Using borehole breakouts to constrain the complete

stress tensor Results from the Sijan Deep Drilling

Project and oshore Santa Maria Basin California

Blair J Zajac and Joann M Stock

Seismological Laboratory California Institute of Technology Pasadena

Abstract We use borehole breakouts in nonvertical drill holes to constrainthe directions of the three principal stresses and their relative magnitudesIn this paper we start by modifying previously published breakout selectioncriteria to work with highly deviated borehole data We present a forwardmodeling technique using genetic algorithms and a nongradient Ndimensionaloptimizer to nd the best tting stress state for a set of breakout data Thestress state is parameterized by three Euler angles and the stress state ratio A technique is developed to determine the condence weighted mistbetween a model and the data We then map out the mist condencelimits on the best tting stress state This technique is applied to datapublished by Qian and Pedersen Removing their constraint of avertical principal stress direction reduces the mist between the stress stateand the breakout data We nd that the best tting stress state they reportwith a vertical principal stress direction lies outside our condence limitsWe also invert breakouts in the oshore Santa Maria Basin California Thesedata show a thrust faulting stress state with the maximum principal stressS at N

E plunging The condence range for the azimuthof S ranges from N E to NE The stress ratio was found to be

Introduction

Stressinduced borehole breakouts observed on fourand sixarm dipmeter logs or acoustic televiewer logsfrom drill holes are shear failures of the borehole wallthat form centered at the azimuth of greatest compressive stress at the borehole wall provided that this compressive stress exceeds the rock strength If the boreholeis vertical and aligned parallel to one of the principalstress axes the breakouts will form at the azimuth ofthe fareld least horizontal principal stress Sh Theorientation of borehole breakouts thus constrains thedirections of the horizontal principal stresses Sh andSH Breakout analyses in vertical holes are commonlyused to determine the directions of the principal horizontal stresses under the assumption that one of theprincipal stresses is vertical Sv The results have beenshown to be consistent with other indicators of stressdirection such as hydraulic fractures earthquake focal

mechanisms and overcoring measurements on both local and regional scales eg Bell and Gough Zoback and Zoback Gough and Bell Plumband Hickman Stock et al Zoback et al Zoback and Healy Zoback

Commonly breakout data are used to constrain directions of stress but not their relative magnitudesHowever in principle breakouts can yield informationon stress magnitudes both because the presence of abreakout indicates that the hoop stress exceeds the yieldstrength of the rock and because when the borehole isnot aligned with a principal stress axis the breakout orientation depends on the relative magnitudes of all threeprincipal stresses as well as on the orientations of thestresses Thus additional information about the stresstensor can be obtained from the directions of breakoutsin deviated nonvertical drill holes one can estimatethe directions of all three principal stresses and provide

Zajac and Stock

some constraint on their relative magnitudes Such atechnique provides a more complete knowledge of thestress tensor and works even if none of the principalstresses is vertical

In this paper we describe our technique of usingmany dierently oriented nonvertical boreholes to constrain the principal stress directions and magnitudesof the stress tensor The smallest data set applicableto this technique would most likely consist of a fewvariably oriented wells logged with oriented caliper armdata Oshore oil platforms provide good sources ofthese types of data since the wells are drilled in manydirections and at nonvertical deviations to fully developthe oil elds Depending upon the number of wells andthe horizontal extent of the caliper data our techniquedetermines a regional stress state localized over a spatial extent of kilometers to tens of kilometers

Other authors have presented dierent techniques ofusing deviated boreholes to determine the local stressstate Qian and Pedersen applied a nonlinearinversion technique to a set of borehole breakout dataAadnoy a b examined fractures in deviated boreholes Peska and Zoback developed a techniqueusing leako tests microfracture measurements porepressure measurements breakout and tensile fractureswith theoretical borehole failure calculations to determine the stress directions and magnitudes from a single deviated well Neither Peska and Zoback northe technique presented in this paper assumes a vertical principal stress direction Multiple applications oftheir technique over a region would yield a more regionalstress state tensor

Theory

The theoretical basis of this technique has been recognized for nearly a decade but it has only been appliedin a few areas perhaps because of the lack of stronglydeviated drill holes in most regions It is based on theelastic equations for stress surrounding an arbitrarilyoriented cylindrical hole in a medium with known fareld stresses Equations for the stress eld surroundinga circular hole in an elastic plate subjected to planestrain are given by Kirsch and various other authors eg Hubbert and Willis Timoshenko and

Goodier Jaeger and Cook The equationsfor the stress components at the wall of a cylindrical hole in a polyaxial stress eld are given by Hira

matsu and Oka Youngdahl and Sternberg Fairhurst Daneshy Richardson Mastin and Qian and Pedersen

If a drill hole is parallel to one of the principal stressdirections the rock strength and the relative magnitudes of the remaining two stresses aect the presence orabsence of breakouts but not their azimuth in the borehole reference frame The magnitudes of the principalstresses must be inferred by another technique ratherthan just by observations of breakout orientation

The breakout width of the hole after breakout formation is controlled by the relative magnitudes of theprincipal stresses so that if one of the principal horizontal stresses is known and the rock strength is knownor estimated the magnitude of the other principal horizontal stress can be estimated Moos and Zoback Vernik and Zoback

However in a nonvertical hole or a hole orientedobliquely to the three principal stress directions onedoes not necessarily need independent measurements ofone principal stress to infer the relative magnitudes ofanother principal stress because the magnitudes of theprincipal stresses as well as their directions inuencethe position of the maximum compressive stress at theborehole wall Richardson and hence the positionat which breakouts would form This fact was furtherelaborated by Mastin who showed stereographicprojections Hobbs et al indicating the directionof breakouts expected in variably oriented drill holes fordierent stress orientations and principal stress magnitudes Thus if several drill holes of dierent deviationsare present in a given area and if these drill holes aresubject to the same stress tensor the orientations ofthe breakouts in these holes may provide strong constraints on the orientations and magnitudes of the principal stresses at that location Zajac and Stock

To plot the breakout data from arbitrarily orientedboreholes and the calculated breakout positions for theoretical fareld stress states lower hemisphere stereographic projections of the borehole azimuth and deviation eg Figure are used

This dependence of breakout position on the fareldtectonic stress is illustrated in Figure where we showpatterns of breakout orientations that would be predicted for arbitrarily oriented drill holes subjected tocertain characteristic stress elds These are similarto the quadrant plots of Mastin and illustratethe degree to which an inversion or forward modelingof borehole breakout observations would constrain thestress regime for a given distribution of borehole orientations The characteristic stress elds are dened bythe orientations of the principal stresses S S and S

Breakouts Constraining Stress Tensors

North

Up

East

Breakout

δBorehole azimuth

Boreholedeviation

EW

N

S

δ

Ω

Figure Relationship between an arbitrarily oriented borehole containing a breakout and how this borehole andits breakout orientation would be plotted on a lower hemisphere stereograph of borehole azimuth and deviation

and the stress ratio

S SS S

where S is the maximum stress S is the intermediatestress and S is the minimum stress

Note that given enough variation in borehole orientations the stress state can be reasonably constrainedsince the patterns vary continuously from an entirelyradial distribution of breakout azimuths for degenerate thrust faulting SH Sh Sv to an entirelycircumferential distribution of breakout azimuths fordegenerate normal faulting Sv SH Sh Here adegenerate case is dened when two of the principalstresses are equal in magnitude The nodal points represent borehole orientations where circumferential stresson the borehole wall is uniform and there is no preferred direction of breakouts at these orientations themaximum stress at the borehole wall is relatively lowand less likely to exceed rock strength so that breakouts may be absent altogether Note that the nodalpoint for degenerate thrust faulting is in the center ofthe projection as the ratio of SH to Sh increases twonodal points appear and move radially away from thecenter along the SH direction The nodal points reachhorizontal at the degenerate case when Sh Sv combined thrust and strikeslip faulting as Sh continuesto decrease the nodal points split again and move alongthe circumference of the plot For the degenerate caseof SH Sv Sh combined strikeslip and normalfaulting the two nodal points are horizontal again andaligned along the direction of Sh as the stress ratiosprogress through the normal faulting stress regime thenodal points again approach the center of the plot

Although in theory these patterns will vary continuously as a function of the stress regime our abilityto resolve them depends on the distribution and quality of the data particularly on the available boreholeorientations Since few drill holes approach the horizontal data near the nodal points for some patternsmay be hard to obtain If the boreholes within a studyregion are all within of vertical then our abilityto resolve the stress ratios will depend on the stressregime stress ratios in normal faulting or thrust faulting stress regimes will be better resolved than those instrikeslip faulting stress regimes Mastin However in recent industry drilling programs it is commonfor boreholes to be deviated more than Our abilityto resolve the stress state will also depend on the consistency of the data since we assume that all data plottedor inverted together correspond to the same stress tensor including both the orientations of the principal axesand the stress ratio

Reference Frames

Throughout this work we use two reference framesthe geographic reference frame and the borehole reference frame Figure The geographic reference frameis an orthonormal reference frame with its X axis horizontal and pointing due east The Y axis is also horizontal and points due north The Z axis is perpendicular to both the X and Y axis and points up Thisgeographic reference frame will often be referred to asthe XYZ reference frame for short

The borehole reference frame stays aligned with theborehole axis as the borehole orientation changes with

Zajac and Stock

Figure Relationship of breakout orientations tostress directions and magnitudes in arbitrarily orienteddrill holes Mastin Lower hemisphere stereographic projections show the breakout orientations projected onto the horizontal plane for a variety of drillhole orientations and stress regimes Solid circles arepoints at which the stress anisotropy is zero and corresponds to borehole orientations with no preferred breakout direction The low maximum compressive stressat the borehole wall at these positions indicates thatbreakouts might be absent If breakouts are presentnear the nodal points however they will change orientation rapidly as borehole orientations vary In thesegures Poissons ratio was taken to be and theorientation of the maximum horizontal principal stressis always eastwest for nondegenerate stress regimestop left Degenerate thrust stress state top centerthrust regime for and top right combination of thrust and strikeslip faulting middleleft Strikeslip stress regime middle centerstrikeslip stress regime and middle rightcombination of normal and strikeslip regimes bottom left Normal stress regime bottommiddle normal stress regime and bottomright degenerate normal stress regime

IX

Y

ZK

J

High side of the hole

Boreholeazimuth

δ

Boreholedeviation

Ω

Figure View of the two coordinate systems associated with the borehole The X Y and Z axes arealigned with the geographic coordinate system TheIJK coordinate system rotates as the borehole orientation changes

depth So the orientation of this reference frame moveswith respect to the geographic XYZ reference frameThe axes are named I J and K and are oriented asfollows The K axis points up along the borehole axisthe J axis is perpendicular to theK axis and in the sameplane as the K and Z axes while I is perpendicularto both the J and K axes and lies in the horizontalplane With this denition the J axis always points tothe high side of the hole If the borehole is vertical thenthis coordinate system coincides with the geographiccoordinate system The borehole reference frame willbe referred to as the IJK reference frame for short

Two rotations are required to rotate a coordinatesystem that is initially aligned with the XYZ coordinate system into alignment with the IJK coordinatesystem The rst rotation about the Z axis rotates thegeographic coordinate system clockwise by the angle tillX coincides with I The resulting coordinate systemwill be referred as the axes The second step rotatesthe axes about the axis by a counterclockwiseangle producing the IJK coordinate system Thetwo angles and in geological terms are the borehole trend and deviation respectively See Figure and

Breakouts Constraining Stress Tensors

equation

The denition of angular measurements requires aspecial note for the two reference frames We use theterm azimuth to refer to a direction measured eastfrom north in the geographic coordinate system Wedo not use the term azimuth in the borehole coordinate system since north is not a special direction inthat coordinate system Rather all angles are measuredcounterclockwise from the I axis when one looks downthe K axis We use the word azimuth to refer to angles measured only in the geographic coordinate systemand the word angle to refer to angles measured in theborehole coordinate system When the two coordinatesystems coincide then the azimuth is calculated bysubtracting the angle from

azimuth angle

Identication of Breakouts

Here we discuss criteria for identication of breakouts and calculation of breakout azimuths given either oriented four or sixarm caliper data Orientedfour or sixarm caliper data are measured on a variety of dierent welllogging tools such as lowangledipmeters highangle dipmeters formation microscannersmicroimagers and stratigraphic highresolutiondipmeter tools SHDTs

Plumb and Hickman examined the validity ofusing fourarm dipmeters to identify borehole breakouts They logged a well in Auburn New York twicewith a fourarm dipmeter and once with a borehole televiewer The borehole televiewer is an acoustic loggingtool that provides highresolution information aboutborehole elongation and the distribution of natural fractures in wells Zemanek et al The acoustic transit time can be used to construct detailed borehole crosssections By comparing the orientation of breakoutsfrom the fourarm dipmeter with borehole cross sectionsconstructed from the televiewer data they found thatthe fourarm dipmeters generally had their long axesaligned with the breakouts This is a critical observation since borehole televiewer logs are not commonlyrun in most drill holes and many wells logged withfourarm dipmeters lack the more detailed televiewerobservations

Plumb and Hickman dened ve criteria theyused to identify zones of breakouts from fourarm dipmeter logs

The tool rotation stops in the zone of elongation

The caliper arm dierence is greater than cm

The smaller of the caliper readings is close tobit size or if the smaller caliper reading is greater thanbit size it should exhibit less variation than the largercaliper

The length of the breakout zone is greaterthan cm

The direction of elongation should not consistently coincide with the azimuth of the high side ofthe borehole when the hole deviates from vertical

From our experience with examining deviated boreholes we have slightly modied Plumb and Hickmans breakout selection criteria The new criteria usedare as follows

The tool rotation stops in the zone of elongation Dipmeter logs record either pad azimuth or therelative bearing of pad or both The pad azimuth isthe XYZ azimuth of the number dipmeter pad Therelative bearing is the IJK angle between the high sideof the hole and pad number Both of these measurethe orientation of the tool in the hole Our selectioncriterion for nondigital data sets is that the maximumvariation of either measurement should be less than When digital data are available the standard deviationof either data type must be less than We useMardias work in the statistics of angular datato calculate the average borehole elongation directionand its standard deviation for a particular section ofhole

The largest and smallest caliper arms shouldbe at least dierent from each other

The smallest caliper arm should be largerthan or equal to the bit size and smaller than timesthe bit size

The standard deviation of each caliper armover a breakout interval should be less than cm inch

The length of the breakout zone should be atleast m

The maximum dierence between the bit sizeand the largest caliper arm should be cm inches

Criterion of Plumb and Hickman was tightened to be more quantitative regarding the variation inthe quality of the breakout direction Criterion waschanged to demand a dierence between the caliperarms instead of a cm dierence This criterion ismore exible for boreholes of dierent radii Criterion was slightly changed to state that the smallest caliper

Zajac and Stock

arm is always as large as the bit size and no larger than times the bit size The minimum breakout lengthof criterion was increased to m to nd longer moreconsistent breakout intervals Criterion of Plumb andHickman discards breakouts that t all other selectioncriteria but are parallel to the high and low sides ofthe hole We keep such data initially since the theoretical breakout patterns for most stress regimes Figure show that in certain borehole orientations the breakoutsare expected to be aligned with the high and low sideof the hole After initial data analysis if we determinethat some of these elongation directions are probablydue to key seats we remove them manually from thedata set We also note that Qian and Pedersen performed an inversion of a set of breakouts measuredwith fourarm dipmeter data containing radial breakouts Figure Finally we add one more criterionwhich discards breakouts with very large spalled regions since other processes such as wholesale failurealong fault zones might account for large spalled regions

Calculation of Elongation Azimuths

Calculation of the elongation direction for a fourarmdipmeter is straightforward We assume that the centerof the dipmeter is located at the center of the boreholeso the elongation direction is parallel to the direction ofthe longer caliper arm We also make the assumptionthat the borehole is symmetric about so that allelongation orientations lie between and withoutany loss of generality

The sixarm dipmeter is similar to the fourarm dipmeter except that it has two extra arms and all of thearms are separated by intervals Calculation of theborehole elongation direction from a sixarm dipmeter ismore complicated since it is not as obvious that any oneset of caliper arms will be aligned with the breakout Inaddition the extra two pads on the borehole wall maychange the threshold of detection of breakouts becausethe increased friction on the borehole wall requires lessellipticity to counteract tool torque We are not awareof any study in which the breakout directions calculatedfrom a sixarm dipmeter have been compared to thosefound either from a borehole televiewer tool or from afourarm dipmeter

Here we describe two methods of calculating theelongation direction from sixarm dipmeter data Therst method uses the orientation of the longest caliperarm as the elongation direction However the longestcaliper arm may not track the breakout so the second

Caliperarm #1

Caliperarm #2

Caliperarm #3

θX

Y

X’Y’

Boreholewall

Breakout

Breakout

Figure Geometry of the caliper arms in the sixarmdipmeter looking down onto the tool and ellipse used tond the breakout orientation

method which we use on our data ts an ellipse tothe caliper arm data and takes the orientation of thesemimajor axis as the elongation direction

We briey describe the second method here Thesixarm dipmeter measures three independent diameters that are separated by These three values dene three vectors which originate at the origin of asuitable coordinate system and are separated by intervals We assume that the three vectors constrain anellipse centered at the origin of the coordinate systemWe parameterize the orientation of this ellipse using the angle between the semimajor axis of the ellipseand the direction of caliper arm Figure To determine we write three equations describing the ellipseinserting the vector positions of the endpoints of thecaliper arms The three equations are then written inmatrix form As in many linear algebra problems for asolution to exist the determinant of the matrix must bezero Here the matrix will have zero determinant whenthe angle of rotation of the ellipse is found Once thisangle of rotation is found the lengths of the semimajorand semiminor axes are easily calculated Using thisinformation with the rotation angle allows calculationof the elongation direction with respect to azimuth ofpad

One potential problem with this method is that thethree caliper arm vectors might be tted by a hyperbolainstead of an ellipse We have observed this problemwith some of the data from Point Pedernales and we

Breakouts Constraining Stress Tensors

choose to not select any breakouts where this happens

Theoretical Breakout Directions in

Arbitrary Stress Fields

Here we discuss how the theoretical breakout direction is calculated given an arbitrary borehole orientation and fareld stress state First the fareld tectonic stress is transformed or rotated into the coordinate system associated with the borehole as the stresstensor

SIJK

Sii Sij Ski

Sij Sjj SjkSki Sjk Skk

A

with compressional stress positive

Hiramatsu and Oka and Fairhurst derived the relationship between fareld stresses andstresses on the wall of a cylindrical hole assuming thatthe medium is isotropic homogeneous and linearlyelastic with constant uid pressure in the borehole Thestresses on the borehole wall are given by

kk Skk Sii Sjj cos

Sij sin

Sii Sjj Sii Sjj cos

Sij sin P

k Sjk cos Ski sin

kk

qkk k

Here is the greatest compressive stress on the borehole wall at a particular position and P is the dierence between the borehole uid pressure and the in situpore pressure Throughout this work we take P tobe The angle is measured from the I axis towardthe J axis Figure Here is Poissons ratio of therock and is taken to be To nd the position atwhich the breakouts should form on the borehole wallwe maximize with respect to the angle

Inversion of Borehole Breakout Data

In this section we describe the steps taken after adipmeter data set is gathered to determine the stressstate that best describes the data Briey the steps areas follows

Calculate the IJK elongation angle for allwells at m depth spacing

SKK

I

J

K

α

ατK

αα

KKσ

σ

SJKS

KI

SKI

SII

SIJ

SJK

SIJ

SJJ

Figure View of the borehole and the components ofthe stress tensor in the borehole coordinate system Theangle is used in nding the cylindrical components ofthe stress tensor on the borehole wall After Qian and

Pedersen

Use the breakout selection criteria listed aboveto identify breakouts

Optionally bin the breakouts into bins ofXYZ borehole azimuth and borehole deviation Wecalculate a weighted average breakout direction anda weighted standard deviation of the breakout anglesMardia We weight individual breakouts linearlywith their length and inversely with the standard deviation of the breakout angle over the breakouts length

Use a genetic algorithm GA to identify theregion of the stress state solution space that most likelycontains the minimum weighted onenorm mist between a stress state and the possibly binned breakoutdata

Run a Ndimensional optimizer with the results from the GA to nd the minimum mist

Calculate error bounds on the best ttingstress state

Calculation of Borehole Elongation Directions

Each data point used in the inversion consists ofthree values the XYZ borehole azimuth the borehole deviation and IJK elongation angle The IJKelongation angle is used in preference to the XYZ elongation azimuth because at large borehole deviationsXYZ elongation azimuths will lie predominantly

away from the borehole azimuth The relationship between the IJK elongation angle and the XYZ elongation azimuth is not linear so calculation of the mist

Zajac and Stock

between XYZ elongation azimuths would not be representative of the real dierence between IJK elongationangles

Typically calculating the IJK elongation angle involves several dierent steps depending on the typesof available data Dierent dipmeters measure dierent angular quantities recording the orientation of thedipmeters caliper arms either the relative bearing orthe pad azimuth If the relative bearing is measuredthen we subtract the relative bearing from to getthe IJK elongation angle since the relative bearing ismeasured with respect to the J axis However if thepad azimuth is measured we calculate the IJK anglethat expresses the same vector direction as the pad azimuth for the caliper arm Depending on the age ofthe logging tool and the type of tool used to log the holewe might need to make a magnetic declination correction for all geographic azimuths that the tool measuressuch as the pad azimuth and the borehole azimuth

Binning of Breakout Data

After identication of the breakouts from a set ofwells lower hemisphere stereographic projection plotsof the breakouts may show a large degree of scatterin the breakout angle In this case to simplify thesubsequent inversion we may bin the data into equalarea borehole azimuth versus borehole deviation binsWe use the technique of Mardia to calculate aweighted average and a weighted standard deviation of aset of breakout angles We weight the data linearly withthe breakouts length and inversely with the standarddeviation of the IJK breakout angles over the length ofan individual breakout

Stress State Parameterization

A stress tensor can be described in a number of waysWe would like a parameterization of the stress tensorthat separates the magnitudes of the principal stressesfrom the orientation of the principal stress directions

We choose to parameterize the stress state with fourparameters three Euler angles and the stress state ratio equation Euler angles are three angles thatdescribe three successive rotations about various coordinate axes and are a natural representation often usedto completely describe the orientation of a set of axesattached to a body in space Goldstein We usethis formalism to describe the orientation of the eigenvectors of a stress tensor

Since the stress tensor is completely described by sixparameters and three of the parameters describe the

stress state orientation the two parameters we are notparameterizing describe the magnitudes of the principalstresses We can parameterize the magnitudes of theprincipal stress directions as

S ab

S ab

S ab

Examination of equations through shows thatthe location of the greatest compressive stress isunaected by the constant multiplicative factor a inequations Therefore the only parameter thatwe choose to ignore in our parameterization is b whichwe set to

Calculation of the Mist

In optimization of a problem a mist scheme mustbe chosen that describes how well a particular modelts the data Following Parker and McNutt andGephart and Forsyth we use a onenormmeasureof the mist

m

nXj

jj oj j

j

where j is a theoretical value for an observation oj andj is the measure of error in the observation We usethe least absolute values L norm measure instead ofthe more familiar least squares L norm approach because our data show a propensity of observations thatare quite a distance away from the predicted orientations from the best tting model In other words thedistribution of angular dierences about the best tting model is not Gaussian and we do not wish toweight those particular measurements as heavily as aleast squares or mist would do It should be notedthat the individual observations oj and standard deviations j appearing in equation would be eitherthe standard deviation of the IJK elongation angle overthe breakout or from the statistics of the binned breakouts

Fitting the Breakout Data

Since the determination of the best tting stress statefor a set of borehole breakout data is inherently nonlinear we choose to use forward modeling instead of anyinverse technique Inversions have been used by others eg Qian and Pedersen We use a twostepapproach First a genetic algorithm Holland Davis Goldberg is applied to the problemto nd an approximate best tting stress state using

Breakouts Constraining Stress Tensors

the stress state parameterization described above consisting of the variables Here the boreholeazimuth and the borehole deviation are the rsttwo Euler angles and is the third Euler angle A Ndimensional optimizer is then initialized with the GAsresults to nd the best tting stress state

Genetic algorithms are an attractive approach tosolving hard nonlinear problems in which the forwardcalculation is straightforward but more traditional techniques might fail Some of the advantages of GAs arethat they eciently search the problem space do notrequire a good starting model and do not get trappedin local minima unlike gradient search methods

GAs operate on a population of models The models are typically binary coded just as oating pointnumbers are encoded in a computer An individualbinaryencoded model is termed a chromosome Therst generation of chromosomes is randomly generatedEach chromosome has associated with it an objectivevalue which is a problemspecic measure of how wellthe chromosome solves the problem From the objective value is calculated a tness value the higher thetness the better the chromosome In our work eachchromosome represents a distinct regional stress stateand the objective value is the weighted onenorm mist between the observed and theoretical IJK breakoutangles Since smaller onenorm mists correspond tohigher levels of tness we use the following equation torelate the two

fi o o oi

o

where o is the populations mean objective value ois the standard deviation of the objective values oi isthe ith chromosomes objective value and fi is the ithtness value Holland

After the tness values have been calculated chromosomes are selected to mate to create the next generation of chromosomes Chromosomes with higher tness values proportionately mate more often Matingbetween two chromosomes is performed by randomlyexchanging part of the binary patterns of both parentchromosomes This operation is known as crossoverCrossover is performed only roughly of the timebetween two chromosomes In the other of matings the two chromosomes are carried directory into thenext generation without crossover

The nal operation of the GA is mutation wherebya small fraction of the bits of a chromosome are ippedThis process introduces variability into the populationand allows broader searching of the solution space The

next generation of chromosomes has now been createdand the cycle begins anew with the calculation of thepopulations objective values

We run our GAs with populations of several hundredchromosomes for several hundred generations SinceGAs do not guarantee that they nd the optimal solution to a problem we take the particular stress statecorresponding to the minimummist observed for all ofthe GA runs completed as the starting point for a general Ndimensional optimizer routine We use the Powell optimizer as described and coded by Brent This particular optimizer does not require the use ofderivatives of the function it is trying to optimize afeature that makes it attractive for the borehole breakout problem

Experience with various breakout data sets has shownthat starting the Powell optimizer with a purely random stress state does not lead to a global minimumof the onenorm mist even for theoretically generatedbreakout data The GA thus supplies an important rststep in eciently searching the space of stress states forgood starting stress states for the Powell optimizer Inpractice we run roughly several hundred separate GAseach with dierent random numbers to make sure wend the global onenorm mist minimum Even thoughwe use the Powell optimizer after the GA the mistis not a smooth function of the stress state since different GA and Powell runs nd dierent local minimaWe could have chosen to start a GA search around theresults of the Powell search but we choose not to doso since such a search would probably not improve theresults all that much

Condence Limits

For the results of the stress state optimization to beof any worth the condence limits in the results must bedetermined We determine the condence limits on thebest tting stress state using a modication of Gephartand Forsyths technique for dening condencelimits on stress state determinations from focal mechanisms

We want to nd the condence limit on themodel we obtained that minimized the onenorm mistequation We do this by assuming that by performing this one experiment an experiment in the sensethat oil wells were drilled and borehole breakout orientations were measured we know something about theensemble of onenorm mists of the best tting modelsthat would result if we were able to perform the experiment again as many times as we liked From thisassumption we can say that our particular mist lies

Zajac and Stock

somewhere in a probability distribution of mists fromall of the possible experiments From this distributionof mists we can nd the mist value M such that acertain percentage of all mists would have mists lessthan M If we know that of all of the mists fromthese experiments have mists m M then we havefound the condence interval

It should be stated that we are not considering thedistributions of m values calculated in searching for thebest tting model for our single experiment In our single experiment the best model has an mist m at oneend of a distribution of mists corresponding to slightlydierent stress states Instead consider that when thebest model for our experiment has been identied allof the other possible stress states for the data in thisexperiment are ignored Here we consider the distribution of best tting models or mists that would befound if we were able to redrill all of the holes along thesame well paths and measure the breakout orientationsin those new holes

We will use the results of Parker and McNutt who calculated the statistics of the onenorm mist

m

nXj

jxj j j

when xj are independent normal random variables withzero mean and standard deviations j The expectedvalue of m is

m

r

n n

Parker and McNutt also wrote a program whichgiven a probability P and n calculates the mist Msuch that the probability is P that m M In otherwords this programndsMP n such that the integralfrom to M of the onenorm distribution is P for aonenorm distribution of the order of n We useMP nlater to calculate the condence limit

To nd the condence limit we begin by deningP

min to be the minimum mist observed in oursearch for the best tting model We then assume thatthis particular onenormmist corresponds to the meanmist of a onenorm distribution Of course the meanmist and the minimized mist will dier so we assumethat we incorrectly estimated the errors j in our mist calculation To correct this estimation we introducea new constant factor f by which we multiply all ofthe standard deviations j such that a new mist sumis equal to the expected mist Since we have alreadyfound the best tting model to our data we have lost k

degrees of freedom where k is the number of variablesin the problem We then should use the expected onenorm mean for nk observations Mathematically thisequation becomes

Xmin

nXj

jj oj j

j

n k nXj

jj oj j

fjfrom

Pmin

f

f

Pmin

n k

Finally to nd the P condence mist valueP

P we use the following expression

PP

f MP n k

where MP n k is the function given by Parker and

McNutt We still must subtract k from n to ndM since we lost k degrees of freedom in the search forthe best tting model Solving for

PP we nd

XP

MP n k

n k

Xmin

Gephart and Forsyth nd a similar expression intheir equation except they assumed that for large none can replace the onenorm inverse cumulative function MP n with values from standard tables of Gaussian statistics Since we have a code that calculatesMP n there was no reason to make this assumption

Error Analysis

Our forward modeling approach does not give anyerror estimates by default unlike other inversion methods We determine our error bounds by varying different parameters of the best tting stress state andevaluating how much variation is required to raise themist to the onenorm condence level Because ofthe nature of successively rotating the principal stressdirections for each Euler angle error bounds for theEuler angles are not of much use To see how well constrained the principal stress directions are we x thestress state ratio iterate the borehole azimuth from to and iterate the borehole deviation from to in increments This iteration rotates the Jaxis across the lower hemisphere of a stereographic plot

Breakouts Constraining Stress Tensors

N

S

EW

25.0

50.0

-6050

-1450

Figure Compiled and processed data from the SijanDeep Drilling Project in Sweden from Qian and Peder

sen plotted on top of the theoretical breakoutpattern for their best tting stress state of SHSv

and ShSv

where SH lies

east of north The nodal points for this stress state lieat a deviation of hence they do not show in thisplot A Poissons ratio of was used to calculate thebreakout pattern The vertical depth scale is in meters

Since there are three error spaces one for S S andS we set Sjj to and We then nd the thirdEuler angle that minimizes the mist This then generates three onenorm mist data sets as a function of and which we use to generate lower hemisphere contour maps of the condence mists for the S Sand S axes

When we vary the stress state ratio away from theminimummist and make the lower hemisphere contourplots of the condence mist we observe smallerenclosed contour regions that are sometimes rotated away from their minimized locations

Analysis of Qian and Pedersens

Borehole Breakout Data

As one test of our technique we reanalyze the borehole breakout data presented by Qian and Pedersen

0

5

10

15

Num

ber

Bre

akou

ts

-50 0 50 100

High Side Angle - Breakout Angle

Figure Histogram of the angular dierence measured in the plane perpendicular to the borehole axisbetween the breakout orientation and the high and lowsides of the borehole for the Qian and Pedersen data

from the Sijan Deep Drilling Project in SwedenThe data are taken from Table of Qian and Peder

sen and are plotted in Figure This table listsa depth range and the average borehole azimuth deviation and breakout orientation for that range Alsolisted is the variance of the breakout direction in degrees over that interval Figure plots a histogram ofthe angular dierences between the location of the highside of the borehole and the breakout angle as measuredin the plane perpendicular to the borehole axis for allof the data We see that the breakouts are clusteredaround the high side of the hole and could possibly bedue to tool drag Since we do not have the originalcaliper curves we cannot apply our breakout selectiontechnique to this data set and we use the data set unmodied

Qian and Pedersen applied a nonlinear inversion technique to their data assuming a vertical principal stress direction with Poissons ratio equal to They found the stress state to be a strikeslipregime SH Sv Sh with the maximum horizon

tal stress located

east of north The ratioof stresses was SHSv

and ShSv

which corresponds to a stress ratio of The errorbounds listed here are nonlinear error bounds calculatedby varying the stress parameters until a large enough

Zajac and Stock

Table Characteristics of the Stress State That Minimized the Weighted OneNorm Mist of the Qian and

Pedersen Borehole Breakout Data

S S SAzimuth N E NE NEPlunge

Value

Optimized minimum weighted onenorm

mist condence level for weighted onenorm mist

mist was observed

In Figure we plot Qian and Pedersens datawith the theoretical breakout pattern expected for thestress state resulting from their inversion A Poissonsratio of was used in this calculation The datashow almost constant breakout azimuths regardless ofthe borehole orientation Because of the relatively restricted range of borehole azimuths present in their dataa large number of nondegenerate normal and strikeslipfaulting stress regimes could t their observations withnearly constant breakout orientations up to the maximum deviation of Figure In this case the dataare not well distributed in order to constrain the complete stress tensor Qian and Pedersens highuncertainty in the ShSv ratio reects exactly this problem with the data distribution This will be a commonproblem in strikeslip stress regimes if highly deviatedholes are not available The only way to remedy this isto either nd or drill boreholes that are nearly horizontal

To compare the inversion technique of Qian and Ped

ersen with the genetic algorithm and Powell optimizer technique described above we inverted theirdata set using our technique The data set was notgridded since it shows a large amount of consistencyThe breakout data are best described by a thrust faulting stress state in which S is oriented N E plunging S is also almost horizontal oriented NEplunging Finally S is almost vertical plunging at an azimuth of NE Table The valuefor our solution is

Error bounds on our stress state determination areshown in Figures and Figure shows contour plotsof the condence mists for the three principal directions on a lower hemisphere stereographic projectionkeeping constant The direction of S is very tightlyconstrained but the orientations of S and S are al

0˚

315˚

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225˚

180˚

135˚

90˚

45˚

S1

S2

S3

S1

S2

S3

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Figure Results from the reanalysis of the Qian

and Pedersen borehole breakout data using ourGA and Powell optimization technique Lower hemisphere stereographic projection shows contours of the weighted onenorm mist condence limits for thethree principal stress directions The stress state ratio is held constant at Note that the direction of Sis very well constrained but S and S can lie virtuallyanywhere within a vertical plane striking NE

Breakouts Constraining Stress Tensors

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Stress Ratio φ

Figure Results from the reanalysis of the Qian and

Pedersen borehole breakout data Plot of theweighted onenorm mist as a function of where thethick solid line is the condence limit thethin solid line is the minimized mist when is heldconstant and the principal stress directions are unconstrained and the dotted line is the mist using the principal stress directions from the best tting model

most unconstrained about an arbitrary rotation aboutthe S axis

In Figure we plot the condence limit thicksolid line the weighted onenorm mist as a functionof using the Euler angles from the best tting modeldotted line and the weighted onenorm mist allowing the Euler angles to vary to minimize the mist fora particular thin solid line The plot demonstratesthat the stress ratio is unconstrained since there exists a stress state that can be rotated in such a wayto t the borehole breakout data within the condence limits for any We have created lower hemisphere stereographic projection plots of the condence mist for nonoptimal values These plots donot show any smaller condence regions a nding thatis not surprising given that the minimum mist as afunction of is almost constant Given the sparsenessof the borehole orientations sampled by these data thelack of constraint on is not surprising However thestress states that t the data for any may not have aprincipal stress direction anywhere near vertical so themathematics of this t does not take into account someof the geophysics of the problem

The orientation of SH determined by Qian and Ped

ersen and determined in this paper dier by only However the remaining components of the stressstate dier considerably Qian and Pedersen found a strikeslip stress state with We founda thrust faulting stress state with The stressdirection contour map Figure also shows that Sand S may be allowed to rotate almost freely aboutthe S axis thereby switching from a thrust faultingto a strikeslip environment This nding is consistentwith the observation that nondegenerate thrust andstrikeslip faulting stress regimes can have very similarnearly constant breakout orientations for small deviations Figure Qian and Pedersen state thattheir inversion clearly shows a strikeslip stress regimeWe feel that their conclusion is too strong given theirbreakout data and the analysis presented here

Qian and Pedersen also state that it is notfeasible to relax the assumption of a vertical principalstress direction given their analysis of inversions of theoretical breakout data However there are several reasons why stress states with nonvertical principal stressesshould be considered First the extra degree of freedom gives a clearer sense of how poorly or well constrained the stress state is from Qian and Pedersens data Second it allows a much better t of thedata The weighted onenorm mist of the data using Qian and Pedersens best tting stress state

Zajac and Stock

238˚

238˚

239˚

239˚

240˚

240˚

241˚

241˚

242˚

242˚

33˚ 33˚

34˚ 34˚

35˚ 35˚

36˚ 36˚

LA

SB

SLBF

HF

Pacific Ocean

Figure Location star of the Point Pedernaleseld in the oshore borderland along with some of themajor Quaternary faults in the southern California region LA downtown Los Angeles SB Santa BarbaraSLBF Santa Lucia Bank Fault and HF Hosgri Fault

is which is outside our condence limit of Finally there might be some breakout data setsin which the assumption of a vertical principal stress direction is invalid leading to an improper understandingof a regions stress state The cost of this extra degreeof freedom is slight in comparison with the potentialgain in understanding of stress states

Borehole Breakouts in Point Pedernales

California

We applied our technique to determine the stressstate in the oshore Santa Maria Basin California fromwells drilled by the Unocal Company from an oshoreplatform in the Point Pedernales Oil eld Figure The oshore Santa Maria Basin is an elongated structural basin parallel to the California coast northwest ofPoint Arguello McCulloch It lies between twoNNW trending structural boundaries a zone of eastdipping normal faults including the Santa Lucia Bankfault on the west side and the Hosgri fault on the eastside However it appears to be stratigraphically continuous with the onshore Santa Maria Basin east of theHosgri fault and to have experienced a similar Miocenedeformational history Sorlien

The basal Tertiary section in the oshore SantaMaria Basin comprises volcanic rocks of probable earlyMiocene age which rest on basement and have been dis

placed by normal faults McCulloch This volcanism and a subsequent phase of subsidence Madocumented by use of backstripping techniques in several wells in the region have been attributed to thecapture of the Monterey microplate by the Pacic plateSorlien and the beginning of clockwise rotation of the Western Transverse Ranges McCrory et al This development was followed by slow thermal subsidence from to Ma McCrory et al Since Ma the tectonics of this oshore region has been locally complicated because of an overall transpressional regime which produced NESW directed shortening between and Ma and much slowerdeformation in Quaternary time eg Clark et al Crouch et al

Unocal provided us with ve paper logs of four wellsfrom the Point Pedernales eld All four wells weredrilled from the same platform and all of the dipmeter data lies within a km radius of the platformTable lists the type of tool used to log the hole andsome of the properties of the wells including the loggeddepth interval the depth interval of processed dipmeterdata and the maximum deviation of the well over theprocessed interval

To analyze the well log data for the ambient tectonicstress we ran the raw dipmeter data through a seriesof steps The steps were as follows

Digitize the paper logs and resample them to m intervals

Apply caliper calibration corrections to thecaliper arm data for those wells in which the well logshows a caliper correction

Compare the dipmeters borehole azimuthdata with the data from an independent directional survey of the hole singleshot deviation surveys or gyroscopic logs If the two data sets dier by roughly themagnetic declination E for this location then apply the declination correction to the dipmeters borehole azimuth and pad azimuth data We applied thedeclination correction to A and A The data fromA and A agreed with their directional surveys towithin and hence were not further corrected

Calculate the borehole elongation directionusing the four and sixarm technique described aboveThese data are plotted at every meter in Figure aslower hemisphere stereographic projections of the borehole elongation Also shown are enlargements of certainregions of the plot to better show the borehole elongation directions

Select breakouts from the borehole elonga

Breakouts Constraining Stress Tensors

A-7

N

S

EW

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45

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75

90-1802.3

-604.7

A-1

A-13

A-16

A-1 A-7

A

A-13

A-16

Figure Lower hemisphere stereographic projection plots of the azimuth of borehole elongation at m logdepth intervals from the hour wells drilled in the Point Pedernales eld top Lower hemisphere with all the welldata plotted bottom Enlargements of the top gure The graduated depth scale shows the true vertical depthin meters

Zajac and Stock

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Figure Plots of the calipercalibrated and declinationcorrected digitized dipmeter data and derived quantitiesas a function of log depth from well A top Borehole elongation direction solid line pad azimuth dottedline and borehole azimuth dashed line middle Borehole deviation bottom Bit size straight solid linecaliper arm solid line and caliper arm dotted line Selected breakout regions are plotted as horizontal barsshowing the depth extent of the breakouts

Breakouts Constraining Stress Tensors

Table Digitized well logs from the Unocal Company of Wells Drilled Oshore From Point Pedernales California in the Santa Maria Basin

Logged Depth Processed True Vertical MaximumWell Name Log Type Interval m Depth Interval m Deviation

A Gearhart four electrode logged

dipmeter survey processedA Gearhart six electrode logged

dipmeter processedA Log Schlumberger SHDT logged

monitor log processedA Log Schlumberger dipmeter logged

monitor log processedA Schlumberger formation logged

microscanner log processed

For well A we received two separate paper logs Logged depth is measured along the well bore but isgreater than the true vertical depth where the borehole is deviated

tion data using the breakout selection criteria describedabove As an example we show the calibrated caliperand declinationcorrected digitized dipmeter data andderived quantities as a function of log depth with theselected breakouts from well A in Figure Figure shows all of the selected breakouts in the Point Pedernales data We did not manually remove any breakoutsdue to key seats from the breakouts selected by computer Note that no breakouts were found in the datafrom well A because of the odd character of the datafrom the caliper arm which routinely showed caliperarm diameters quite a bit larger and smaller than thebit size and caused the data to fail at matching criterion described earlier Since well A was the onlywell logged with a sixarm dipmeter the particular technique used to calculate sixarm borehole elongation angles becomes moot

In Figure we plot a histogram of the angular differences between the location of the high and low sidesof the borehole and the breakout angle measured in theplane perpendicular to the borehole axis for all of ourselected breakouts There is roughly a spread ofbreakout angles about the high side of the hole Thesedata show the same clustering of breakout angles nearthe high and low sides of the hole as Qian and Peder

sens data Figure

Invert the selected breakout data for the besttting stress state using the combined GA and Powelloptimization technique described above The results ofthe inversion are shown in Table We note that themist between the optimized model is more thantwice as large as the mist from the Qian and

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-80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30

High Side Angle - Breakout Angle

Figure Histogram of the angular dierence measured in the plane perpendicular to the borehole axisbetween the breakout orientation and the high and lowsides of the borehole for the Point Pedernales data

Zajac and Stock

Table Characteristics of the Stress State That Minimized the Weighted OneNorm Mist of the Point Pedernales Borehole Breakout Data

S S SAzimuth NE NE NEPlunge

Value

Optimized minimum weighted onenorm

mist condence level for weighted onenorm mist

N

S

EW

15.0

30.0

45.0

60.0

75.0

90.0-1211.5

-623

Figure Lower hemisphere stereographic projectionplot of the selected Point Pedernales breakouts plottedon top of the theoretical breakout pattern of the best tting stress state The graduated scale shows the depthof the selected breakouts in meters The two solid circlesare the location of the nodal points of the theoreticalbreakout pattern

Pedersen inversion

After inverting the Point Pedernales data we compare the theoretical breakout pattern with the observations of good quality breakouts Figure Both thenodal points of the pattern are along a strike of roughlyNW with the nearvertical one plunging and thenearhorizontal one plunging from horizontal Onenodal point lies between the A and A clusters ofbreakouts which have a distinctly dierent trend Finally it should be noted that the nodal points are notsymmetric about the origin of the plot unlike the simple

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135˚

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45˚

S1

S2

S3

S1

S2

S3

S1

S2

S3

Figure Lower hemisphere stereographic projectionplot of orientations of the principal stress directions ofthe best tting stress state The inner and outer contours are the and condence limits for theeach principal stress direction The stress state ratiowas held constant at

stress state examples shown in Figure because noneof the principal stress directions are vertical Table

The best t orientation of the greatest principalstress direction S is N

E surprisingly more eastof south than is to be expected However the condence levels on the S direction would permit it to liein the azimuth range from N E to NE Figure Given the relative shallowness of the breakoutdata the deepest breakout being km deep the stressstate has no principal vertical stress direction The minimum stress S is the closest principal stress directionto vertical being away from vertical

This nding does not agree well with the regionalbreakout focal mechanism hydraulic and geologicdata presented by Mount and Suppe which showhorizontal principal stress directions oriented NESWIt also does not agree well with the inferred NNW trendsof young folds in the Southern Santa Maria Basin egClark et al which might imply a more NE orENE direction of SH However it is possible that thestress state in the Point Pedernales eld varies locallyfrom place to place so that our result may not be regionally representative of the stress eld beyond the immediate area of the borehole system studied here

Breakouts Constraining Stress Tensors

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Figure Plot of the weighted onenorm mist forthe Point Pedernales data as a function of where thethick solid line is the condence limit thethin solid line is the minimized mist when is heldconstant and the principal stress directions are unconstrained and the dotted line is the mist using the principal stress directions from the best tting model

The stress state ratio is reasonably well constrainedby the data we used at the condence level Figure This nding is in contrast to the Qian and

Pedersen data which were unable to constrainthe stress ratio The notable dierence here is due tothe location of the breakout nodal points one of whichis well constrained in the Point Pedernales data by thedierence in breakout orientation between the wells A and A Although we used all the data we couldobtain from this drilling platform we recognize that useof additional data for this region would be preferable inorder to better constrain the regional stress state Obviously more complete coverage of the borehole orientation space represented by the plot in Figure wouldallow a better estimation of the overall variability of themeasurements and show whether the inferred positionsof the nodal points are likely to be correct on a moreregional scale

Conclusion

We developed a technique using genetic algorithmsand a Powell optimizer whereby borehole breakout datafrom variably deviated boreholes are used to constrainthe orientation of the principal stress directions andtheir relative magnitudes Some of the advantages ofGAs are that they eciently search the problem spacedo not require a good starting model do not get trappedin local minima unlike gradient search methods andonly require a forward modeling calculation of the mist between the theoretical breakouts for a model andthe data However as demonstrated by our inversion ofthe Point Pedernales data when the mist as a functionof the model is not smooth the GA may not nd theglobal minimum requiring many runs of the GA andPowell optimizer to ensure that a reasonable minimumwas found We use a calculation inspired by Gephart

and Forsyth to determine the condencelimit on a weighted mist between a particular modeland the data We then place condence limits onthe results of an inversion by searching the model spacearound the best tting model for model mists that lieat the condence limit

In processing our raw dipmeter data we developeda modied list of borehole breakout selection criteriabased on Plumb and Hickmans criteria Themain dierence between their criteria and ours is thatwe do not discard radial breakouts since theoreticalbreakout patterns show that there are borehole orientations in which radial breakouts are expected for moststress states

Zajac and Stock

If the regional stress state is either normal or thrustfaulting then the breakout pattern nodal points arelikely to be found at low borehole deviations resultingin a good determination of the principal stress directions and the stress state ratio If the regional stressstate is strikeslip or has a large amount of strikeslipfaulting in it then highly deviated boreholes are required to constrain the stress tensor

We applied our technique to the Qian and Pedersen

breakout data where we showed that removingthe constraint of a vertical principal stress direction lowered the total mist between the best tting model andthe data The Qian and Pedersen inversion fallsoutside the condence limits found on our solutionHowever our determination of the horizontal maximumprincipal stress direction agreed with Qian and Peder

sens result to within The data also showedan unconstrained ratio since the borehole orientations did not vary suciently to locate a nodal point inthe breakout pattern

Finally we inverted a breakout data set from the oshore Santa Maria Basin California and determinedthat the maximum principal stress direction is orientedNE and plunges The condence limits on the S azimuth are N E to NE Theinversion also yielded a stress state in which none of theprincipal stresses are vertical The S direction is theprincipal stress closest to vertical making a angleaway from vertical This is a highly nonvertical stressfor being within km of the surface

Because of the highly variable borehole azimuths anddeviations of wells drilled from oshore platforms andbecause oriented caliper data are generally recorded forsuch wells a large quantity of data probably exist thatcan be inverted by using our technique If more detailedborehole imaging is available to constrain breakout orientations eg formation microimaging borehole televiewer this technique can also be applied with a higherdegree of condence regarding the quality of breakoutsused in the analysis

Acknowledgments The authors thank Julie Shemeta

for providing a lot of help with this work Jerry Lohr for

releasing the Santa Maria data to us and Torch Oper

ating Company for allowing us to publish the data We

acknowledge Leonid Germanovich Bezalel Haimson and

Pavel Peska for their constructive reviews of this paper We

also thank Wessel and Smith for creating the GMT

plotting tools This work was supported by the National

Earthquake Hazards Reduction Program of the US Geo

logical Survey USGS and by NSF grant EAR for

J M Stock Research was supported by the USGS Depart

ment of the Interior under USGS award G

The views and conclusions contained in this document are

those of the authors and should not be interpreted as nec

essarily representing the ocial policies either expressed or

implied of the US government Caltech Seismological Lab

oratory Contribution Number

References

Aadnoy B S Inversion technique to determine theinsitu stress eld from fracturing data J Pet SciEng ! a

Aadnoy B S Insitu stress direction from boreholefracture traces J Pet Sci Eng ! b

Bell J S and D I Gough Northeastsouthwest compressive stress in Alberta Evidence from oil wellsEarth Planet Sci Lett !

Bell J S and D I Gough The use of borehole breakouts in the study of crustal stress in Hydraulic Fracturing Stress Measurements edited by M D Zobackand B C Haimson pp ! Nat Acad PressWashington D C

Brent R P Algorithms for Minimization Without

Derivatives PrenticeHall Englewood Clis N J

Clark D H N T Hall D H Hamilton and R GHeck Structural analysis of late Neogene deformation in the Central Oshore Santa Maria Basin California J Geophys Res B !

Crouch J K S B Bachman and J T Shay PostMiocene compressional tectonics along the Central California Margin in Tectonics and Sedimen

tation Along the California Margin Pacic Section

SEPM vol edited by J K Crouch and S BBachman pp ! Soc of Econ Paleontol andMineral Tulsa Okla

Daneshy A A A study of inclined hydraulic fracturesJ Soc Pet Eng !

Davis L Ed Genetic Algorithms and Simulated An

nealing Morgan Kaufmann San Francisco Calif

Fairhurst C Methods of determining in situ rockstresses at great depths Tech Rep TRI MoRiver Div Corps of Eng Omaha Neb

Gephart J W and D W Forsyth An improvedmethod for determining the regional stress tensor using earthquake focal mechanism data Application tothe San Fernando Earthquake sequence J GeophysRes B !

Breakouts Constraining Stress Tensors

Goldberg D E Genetic Algorithms in Search Opti

mization and Machine Learning AddisonWesleyReading Mass

Goldstein H Classical Mechanics nd ed AddisonWesley Reading Mass

Gough D I and J S Bell Stress orientation from oilwell fractures in Alberta and Texas Can J EarthSci !

Hiramatsu Y and Y Oka Stress around a shaft orlevel excavated in ground with a threedimensionalstress state Mem Fac Eng Kyoto Univ !

Hobbs B E W D Means and P F WilliamsAn Outline of Structural Geology John Wiley NewYork

Holland J H Adaptation in Natural and Articial Systems Univ of Mich Press Ann Arbor

Hubbert M K and D G Willis Mechanics of hydraulic fracturing J Pet Technol !

Jaeger J C and N G W Cook Fundamentals of

Rock Mechanics rd ed Chapman and Hall NewYork

Kirsch G Die Theorie der Elastizit"at und die Defurfnisse der Fertigkeirslehre VDI Z

Mardia K V Statistics of Directional Data Academic San Diego Calif

Mastin L Eect of borehole deviation on breakoutorientations J Geophys Res B !

McCrory P A D S Wilson J C Ingle and R GStanley Neogene geohistory analysis of Santa MariaBasin California and its relationship to transfer ofcentral California to the Pacic plate US Geol

Surv Bull J

McCulloch D S Regional geology and hydrocarbonpotential of oshore central California in Geology

and Resource Potential of the Continental Margin of

Western North America and Adjacent Ocean Basins

Beaufort Sea to Baja California vol Earth Sci

Ser edited by D W Scholl A Grantz and J GVedder chap pp ! CircumPac Councfor Energy and Miner Resour Houston Tex

Moos D and M D Zoback Utilization of observationsof well bore failure to constrain the orientation andmagnitude of crustal stresses Application to continental Deep Sea Drilling Project and Ocean DrillingProject boreholes J Geophys Res B !

Mount V S and J Suppe Presentday stress orientations adjacent to active strikeslip faults Californiaand Sumatra J Geophys Res B !

Parker R L and M K McNutt Statistics for theonenorm mist measure J Geophys Res B!

Pe#ska P and M D Zoback Compressive and tensilefailure of inclined well bores and determination ofinsitu stress and rock strength J Geophys Res B !

Plumb R A and S H Hickman Stressinduced borehole elongation A comparison between the fourarmdipmeter and the borehole televiewer in the AuburnGeothermal well J Geophys Res B !

Qian W and L B Pedersen Inversion of borehole breakout orientation data J Geophys Res B ! Correction J Geophys Res B !

Richardson R M Hydraulic fracture in arbitrarily oriented boreholes An analytic approach in Hydraulic

Fracturing Stress Measurements edited by M DZoback and B C Haimson pp ! Nat AcadPress

Sorlien C C Structure and Neogene evolution of theSouthern oshore Santa Maria Basin and WesternSanta Barbara Channel California PhD thesisUniv of Cailf Santa Barbara

Stock J M J H Healy S H Hickman and M DZoback Hydraulic fracturing stress measurements atYucca Mountain Nevada and relationship to the regional stress eld J Geophys Res B !

Timoshenko S and J Goodier Theory of Elasticity McGrawHill New York

Vernik L and M D Zoback Estimation of maximumhorizontal principal stress magnitude from stressinduced well bore breakouts in the Cajon Pass Scientic Research Borehole J Geophys Res B !

Wessel P and W H F Smith New version ofthe generic mapping tools released EoS Trans

AGU Electron Suppl Aug Available ashttpwwwaguorgeos elecehtml

Youngdahl C K and E Sternberg Threedimensionalstress concentration around a cylindrical hole in asemiinnite elastic body J Appl Mech !

Zajac and Stock

Zajac B and J M Stock Using borehole breakouts toconstrain the complete stress tensor abstract EoSTrans AGU Fall Meet Suppl

Zemanek J E E Glenn L J Norton and R LCaldwell Formation evaluation by inspection withthe borehole televiewer Geophysics !

Zoback M D and J H Healy In situ stress measurements to km depth in the Cajon Pass ScienticResearch Borehole Implications for the mechanicsof crustal faulting J Geophys Res B !

Zoback M L storder and ndorder patterns ofstress in the lithosphere The World Stress MapProject J Geophys Res B !

Zoback M L and M D Zoback State of stress inthe conterminous United States J Geophys Res B !

Zoback M L et al Global patterns of intraplatestress A status report on the world stress mapproject of the International Lithosphere ProgramNature !

B J Zajac and J M Stock SeismologicalLaboratory California Institute of TechnologyPasadena CA email blair$gpscaltechedujstock$gpscaltechedu

August revised November accepted December

This preprint was prepared with AGUs LATEX macros v

with the extension package AGU by P W Daly version b

from