsquos ratioVertical stress

Minimum horizontal stressPore pressure

CohesionFriction

Maximum horizontal stress

Figure 8 Sensitivity analysis in the strike slip regime (tornado chartfor Mohr-Coulomb) [4]

to prevent borehole collapse) in deviated boreholes thestresses at the borehole wall should be calculated using (4)and then compared with Mogi-Coulomb failure criterionHowever the resulting tangential and axial stresses are notprincipal stresses as the shear stress 120590120579119911 may not be equal tozero As a result the tangential and the axial stresses shouldbe determined using the following

1205901 = 12 (120590119909 + 120590119910) + radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

1205902 = 12 (120590119909 + 120590119910) minus radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

(19)

where 1205901 and 1205902 are tangential and axial stresses respectively120590119909 and 120590119910 are normal stresses in 119909 and 119910 directions respec-tively and 120591119909119910 is the shear stress in 119909-119910 direction

Introducing the resulting principal stresses to the Mogi-Coulomb failure criterion a 4th-degree equation needs tobe solved in order to find the collapse pressure This meansthere is no closed-form analytical solution that can be derivedfor collapse pressure in deviated boreholes utilizing Mogi-Coulomb failure criterion similar to vertical boreholes or ifMohr-Coulomb failure criterion had been utilized [12]

6 Journal of Petroleum Engineering

Figure 9 Interface of the deterministic instability model in Excelfor nonvertical boreholes

Table 1 Vertical wellbore failure for different stress regimes

Input Value120590vg (psift) 11120590Hg (psift) 1120590hg (psift) 091198750 (psift) 046Cohesion (psi) 870Friction angle (deg) 313Poissonrsquos ratio 033

3 Probabilistic Numerical Models forNonvertical Wellbores

The previous section illustrates that there is no closed-formsolution to model deviated wellbore stability However Al-Ajmi [12] has applied iterative loops in a computer programto calculate the collapse pressure for nonvertical boreholesMathCAD program was used to perform these iterated loopsto find the critical mud pressure (see the Appendix)

This model was utilized to build a stability model in theExcel program In the developed model equations of stressesand Mogi-Coulomb failure criterion have been convertedfrom MathCAD language to the language of the Excelprogram Also the macro tool (which is using Visual BasicProgramming Language) in the Excel has been utilized to dothe iterated loops to calculate the collapse pressure Figure 9illustrates the interface of the developed model

The input parameters of the developed model are depth(ℎ) vertical stress gradient (120590vg) maximum horizontal stressgradient (120590Hg) minimum horizontal stress gradient (120590hg)pore pressure gradient (1198750119892) cohesion (119888) friction angle (120601)and Poissonrsquos ratio (v)

The purpose of this model is to calculate the collapsepressure for a given set of input parameters that are con-stant (having static values) for 100 different well trajectoryscenarios In other words themodel will find out the collapsepressure for 10 values of inclination angle (119894) ranging fromzero to 90 degrees multiplied by 10 values of azimuth angle(119896) ranging from zero to 90 degrees too

To ensure the validity of this model it was run for acertain case (inputs are listed inTable 1) and compared againstAl-Ajmirsquos MathCADmodel The output result was absolutelyidentical as illustrated in Figures 10 and 11

Figure 10 Resulting collapse pressure from the developed Excelmodel

Figure 11 Resulting collapse pressure from the MathCAD model

The model is consistent with Al-Ajmirsquos MathCAD modelin calculating collapse pressure for static input parametershowever the target is to end up with a probabilistic modelin order to conduct instability risk analysis for nonvertical(deviated) wellbores As a result the Risk program wasutilized with the new Excel based instability model to give aprobabilistic model (see Figure 12)

31 An Overview of Risk Program Risk is an add-in pro-gram to Microsoft Excel It is using Monte Carlo simulationthat allows user to test variety of scenarios in a given modeland determine the probability distribution of each outcomeWhen combined with a well-built model it can give you agood feel for how likely a given outcome is for amodeled planor business venture In this model the sixth version of Riskprogram is used to run Monte Carlo simulation It allowsusers to make decisions using the sensitivity risk analysis Inthis program parameters can be set fixed or can be given adesired range and distribution Due to the limitation of in situstresses and rock properties data uniform distribution wasused mostly in data simulations

4 Probabilistic Collapse Pressure Analysis

Thedeveloped probabilistic wellbore collapse pressuremodelwas run to calculate the critical mud pressure for different

Journal of Petroleum Engineering 7

Figure 12 Interface of the probabilistic instability model in Excelfor deviated boreholes

casesThemodel utilizes theRisk program and (as has beenmentioned in the previous chapter) the model conductedMonte Carlo simulation for each case to capture the effectof uncertainty in input variables (in situ stresses and rockproperties) in collapse mud pressure In situ stresses androck properties data of a sandstone formation provided inliterature are used in this analysis Uniform distributionwas applied due to limitation of data Then for each casea sensitivity analysis was carried out to figure out whichparameter has more impact on collapse pressures

41 Collapse Pressure Risk Analysis for Normal Faulting StressRegime In this case Monte Carlo simulation was run tocalculate the collapse mud pressure at normal faulting in situstress regime The borehole was assumed to be drilled in asandstone formation with a depth of 3048 meters The usedfield stresses condition and rock properties are described inTables 2 and 3 respectively

In the first simulation all input values were given rangesof values with uniform distributions including well trajecto-ries (inclination angle 119894 and azimuth angle 119896) To calculate thecollapse pressures the developed probabilisticmodel was runapplying Monte Carlo simulation with 10000 iterations usingRisk program [14]

The histogram in Figure 13 illustrates the resulting valuesof possible overbalance pressures (collapse mud pressure)and their occurrence frequencies within the given rangesof input parameters The result shows that the minimumpossible value of collapse pressure is 36780 kPa whereas themaximum possible value is 44790 kPa Table 4 summarizesthe statistics of the simulation

To study the relations between input parameters and theresulting collapse pressures scattered plots of collapse pres-sure versus input parameters were constructed (Figures 14ndash16) The scattered plots of 119875119908 versus input parameters showthat 119875119908 increases with deviation angle (119894) in a relation similar

Table 2 Normal faulting in situ stress regime [4]

Field stresses (kPam) Minimum Maximum120590vg 2194 2262120590Hg 1934 2036120590hg 1876 19341198750 1011 1052

Table 3 Rock properties for a sandstone formation [4]

Rock properties Minimum MaximumCohesion (kPa) 4500 5500Friction angle (deg) 31 33Poissonrsquos ratio 02 03

Table 4 Normal faulting Monte Carlo simulation statistics (chang-ing all input parameters)

Statistics 119875119908 (kPa)Minimum 36780Maximum 44792Mean 40598Standard deviation 168790 confidence interval 38310ndash43250

Table 5 Normal faulting Monte Carlo simulation statistics (con-stant input parameters except 119894)Statistics 119875119908 (kPa)Minimum 38360Maximum 40866Mean 39480Standard deviation 9287190 confidence interval 38363ndash40852 thousand

to (S) shape However there are a lot of noises associated withthis relation due to the effect of other parameters For the restof the parameters nowell-defined relations can be figured outfrom plots even though the overall trends of plotted valuesmay give indications of general relations between 119875119908 andinput parameters

Another type of simulation was conducted while settingall input parameters constant except one in order to studythe effect of uncertainty of unfixed parameter on the criticalmudpressureThen the simulationwas repeated several timesand in each time a different parameter was set variably Forthe first case deviation angle (119894) was the only parameterthat is given a range (from 0 to 90 degrees) whereas therest of parameters were kept constant at the mid values ofthe given ranges The occurrence frequencies of possiblecollapse pressures are illustrated in Figure 17 And Table 5summarizes the statistics of the simulation In order to studythe relation between deviation angle (119894) and collapse pressure119875119908 a scattered plot has been constructed The analysis ofthe resulting collapse pressure shows that deviation angle(119894) has an (S) shape relation with the critical mud pressure(see Figure 18) ) Figures 19 and 20 illustrate the relation of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

4 Journal of Petroleum Engineering

Triaxial

Triaxial

Uniaxial

Uniaxial

tension

compression

compression

120591oct

c cot120601 T02 C02 120590m

l

a

b

Figure 4 Mogi-Coulomb failure envelope [12]

Combining (8) and (9) and (7) and rearranging theequation the failure function 119865 will be

119865 = 119886 + 119887 (1205901 + 12059032 )minus 13radic(1205901 minus 1205902)2 + (1205902 minus 1205903)2 + (1205903 minus 1205901)2

(11)

Failure occurs if 119865 le 0The Mogi-Coulomb parameters (119886 119887) can be related to

Coulomb failure parameters (119888 120601) as follows119886 = 2radic23 119888 cos120601119887 = 2radic23 sin120601

(12)

where 119888 is the shear strength or cohesion of the material and120601 is the internal friction angle [13]

2 Modeling Wellbore Stability

21 Vertical Wellbore Failure To ensure mechanical wellborestability in drilling vertical boreholes the lower limit of themud pressure (119875119908) must be considered to be the collapsepressure To find the collapse pressure Kirsch solution isapplied to find stresses around a borehole radial (120590119903) axial(120590119911) and tangential (120590120579) This solution then must be coupledwith a failure criterion (in this research the Mogi-Coulombcriterion will be used) to end up with collapsed pressuremodel

UsingKirsch solution the stresses at the boreholewall canbe defined by the following

120590119903 = 119875119908120590120579 = 119860 minus 119875119908120590119911 = 119861

(13)

119860 and 119861 are constants and can be defined by

119860 = 3120590119867 minus 120590ℎ119861 = 120590V + 2V (120590119867 minus 120590ℎ) (14)

where 120590V is vertical in situ stress 120590119867 and 120590ℎ aremaximum andminimum horizontal in situ stresses respectively And V is amaterial constant called Poissonrsquos ratio

A closed-form solution for collapsed pressure can beachieved by applying (13) and (14) to the Mogi-Coulombcriterion This solution can be used as a borehole failuremodel However in this model there are three differentpermutations of the three principle stresses (120590119903 120590119911 and 120590120579)which need to be studied in order to determine the collapsepressureThese permutations are as follows (1) 120590119911 ge 120590120579 ge 120590119903(2) 120590120579 ge 120590119911 ge 120590119903 and (3) 120590120579 ge 120590119903 ge 120590119911 (Figure 5)

Equations (15) illustrate closed-form solutions to calcu-late collapse pressures in vertical wellbore for in situ stressregimes normal faulting (120590119911 gt 120590120579 gt 120590119903) reverse faulting(120590120579 gt 120590119911 gt 120590119903) and strike slip (120590120579 gt 120590119903 gt 120590119911) respectively1198751199081198871= 16 minus 211988710158402 [(3119860 + 21198871015840119896) minus radic119867 + 12 (1198702 + 1198871015840119860119870)] 1198751199081198872= 12119860 minus 16radic12 [1198861015840 + 1198871015840 (119860 minus 21198750)]2 minus 3 (119860 minus 2119861)21198751199081198873= 16 minus 211988710158402 [(3119860 + 21198871015840119866) minus radic119867 + 12 (1198662 + 1198871015840119860119866)]

(15)

In these equations failure occurs if 119875119908 (internal wellborepressure) le 119875119908119887 (the lower limit of the mud pressure) where

119867 = 1198602 (411988710158402 minus 3) + (1198612 minus 119860119861) (411988710158402 minus 12) (16)

119870 = 1198861015840 + 1198871015840 (119861 minus 21198750) (17)

119866 = 119870 + 1198871015840119860 (18)

It has been proven in literature that the most commonstress distribution is the second case when 120590120579 ge 120590119911 ge 120590119903 andit is called reverse faulting stress regime (eg [4ndash6 8ndash10])

22 Probabilistic Approach Risk Analysis in Vertical WellboreStability In 2010Al-Harthi andAl-Ajmi applied risk analysisto wellbore stability They utilized the Mogi-Coulomb stabil-ity model (see (15) (19) and (20)) to conduct a sensitivityanalysis in vertical wellbores They carried out Monte Carlosimulation using Risk program They also conducted asensitivity analysis using Mohr-Coulomb stability modelThe comparison between the two analyses showed that theMohr-Coulomb criterion tends to overestimate minimumoverbalance mud pressure that is required to keep boreholestable during drilling (as illustrated in Figure 6) They alsoconcluded that risk analysis is a good tool to estimate thecritical mud pressure of drilling fluid with existence ofuncertainties in input parameters The sensitivity analysis ofthe Mogi-Coulomb criterion (which is taking in account theeffect of the intermediate principal stress) showed that themaximum horizontal stress (120590119867) is the most critical factor

Journal of Petroleum Engineering 5

1205901

1205902

1205903

(a)

1205901

1205902

1205903

(b)

1205901

1205902

1205903

(c)

Figure 5 In situ stress regimes (a) normal fault (b) reverse fault and (c) strike slip [12]

times10minus4

Mohr-CMogi-C

0

05

1

15

2

25

3

35

4

45

Prob

abili

ty d

istrib

utio

n fu

nctio

n

12 13 14 15 16 17 18 19 20 2111

Overbalance pressure (times103 kPa)

Figure 6 Overbalance pressure in the strike slip regimeTheMogi-Coulomb criterion on the left side of the graph and the Mohr-Coulomb criterion on the right side [4]

1800

0

1700

0

1600

0

17500

16500

18500

15500

Mean of overbalance (kPa) (Mogi-C)

Vertical stressPoissonrsquos ratio

Minimum horizontal stressPore pressure

CohesionFriction angle

Maximum horizontal stress

Figure 7 Sensitivity analysis in the strike slip regime (tornado chartfor Mogi-Coulomb) [4]

in vertical wellbore stability analysis under any in situ stressregime (as illustrated in Figures 7 and 8)

23 Nonvertical Wellbore Failure In order to determine thecritical mud pressure (the minimum mud pressure required

2100

0

2200

0

2000

0

2300

0

22500

20500

23500

21500

Mean of overbalance (kPa) (Mohr-C)

Poissonrsquos ratioVertical stress

Minimum horizontal stressPore pressure

CohesionFriction

Maximum horizontal stress

Figure 8 Sensitivity analysis in the strike slip regime (tornado chartfor Mohr-Coulomb) [4]

to prevent borehole collapse) in deviated boreholes thestresses at the borehole wall should be calculated using (4)and then compared with Mogi-Coulomb failure criterionHowever the resulting tangential and axial stresses are notprincipal stresses as the shear stress 120590120579119911 may not be equal tozero As a result the tangential and the axial stresses shouldbe determined using the following

1205901 = 12 (120590119909 + 120590119910) + radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

1205902 = 12 (120590119909 + 120590119910) minus radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

(19)

where 1205901 and 1205902 are tangential and axial stresses respectively120590119909 and 120590119910 are normal stresses in 119909 and 119910 directions respec-tively and 120591119909119910 is the shear stress in 119909-119910 direction

Introducing the resulting principal stresses to the Mogi-Coulomb failure criterion a 4th-degree equation needs tobe solved in order to find the collapse pressure This meansthere is no closed-form analytical solution that can be derivedfor collapse pressure in deviated boreholes utilizing Mogi-Coulomb failure criterion similar to vertical boreholes or ifMohr-Coulomb failure criterion had been utilized [12]

6 Journal of Petroleum Engineering

Figure 9 Interface of the deterministic instability model in Excelfor nonvertical boreholes

Table 1 Vertical wellbore failure for different stress regimes

Input Value120590vg (psift) 11120590Hg (psift) 1120590hg (psift) 091198750 (psift) 046Cohesion (psi) 870Friction angle (deg) 313Poissonrsquos ratio 033

3 Probabilistic Numerical Models forNonvertical Wellbores

The previous section illustrates that there is no closed-formsolution to model deviated wellbore stability However Al-Ajmi [12] has applied iterative loops in a computer programto calculate the collapse pressure for nonvertical boreholesMathCAD program was used to perform these iterated loopsto find the critical mud pressure (see the Appendix)

This model was utilized to build a stability model in theExcel program In the developed model equations of stressesand Mogi-Coulomb failure criterion have been convertedfrom MathCAD language to the language of the Excelprogram Also the macro tool (which is using Visual BasicProgramming Language) in the Excel has been utilized to dothe iterated loops to calculate the collapse pressure Figure 9illustrates the interface of the developed model

The input parameters of the developed model are depth(ℎ) vertical stress gradient (120590vg) maximum horizontal stressgradient (120590Hg) minimum horizontal stress gradient (120590hg)pore pressure gradient (1198750119892) cohesion (119888) friction angle (120601)and Poissonrsquos ratio (v)

The purpose of this model is to calculate the collapsepressure for a given set of input parameters that are con-stant (having static values) for 100 different well trajectoryscenarios In other words themodel will find out the collapsepressure for 10 values of inclination angle (119894) ranging fromzero to 90 degrees multiplied by 10 values of azimuth angle(119896) ranging from zero to 90 degrees too

To ensure the validity of this model it was run for acertain case (inputs are listed inTable 1) and compared againstAl-Ajmirsquos MathCADmodel The output result was absolutelyidentical as illustrated in Figures 10 and 11

Figure 10 Resulting collapse pressure from the developed Excelmodel

Figure 11 Resulting collapse pressure from the MathCAD model

The model is consistent with Al-Ajmirsquos MathCAD modelin calculating collapse pressure for static input parametershowever the target is to end up with a probabilistic modelin order to conduct instability risk analysis for nonvertical(deviated) wellbores As a result the Risk program wasutilized with the new Excel based instability model to give aprobabilistic model (see Figure 12)

31 An Overview of Risk Program Risk is an add-in pro-gram to Microsoft Excel It is using Monte Carlo simulationthat allows user to test variety of scenarios in a given modeland determine the probability distribution of each outcomeWhen combined with a well-built model it can give you agood feel for how likely a given outcome is for amodeled planor business venture In this model the sixth version of Riskprogram is used to run Monte Carlo simulation It allowsusers to make decisions using the sensitivity risk analysis Inthis program parameters can be set fixed or can be given adesired range and distribution Due to the limitation of in situstresses and rock properties data uniform distribution wasused mostly in data simulations

4 Probabilistic Collapse Pressure Analysis

Thedeveloped probabilistic wellbore collapse pressuremodelwas run to calculate the critical mud pressure for different

Journal of Petroleum Engineering 7

Figure 12 Interface of the probabilistic instability model in Excelfor deviated boreholes

casesThemodel utilizes theRisk program and (as has beenmentioned in the previous chapter) the model conductedMonte Carlo simulation for each case to capture the effectof uncertainty in input variables (in situ stresses and rockproperties) in collapse mud pressure In situ stresses androck properties data of a sandstone formation provided inliterature are used in this analysis Uniform distributionwas applied due to limitation of data Then for each casea sensitivity analysis was carried out to figure out whichparameter has more impact on collapse pressures

41 Collapse Pressure Risk Analysis for Normal Faulting StressRegime In this case Monte Carlo simulation was run tocalculate the collapse mud pressure at normal faulting in situstress regime The borehole was assumed to be drilled in asandstone formation with a depth of 3048 meters The usedfield stresses condition and rock properties are described inTables 2 and 3 respectively

In the first simulation all input values were given rangesof values with uniform distributions including well trajecto-ries (inclination angle 119894 and azimuth angle 119896) To calculate thecollapse pressures the developed probabilisticmodel was runapplying Monte Carlo simulation with 10000 iterations usingRisk program [14]

The histogram in Figure 13 illustrates the resulting valuesof possible overbalance pressures (collapse mud pressure)and their occurrence frequencies within the given rangesof input parameters The result shows that the minimumpossible value of collapse pressure is 36780 kPa whereas themaximum possible value is 44790 kPa Table 4 summarizesthe statistics of the simulation

To study the relations between input parameters and theresulting collapse pressures scattered plots of collapse pres-sure versus input parameters were constructed (Figures 14ndash16) The scattered plots of 119875119908 versus input parameters showthat 119875119908 increases with deviation angle (119894) in a relation similar

Table 2 Normal faulting in situ stress regime [4]

Field stresses (kPam) Minimum Maximum120590vg 2194 2262120590Hg 1934 2036120590hg 1876 19341198750 1011 1052

Table 3 Rock properties for a sandstone formation [4]

Rock properties Minimum MaximumCohesion (kPa) 4500 5500Friction angle (deg) 31 33Poissonrsquos ratio 02 03

Table 4 Normal faulting Monte Carlo simulation statistics (chang-ing all input parameters)

Statistics 119875119908 (kPa)Minimum 36780Maximum 44792Mean 40598Standard deviation 168790 confidence interval 38310ndash43250

Table 5 Normal faulting Monte Carlo simulation statistics (con-stant input parameters except 119894)Statistics 119875119908 (kPa)Minimum 38360Maximum 40866Mean 39480Standard deviation 9287190 confidence interval 38363ndash40852 thousand

to (S) shape However there are a lot of noises associated withthis relation due to the effect of other parameters For the restof the parameters nowell-defined relations can be figured outfrom plots even though the overall trends of plotted valuesmay give indications of general relations between 119875119908 andinput parameters

Another type of simulation was conducted while settingall input parameters constant except one in order to studythe effect of uncertainty of unfixed parameter on the criticalmudpressureThen the simulationwas repeated several timesand in each time a different parameter was set variably Forthe first case deviation angle (119894) was the only parameterthat is given a range (from 0 to 90 degrees) whereas therest of parameters were kept constant at the mid values ofthe given ranges The occurrence frequencies of possiblecollapse pressures are illustrated in Figure 17 And Table 5summarizes the statistics of the simulation In order to studythe relation between deviation angle (119894) and collapse pressure119875119908 a scattered plot has been constructed The analysis ofthe resulting collapse pressure shows that deviation angle(119894) has an (S) shape relation with the critical mud pressure(see Figure 18) ) Figures 19 and 20 illustrate the relation of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Petroleum Engineering 5

1205901

1205902

1205903

(a)

1205901

1205902

1205903

(b)

1205901

1205902

1205903

(c)

Figure 5 In situ stress regimes (a) normal fault (b) reverse fault and (c) strike slip [12]

times10minus4

Mohr-CMogi-C

0

05

1

15

2

25

3

35

4

45

Prob

abili

ty d

istrib

utio

n fu

nctio

n

12 13 14 15 16 17 18 19 20 2111

Overbalance pressure (times103 kPa)

Figure 6 Overbalance pressure in the strike slip regimeTheMogi-Coulomb criterion on the left side of the graph and the Mohr-Coulomb criterion on the right side [4]

1800

0

1700

0

1600

0

17500

16500

18500

15500

Mean of overbalance (kPa) (Mogi-C)

Vertical stressPoissonrsquos ratio

Minimum horizontal stressPore pressure

CohesionFriction angle

Maximum horizontal stress

Figure 7 Sensitivity analysis in the strike slip regime (tornado chartfor Mogi-Coulomb) [4]

in vertical wellbore stability analysis under any in situ stressregime (as illustrated in Figures 7 and 8)

23 Nonvertical Wellbore Failure In order to determine thecritical mud pressure (the minimum mud pressure required

2100

0

2200

0

2000

0

2300

0

22500

20500

23500

21500

Mean of overbalance (kPa) (Mohr-C)

Poissonrsquos ratioVertical stress

Minimum horizontal stressPore pressure

CohesionFriction

Maximum horizontal stress

Figure 8 Sensitivity analysis in the strike slip regime (tornado chartfor Mohr-Coulomb) [4]

to prevent borehole collapse) in deviated boreholes thestresses at the borehole wall should be calculated using (4)and then compared with Mogi-Coulomb failure criterionHowever the resulting tangential and axial stresses are notprincipal stresses as the shear stress 120590120579119911 may not be equal tozero As a result the tangential and the axial stresses shouldbe determined using the following

1205901 = 12 (120590119909 + 120590119910) + radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

1205902 = 12 (120590119909 + 120590119910) minus radic1205912119909119910 + 14 (120590119909 minus 120590119910)2

(19)

where 1205901 and 1205902 are tangential and axial stresses respectively120590119909 and 120590119910 are normal stresses in 119909 and 119910 directions respec-tively and 120591119909119910 is the shear stress in 119909-119910 direction

Introducing the resulting principal stresses to the Mogi-Coulomb failure criterion a 4th-degree equation needs tobe solved in order to find the collapse pressure This meansthere is no closed-form analytical solution that can be derivedfor collapse pressure in deviated boreholes utilizing Mogi-Coulomb failure criterion similar to vertical boreholes or ifMohr-Coulomb failure criterion had been utilized [12]

6 Journal of Petroleum Engineering

Figure 9 Interface of the deterministic instability model in Excelfor nonvertical boreholes

Table 1 Vertical wellbore failure for different stress regimes

Input Value120590vg (psift) 11120590Hg (psift) 1120590hg (psift) 091198750 (psift) 046Cohesion (psi) 870Friction angle (deg) 313Poissonrsquos ratio 033

3 Probabilistic Numerical Models forNonvertical Wellbores

The previous section illustrates that there is no closed-formsolution to model deviated wellbore stability However Al-Ajmi [12] has applied iterative loops in a computer programto calculate the collapse pressure for nonvertical boreholesMathCAD program was used to perform these iterated loopsto find the critical mud pressure (see the Appendix)

This model was utilized to build a stability model in theExcel program In the developed model equations of stressesand Mogi-Coulomb failure criterion have been convertedfrom MathCAD language to the language of the Excelprogram Also the macro tool (which is using Visual BasicProgramming Language) in the Excel has been utilized to dothe iterated loops to calculate the collapse pressure Figure 9illustrates the interface of the developed model

The input parameters of the developed model are depth(ℎ) vertical stress gradient (120590vg) maximum horizontal stressgradient (120590Hg) minimum horizontal stress gradient (120590hg)pore pressure gradient (1198750119892) cohesion (119888) friction angle (120601)and Poissonrsquos ratio (v)

The purpose of this model is to calculate the collapsepressure for a given set of input parameters that are con-stant (having static values) for 100 different well trajectoryscenarios In other words themodel will find out the collapsepressure for 10 values of inclination angle (119894) ranging fromzero to 90 degrees multiplied by 10 values of azimuth angle(119896) ranging from zero to 90 degrees too

To ensure the validity of this model it was run for acertain case (inputs are listed inTable 1) and compared againstAl-Ajmirsquos MathCADmodel The output result was absolutelyidentical as illustrated in Figures 10 and 11

Figure 10 Resulting collapse pressure from the developed Excelmodel

Figure 11 Resulting collapse pressure from the MathCAD model

The model is consistent with Al-Ajmirsquos MathCAD modelin calculating collapse pressure for static input parametershowever the target is to end up with a probabilistic modelin order to conduct instability risk analysis for nonvertical(deviated) wellbores As a result the Risk program wasutilized with the new Excel based instability model to give aprobabilistic model (see Figure 12)

31 An Overview of Risk Program Risk is an add-in pro-gram to Microsoft Excel It is using Monte Carlo simulationthat allows user to test variety of scenarios in a given modeland determine the probability distribution of each outcomeWhen combined with a well-built model it can give you agood feel for how likely a given outcome is for amodeled planor business venture In this model the sixth version of Riskprogram is used to run Monte Carlo simulation It allowsusers to make decisions using the sensitivity risk analysis Inthis program parameters can be set fixed or can be given adesired range and distribution Due to the limitation of in situstresses and rock properties data uniform distribution wasused mostly in data simulations

4 Probabilistic Collapse Pressure Analysis

Thedeveloped probabilistic wellbore collapse pressuremodelwas run to calculate the critical mud pressure for different

Journal of Petroleum Engineering 7

Figure 12 Interface of the probabilistic instability model in Excelfor deviated boreholes

casesThemodel utilizes theRisk program and (as has beenmentioned in the previous chapter) the model conductedMonte Carlo simulation for each case to capture the effectof uncertainty in input variables (in situ stresses and rockproperties) in collapse mud pressure In situ stresses androck properties data of a sandstone formation provided inliterature are used in this analysis Uniform distributionwas applied due to limitation of data Then for each casea sensitivity analysis was carried out to figure out whichparameter has more impact on collapse pressures

41 Collapse Pressure Risk Analysis for Normal Faulting StressRegime In this case Monte Carlo simulation was run tocalculate the collapse mud pressure at normal faulting in situstress regime The borehole was assumed to be drilled in asandstone formation with a depth of 3048 meters The usedfield stresses condition and rock properties are described inTables 2 and 3 respectively

In the first simulation all input values were given rangesof values with uniform distributions including well trajecto-ries (inclination angle 119894 and azimuth angle 119896) To calculate thecollapse pressures the developed probabilisticmodel was runapplying Monte Carlo simulation with 10000 iterations usingRisk program [14]

The histogram in Figure 13 illustrates the resulting valuesof possible overbalance pressures (collapse mud pressure)and their occurrence frequencies within the given rangesof input parameters The result shows that the minimumpossible value of collapse pressure is 36780 kPa whereas themaximum possible value is 44790 kPa Table 4 summarizesthe statistics of the simulation

To study the relations between input parameters and theresulting collapse pressures scattered plots of collapse pres-sure versus input parameters were constructed (Figures 14ndash16) The scattered plots of 119875119908 versus input parameters showthat 119875119908 increases with deviation angle (119894) in a relation similar

Table 2 Normal faulting in situ stress regime [4]

Field stresses (kPam) Minimum Maximum120590vg 2194 2262120590Hg 1934 2036120590hg 1876 19341198750 1011 1052

Table 3 Rock properties for a sandstone formation [4]

Rock properties Minimum MaximumCohesion (kPa) 4500 5500Friction angle (deg) 31 33Poissonrsquos ratio 02 03

Table 4 Normal faulting Monte Carlo simulation statistics (chang-ing all input parameters)

Statistics 119875119908 (kPa)Minimum 36780Maximum 44792Mean 40598Standard deviation 168790 confidence interval 38310ndash43250

Table 5 Normal faulting Monte Carlo simulation statistics (con-stant input parameters except 119894)Statistics 119875119908 (kPa)Minimum 38360Maximum 40866Mean 39480Standard deviation 9287190 confidence interval 38363ndash40852 thousand

to (S) shape However there are a lot of noises associated withthis relation due to the effect of other parameters For the restof the parameters nowell-defined relations can be figured outfrom plots even though the overall trends of plotted valuesmay give indications of general relations between 119875119908 andinput parameters

Another type of simulation was conducted while settingall input parameters constant except one in order to studythe effect of uncertainty of unfixed parameter on the criticalmudpressureThen the simulationwas repeated several timesand in each time a different parameter was set variably Forthe first case deviation angle (119894) was the only parameterthat is given a range (from 0 to 90 degrees) whereas therest of parameters were kept constant at the mid values ofthe given ranges The occurrence frequencies of possiblecollapse pressures are illustrated in Figure 17 And Table 5summarizes the statistics of the simulation In order to studythe relation between deviation angle (119894) and collapse pressure119875119908 a scattered plot has been constructed The analysis ofthe resulting collapse pressure shows that deviation angle(119894) has an (S) shape relation with the critical mud pressure(see Figure 18) ) Figures 19 and 20 illustrate the relation of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Petroleum Engineering

Figure 9 Interface of the deterministic instability model in Excelfor nonvertical boreholes

Table 1 Vertical wellbore failure for different stress regimes

Input Value120590vg (psift) 11120590Hg (psift) 1120590hg (psift) 091198750 (psift) 046Cohesion (psi) 870Friction angle (deg) 313Poissonrsquos ratio 033

3 Probabilistic Numerical Models forNonvertical Wellbores

The previous section illustrates that there is no closed-formsolution to model deviated wellbore stability However Al-Ajmi [12] has applied iterative loops in a computer programto calculate the collapse pressure for nonvertical boreholesMathCAD program was used to perform these iterated loopsto find the critical mud pressure (see the Appendix)

This model was utilized to build a stability model in theExcel program In the developed model equations of stressesand Mogi-Coulomb failure criterion have been convertedfrom MathCAD language to the language of the Excelprogram Also the macro tool (which is using Visual BasicProgramming Language) in the Excel has been utilized to dothe iterated loops to calculate the collapse pressure Figure 9illustrates the interface of the developed model

The input parameters of the developed model are depth(ℎ) vertical stress gradient (120590vg) maximum horizontal stressgradient (120590Hg) minimum horizontal stress gradient (120590hg)pore pressure gradient (1198750119892) cohesion (119888) friction angle (120601)and Poissonrsquos ratio (v)

The purpose of this model is to calculate the collapsepressure for a given set of input parameters that are con-stant (having static values) for 100 different well trajectoryscenarios In other words themodel will find out the collapsepressure for 10 values of inclination angle (119894) ranging fromzero to 90 degrees multiplied by 10 values of azimuth angle(119896) ranging from zero to 90 degrees too

To ensure the validity of this model it was run for acertain case (inputs are listed inTable 1) and compared againstAl-Ajmirsquos MathCADmodel The output result was absolutelyidentical as illustrated in Figures 10 and 11

Figure 10 Resulting collapse pressure from the developed Excelmodel

Figure 11 Resulting collapse pressure from the MathCAD model

The model is consistent with Al-Ajmirsquos MathCAD modelin calculating collapse pressure for static input parametershowever the target is to end up with a probabilistic modelin order to conduct instability risk analysis for nonvertical(deviated) wellbores As a result the Risk program wasutilized with the new Excel based instability model to give aprobabilistic model (see Figure 12)

31 An Overview of Risk Program Risk is an add-in pro-gram to Microsoft Excel It is using Monte Carlo simulationthat allows user to test variety of scenarios in a given modeland determine the probability distribution of each outcomeWhen combined with a well-built model it can give you agood feel for how likely a given outcome is for amodeled planor business venture In this model the sixth version of Riskprogram is used to run Monte Carlo simulation It allowsusers to make decisions using the sensitivity risk analysis Inthis program parameters can be set fixed or can be given adesired range and distribution Due to the limitation of in situstresses and rock properties data uniform distribution wasused mostly in data simulations

4 Probabilistic Collapse Pressure Analysis

Thedeveloped probabilistic wellbore collapse pressuremodelwas run to calculate the critical mud pressure for different

Journal of Petroleum Engineering 7

Figure 12 Interface of the probabilistic instability model in Excelfor deviated boreholes

casesThemodel utilizes theRisk program and (as has beenmentioned in the previous chapter) the model conductedMonte Carlo simulation for each case to capture the effectof uncertainty in input variables (in situ stresses and rockproperties) in collapse mud pressure In situ stresses androck properties data of a sandstone formation provided inliterature are used in this analysis Uniform distributionwas applied due to limitation of data Then for each casea sensitivity analysis was carried out to figure out whichparameter has more impact on collapse pressures

41 Collapse Pressure Risk Analysis for Normal Faulting StressRegime In this case Monte Carlo simulation was run tocalculate the collapse mud pressure at normal faulting in situstress regime The borehole was assumed to be drilled in asandstone formation with a depth of 3048 meters The usedfield stresses condition and rock properties are described inTables 2 and 3 respectively

In the first simulation all input values were given rangesof values with uniform distributions including well trajecto-ries (inclination angle 119894 and azimuth angle 119896) To calculate thecollapse pressures the developed probabilisticmodel was runapplying Monte Carlo simulation with 10000 iterations usingRisk program [14]

The histogram in Figure 13 illustrates the resulting valuesof possible overbalance pressures (collapse mud pressure)and their occurrence frequencies within the given rangesof input parameters The result shows that the minimumpossible value of collapse pressure is 36780 kPa whereas themaximum possible value is 44790 kPa Table 4 summarizesthe statistics of the simulation

To study the relations between input parameters and theresulting collapse pressures scattered plots of collapse pres-sure versus input parameters were constructed (Figures 14ndash16) The scattered plots of 119875119908 versus input parameters showthat 119875119908 increases with deviation angle (119894) in a relation similar

Table 2 Normal faulting in situ stress regime [4]

Field stresses (kPam) Minimum Maximum120590vg 2194 2262120590Hg 1934 2036120590hg 1876 19341198750 1011 1052

Table 3 Rock properties for a sandstone formation [4]

Rock properties Minimum MaximumCohesion (kPa) 4500 5500Friction angle (deg) 31 33Poissonrsquos ratio 02 03

Table 4 Normal faulting Monte Carlo simulation statistics (chang-ing all input parameters)

Statistics 119875119908 (kPa)Minimum 36780Maximum 44792Mean 40598Standard deviation 168790 confidence interval 38310ndash43250

Table 5 Normal faulting Monte Carlo simulation statistics (con-stant input parameters except 119894)Statistics 119875119908 (kPa)Minimum 38360Maximum 40866Mean 39480Standard deviation 9287190 confidence interval 38363ndash40852 thousand

to (S) shape However there are a lot of noises associated withthis relation due to the effect of other parameters For the restof the parameters nowell-defined relations can be figured outfrom plots even though the overall trends of plotted valuesmay give indications of general relations between 119875119908 andinput parameters

Another type of simulation was conducted while settingall input parameters constant except one in order to studythe effect of uncertainty of unfixed parameter on the criticalmudpressureThen the simulationwas repeated several timesand in each time a different parameter was set variably Forthe first case deviation angle (119894) was the only parameterthat is given a range (from 0 to 90 degrees) whereas therest of parameters were kept constant at the mid values ofthe given ranges The occurrence frequencies of possiblecollapse pressures are illustrated in Figure 17 And Table 5summarizes the statistics of the simulation In order to studythe relation between deviation angle (119894) and collapse pressure119875119908 a scattered plot has been constructed The analysis ofthe resulting collapse pressure shows that deviation angle(119894) has an (S) shape relation with the critical mud pressure(see Figure 18) ) Figures 19 and 20 illustrate the relation of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Petroleum Engineering 7

Figure 12 Interface of the probabilistic instability model in Excelfor deviated boreholes

casesThemodel utilizes theRisk program and (as has beenmentioned in the previous chapter) the model conductedMonte Carlo simulation for each case to capture the effectof uncertainty in input variables (in situ stresses and rockproperties) in collapse mud pressure In situ stresses androck properties data of a sandstone formation provided inliterature are used in this analysis Uniform distributionwas applied due to limitation of data Then for each casea sensitivity analysis was carried out to figure out whichparameter has more impact on collapse pressures

41 Collapse Pressure Risk Analysis for Normal Faulting StressRegime In this case Monte Carlo simulation was run tocalculate the collapse mud pressure at normal faulting in situstress regime The borehole was assumed to be drilled in asandstone formation with a depth of 3048 meters The usedfield stresses condition and rock properties are described inTables 2 and 3 respectively

In the first simulation all input values were given rangesof values with uniform distributions including well trajecto-ries (inclination angle 119894 and azimuth angle 119896) To calculate thecollapse pressures the developed probabilisticmodel was runapplying Monte Carlo simulation with 10000 iterations usingRisk program [14]

The histogram in Figure 13 illustrates the resulting valuesof possible overbalance pressures (collapse mud pressure)and their occurrence frequencies within the given rangesof input parameters The result shows that the minimumpossible value of collapse pressure is 36780 kPa whereas themaximum possible value is 44790 kPa Table 4 summarizesthe statistics of the simulation

To study the relations between input parameters and theresulting collapse pressures scattered plots of collapse pres-sure versus input parameters were constructed (Figures 14ndash16) The scattered plots of 119875119908 versus input parameters showthat 119875119908 increases with deviation angle (119894) in a relation similar

Table 2 Normal faulting in situ stress regime [4]

Field stresses (kPam) Minimum Maximum120590vg 2194 2262120590Hg 1934 2036120590hg 1876 19341198750 1011 1052

Table 3 Rock properties for a sandstone formation [4]

Rock properties Minimum MaximumCohesion (kPa) 4500 5500Friction angle (deg) 31 33Poissonrsquos ratio 02 03

Table 4 Normal faulting Monte Carlo simulation statistics (chang-ing all input parameters)

Statistics 119875119908 (kPa)Minimum 36780Maximum 44792Mean 40598Standard deviation 168790 confidence interval 38310ndash43250

Table 5 Normal faulting Monte Carlo simulation statistics (con-stant input parameters except 119894)Statistics 119875119908 (kPa)Minimum 38360Maximum 40866Mean 39480Standard deviation 9287190 confidence interval 38363ndash40852 thousand

to (S) shape However there are a lot of noises associated withthis relation due to the effect of other parameters For the restof the parameters nowell-defined relations can be figured outfrom plots even though the overall trends of plotted valuesmay give indications of general relations between 119875119908 andinput parameters

Another type of simulation was conducted while settingall input parameters constant except one in order to studythe effect of uncertainty of unfixed parameter on the criticalmudpressureThen the simulationwas repeated several timesand in each time a different parameter was set variably Forthe first case deviation angle (119894) was the only parameterthat is given a range (from 0 to 90 degrees) whereas therest of parameters were kept constant at the mid values ofthe given ranges The occurrence frequencies of possiblecollapse pressures are illustrated in Figure 17 And Table 5summarizes the statistics of the simulation In order to studythe relation between deviation angle (119894) and collapse pressure119875119908 a scattered plot has been constructed The analysis ofthe resulting collapse pressure shows that deviation angle(119894) has an (S) shape relation with the critical mud pressure(see Figure 18) ) Figures 19 and 20 illustrate the relation of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Petroleum Engineering

times10minus450 50900

3813 4325Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

36779644479159405987316870110000

00

05

10

15

20

25

Prob

abili

ty d

istrip

utio

n fu

nctio

n

37 38 39 40 41 42 43 44 4536

Overbalance pressure (times103 kPa)

Figure 13 Overbalance pressure in normal faulting (changing allinput parameters)

Pw_Mogi C versus i450

40

598

Corr (Pearson)Corr (rank)

45000

25982

405983716870109051

09032

40

46032

468

minus2 0 2 4 6 8 10 12 14minus4

i

34

36

38

40

42

44

46

Pw_M

ogi

C

Pw_Mogi C versus i

X meanX Std devY meanY Std dev

times103

Figure 14 Collapse pressure (kPa) versus deviation angle (degree)

collapse pressure versus (119896) and collapse pressure versus (V)respectively

42 Sensitivity Analysis To study the effects of uncertaintyof input parameters on collapse pressures sensitivity analysiswas carried out The constructed tornado diagram (seeFigure 21) shows that deviation and azimuth angles have themost impact on the drilling fluid pressures Friction angleand cohesion come next This means well planner in such

450

40

598

Pw_Mogi C versus k

minus01068

minus01104

271

229

237

263

34

36

38

40

42

44

46

Pw_M

ogi

C

minus2 0 2 4 6 8 10 12 14minus4

k

Pw_Mogi C versus k

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

45000

25982

4059837168701

times103

Figure 15 Collapse pressure (kPa) versus azimuth angle (degree)

02500

40

598

Pw_Mogi C versus v

254

246

254

246

Pw_Mogi C versus v

X meanX Std devY meanY Std devCorr (Pearson)Corr (rank)

4059837168701

34

36

38

40

42

44

46

Pw_M

ogi

C

0250000

0028869

minus00183

minus00161

018 020 022 024 026 028 030 032 034016

times103

Figure 16 Collapse pressure (kPa) versus Poissonrsquos ratio

situation should pay more attention to well trajectories androck properties

Similar simulations and sensitivity have been conductedfor strike slip and reverse faulting stress regimes Figures22 and 23 illustrate the result of sensitivity analysis of inputparameters on both cases respectively

43 Collapse Pressure Risk Analysis in Terms of In Situ StressesRock Properties and Well Trajectories In this case the targetwas to study the effects of uncertainty of in situ stressesrock properties and well trajectories as groups in boreholestability applying the developed instability model based onMogi-Coulomb criterion

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Petroleum Engineering 9

times10minus450 5090038363 40849

Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

38359404086740395074692589

1000

00000

00005

00010

00015

00020

00025

Prob

abili

ty d

istrib

utio

n fu

nctio

n

385 390 395 400 405 410380

Overbalance pressure (times103 kPa)

Figure 17 Overbalance pressure in normal faulting (constant inputparameters except 119894)

Pw_Mogi C versus i

Pw_Mogi C versus i450

39

507

Corr (Pearson)Corr (rank)

45000

25994

395074692589

09850

10000

19

48100

500

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

i

36

37

38

39

40

41

42

43

Pw_M

ogi

C

Figure 18 Collapse pressure (kPa) versus deviation angle (degree)

First the model was run keeping rock properties and welltrajectories at static values while in situ stresses were givenranges of values (see Figure 24) After that the simulationwas run while rock properties were given ranges of valuesand the rest were fixed again keeping only well trajectoriesvariables

Sensitivity analysis was carried out to find out which ofthe three groups (in situ stresses rock properties or welltrajectories) has more effects on collapse pressure underexistence of uncertaintyThe tornado chart in Figure 25 showsthat well trajectories group has the most impact on collapse

Pw_Mogi C versus k

Pw_Mogi C versus k450

39

024

Corr (Pearson)Corr (rank)

45000

25994

390236536445

51

449

X meanX Std devY meanY Std dev

times103

minus2 0 2 4 6 8 10 12 14minus4

k

375

38

385

39

395

40

405

Pw_M

ogi

Cminus09840

minus09997

500

00

Figure 19 Collapse pressure (kPa) versus azimuth angle (degree)

39

468664

02500

Pw_Mogi C versus v

Pw_Mogi C versus v

Corr (Pearson)Corr (rank)

0250001

0028883

3946866400044102

04280

04280

20658

294442

X meanX Std devY meanY Std dev

394670

394675

394680

394685

394690

394695

394700

394705

Pw_M

ogi

C

018 020 022 024 026 028 030 032 034016

v

Figure 20 Collapse pressure (kPa) versus Poissonrsquos ratio (V)

pressure Rock properties group come next and then in situstresses group

5 Conclusions

(1) A new probabilistic model has been developed forwellbore stability analysis

(2) The developed model has been applied in normalfaulting stress regime and in strike slip stress regime

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Petroleum Engineering

times103

Poissonrsquos ratioMinimum horizontal stress

Vertical stressPore pressure

Maximum horizontal stressCohesion

Friction angleAzimuth angle (k)Deviation angle (i)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)(normal faulting)

Figure 21 Sensitivity analysis in normal fault stress regime (tornadodiagram)

times103

Overbalance mud pressure (kPa) (strike slip)

Deviation angle (i)Azimuth angle (k)

Friction angleCohesion

Maximum horizontal stressPore pressure

Vertical stress

Minimum horizontal stressPoissonrsquos ratio

38 40 42 44 46 48 50

Figure 22 Sensitivity analysis in strike slip stress regime (tornadodiagram)

and reverse faulting stress regimes which are themostcommonly encountered stress regimes in the fieldThe results indicated that the biggest impact factor inwellbore stability is the well trajectories followed bythe rock strength parameters

(3) Rock strength parameters are not precisely estimatedin many field cases while they have a significantimpact in wellbore stability analysis

(4) The critical mud pressure values in strike slip stressregime are higher than those estimated in normalfaulting and reverse faulting stress regimes Thatmeans the instability for a wellbore is high understrike slip stress regime compared to normal faultingand reverse faulting stress regimes

(5) The influence of the formation pressure maximumhorizontal stress and vertical stress on critical mudpressure determination is the same under normalfault and strike slip stress regimes

(6) The increase in deviation angle in normal faultingwill result in more stability However in strike slipregime the higher the angle of drilling direction theless stability

(reverse faulting)Overbalance mud pressure (kPa)

425 43 435 44 445 45 455 46

(times103 kPa)

Poissonrsquos ratio

Minimum horizontal stress

Vertical stress

Deviation angle (i)

Pore pressure

Cohesion

Friction angle

Maximum horizontal stress

Azimuth angle (k)

Figure 23 Sensitivity analysis in strike slip stress regime (tornadodiagram)

times10minus450 50900

39650 40746Pw_Mogi C

Pw_Mogi C

MinimumMaximumMeanStd devValues

393032641113594020069

32700

10000

394 396 398 400 402 404 406 408 410 41239200000

00002

00004

00006

00008

00010

00012

00014fu

nctio

nPr

obab

ility

dist

ribut

ion

Overbalance pressure (times103 kPa)

Figure 24 Overbalance pressure (with variable in situ stresses)

times103

Drilling trajectory (i k)

Rock properties (c 120579 )

In situ stresses (120590120590H 120590h)

38 385 39 395 40 405 41 415

Overbalance mud pressure (kPa)

Figure 25 Sensitivity analysis (tornado diagram)

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Petroleum Engineering 11

(7) Normal distribution gives narrower ranges of collapsepressure than uniform distribution (higherminimumvalues and lower maximum values)

Appendix

MathCAD Program to Calculate CollapsePressure for Any Inclination [12]

Input Parameters Depth (ft) is given as follows

ℎ fl 10000 (A1)

Vertical stress gradient (psift) is given as follows

120590vg fl 095 (A2)

Maximum horizontal stress gradient (psift) is given asfollows

120590Hg fl 09 (A3)

Minimum horizontal stress gradient (psift) is given asfollows

120590hg fl 075 (A4)

Pore pressure gradient (psift) is given as follows

1198750119892 fl 045 (A5)

Rock Properties Cohesion (psi) is given as follows

119888 fl 640 (A6)

Friction angle (deg) is given as follows

120601 fl 352 sdot ( 120587180) (A7)

Poissonrsquos ratio is given as follows

V fl 03 (A8)

Borehole OrientationThe range of azimuth (10x deg) is givenas follows

119896 fl 0 sdot sdot sdot 9 (A9)

The range of deviation (10x deg) is given as follows

119894 fl 0 sdot sdot sdot 9 (A10)

Calculation of In Situ Stresses and Pore Pressure at the Depthof Interest Vertical stress is given as follows

120590V fl 120590vg sdot ℎ (A11)

Maximum horizontal stress is given as follows

120590119867 fl 120590Hg sdot ℎ (A12)

Minimum horizontal stress is given as follows

120590ℎ fl 120590hg sdot ℎ (A13)

Minimum in situ stress is given as follows

120590min fl min (120590V 120590119867 120590ℎ) (A14)

Pore pressure is given as follows

1198750 fl 1198750119892 sdot ℎ (A15)

Determination of Mogi-Coulomb Strength Parameters Onehas

119886 fl (2 sdot radic23 ) sdot 119888 sdot cos (120601) 119887 fl (2 sdot radic23 ) sdot sin (120601)

(A16)

Estimation of In Situ Stresses in the Vicinity of the Borehole forEach Borehole Trajectory The borehole inclination in radianis given as follows

119894119894119894 fl 10 sdot 119894 sdot ( 120587180) (A17)

The borehole azimuth angle in radian is given as follows

120572119896 fl 10 sdot 119896 sdot ( 120587180) (A18)

The in situ stresses in the coordinate system (119909 119910 119911) are givenas follows

120590119909119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot cos (119894119894119894)2+ 120590V sdot sin (119894119894119894)2

120590119910119896 fl 120590119867 sdot sin (120572119896)2 + 120590ℎ sdot cos (120572119896)2 120590119911119911119894119896 fl (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2) sdot sin (119894119894119894)2

+ 120590V sdot cos (119894119894119894)2 120590119909119910119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot cos (119894119894119894) 120590119909119911119894119896 fl 05 sdot (120590119867 sdot cos (120572119896)2 + 120590ℎ sdot sin (120572119896)2 minus 120590V)

sdot sin (2 sdot 119894119894119894) 120590119910119911119894119896 fl 05 sdot (120590ℎ minus 120590119867) sdot sin (2 sdot 120572119896) sdot sin (119894119894119894)

(A19)

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 Journal of Petroleum Engineering

Specifying the Location of the Maximum Stress ConcentrationThe orientation of the maximum and minimum tangentialstresses is given as follows

1205791119894119896

fl

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1205874 if 120590119909119894119896 = 120590119910119896 12 sdot 119886 sdot tan[2 sdot

120590119909119910119894119896120590119909119894119896 minus 120590119910119896 ] otherwise1205792119894119896 fl 1205791119894119896 + 1205872

(A20)

Identifying the angle that is associated with the maximumtangential stress one has

1205901205791119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205791119894119896) minus 4 sdot 120590119909119910119894119896sdot sin (2 sdot 1205791119894119896)

1205901205792119889119894119896 fl 120590119909119894119896 + 120590119910119896 minus 2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 1205792119894119896) minus 4 sdot 120590119909119910119894119896

sdot sin (2 sdot 1205792119894119896) 120590120579119889max119894119896 fl max (1205901205791119889119894119896 1205901205792119889119894119896)

(A21)

The location of the maximum stress concentration is given asfollows

120579max119894119896 fl10038161003816100381610038161003816100381610038161003816100381610038161205791119894119896 if 120590120579119889max119894119896 = 1205901205791119889119894119896 1205792119894119896 otherwise (A22)

The Axial and Shear Stresses in 119911-Plane at max One has

120590119911max119894119896 fl 120590119911z119894119896 minus V sdot [2 sdot (120590119909119894119896 minus 120590119910119896)sdot cos (2 sdot 120579max119894119896) + 4 sdot 120590119909119910119894119896 sdot sin (2 sdot 120579max119894119896)]

120590120579119911max119894119896 fl 2 sdot (120590119910z119894119896 sdot cos (120579max119894119896) minus 120590119909z119894119896sdot sin (120579max119894119896))

(A23)

Evaluation of the critical mud pressure using Mogi-Cou-lomb criterion

120590p2119895if 120590p2119895

ge 120590r119895

120590r119895 otherwise

Pw119895

Pw_Mogi119894119896 = for j isin 1 middot P0 middot middot middot 1 middot 120590min

Pw119895larr997888

j

1

120590120579max119895larr997888 120590120579dmax119894119896

minus Pw119895

120590r119895larr997888 Pw119895

120590p1119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) + radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

120590p2119895larr997888

1

2middot (120590120579max119895

+ 120590zmax119894119896) minus radic(120590120579zmax119894119896

)2 +1

4middot (120590120579max119895

minus 120590zmax119894119896)2

1205901119895larr997888 max (120590p1119895

120590p2119895 120590r119895

)1205903119895

larr997888 min (120590p1119895 120590p2119895

120590r119895)

1205902119895larr997888

120591oct119895 larr9978881

3middot [radic(1205901119895

minus 1205902119895)2 + (1205902119895

minus 1205903119895)2 + (1205903119895

minus 1205901119895)2]

larr997888 a + b middot [[(1205901119895

+ 1205903119895)

2minus P0

]]

120591Mogi119895

break if minus 120591oct119895 ge 0120591Mogi119895

(A24)

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Petroleum Engineering 13

Competing Interests

The authors declare that they have no competing interests

References

[1] S Khan Wellbore stability during underbalanced drilling [MSthesis] 2006

[2] M McLean andM Addis ldquoWellbore stability analysis a reviewof current methods of analysis and their field applicationrdquo inProceedings of the SPEIADC Drilling Conference Society ofPetroleum Engineers Houston Tex USA 1990

[3] J P Lowrey and S Ottesen ldquoAn assessment of the mechanicalstability of wells offshore Nigeriardquo SPE Drilling amp CompletionSPE-26351-PA Society of Petroleum Engineers 1995

[4] A M Al-Ajmi and M H Al-Harthy ldquoProbabilistic wellborecollapse analysisrdquo Journal of Petroleum Science and Engineeringvol 74 no 3-4 pp 171ndash177 2010

[5] A M Al-Ajmi and R W Zimmerman ldquoStability analysis ofvertical boreholes using the Mogi-Coulomb failure criterionrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 43 no 8 pp 1200ndash1211 2006

[6] A Al-Ajmi and R W Zimmerman ldquoStability analysis ofdeviated boreholes using the mogi-coulomb failure criterionwith applications to some north sea and indonesian reservoirsrdquoin Proceedings of the SPE Asia-Pacific Drilling Technology Con-ference SPE 104035-MS Bangkok Thailand November 2006

[7] M McLean and M Addis ldquoWellbore stability the effect ofstrength criteria onmud weight recommendationsrdquo in Proceed-ings of the 65th SPEAnnual Technical Conference and ExhibitionSPE 20405 New Orleans La USA September 1990

[8] B S Aadnoy ldquoModeling of the stability of highly inclined bore-holes in anisotropic rock formationsrdquo SPE Drilling Engineeringvol 3 no 3 pp 259ndash268 1988

[9] E Karstad and B S Aadnoy ldquoOptimization of borehole stabilityusing 3D stress optimizationrdquo in Proceedings of the SPE AnnualTechnical Conference and Exhibition Dallas Tex USA October2005

[10] E Fjaeligr R Holt P Horsrud A Raaen and R Risnes PetroleumRelated RockMechanics Elsevier AmsterdamNetherlands 2ndedition 2008

[11] A M Al-Ajmi and R W Zimmerman ldquoRelation between theMogi and the Coulomb failure criteriardquo International Journal ofRock Mechanics and Mining Sciences vol 42 no 3 pp 431ndash4392005

[12] A Al-Ajmi Wellbore stability analysis based on a new true-tiaxial failure criterion [PhD dissertation] Royal Institute ofTechnology Stockholm Sweden 2006

[13] S K Al-Shaaibi A M Al-Ajmi and Y Al-Wahaibi ldquoThreedimensional modeling for predicting sand productionrdquo Journalof PetroleumScience and Engineering vol 109 pp 348ndash363 2011

[14] Palisade ldquoRisk softwarerdquo 2012 httpwwwpalisadecom

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of