00026600_A New Method to Determine in-Situ Formation Rock Properties During a Microfrac Test

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WE 26600A New Method To Determine In-SituFormation Rock Properties During a

mmSocietyof Petrdwn Er@nwsStress Directions and In-SituMicrofrac Test

Peng Lin and T,G, Flay, Halliburton Energy ServicesSPEMembers

zCopyright 1993. SocIeIy 01Petroleum Ertgineers, Inc.This paper was prapared for Dresenlatlonat the 681hAnnual Tecfmcal Conference and Exfrl&lhon of the Society of Petroleum Englnaare held In Houston, Texas, 3-8 October 1993.TfMapaper was aelacted for presentation by an SPE Program CommlI!ea following review of Information contained in an abatract aubmitled by the author(e). Confents of the paper,as presen!ed, have no! been rawewed by the SocIely of Patroleum Enginears and ara subject to correction by the aulhor(s). The malerial, as presonled, doss not rtecesssrily reflectanyposition of the S~clety of Pelroleum Engmaers,ita oltlcere. or members. Paperapraaanted at SPEmeetings aresubject to publication reviawby Editorial Committaaaof lhe Societyof PalroleumEngineara.Permissionto coPyiarestrictedto anabalractof notmore than300worde.Illustrafionamay notbe Copied.Theabshacl sftould containConsplcuouaacknowFedgmantof where and by whom tha papar lapresented. Write Librarian. SPE, P.O. Box 833636, Richardson, TX 75083.3836, U.S.A. Telex, 163245 SPEUT.

ABSTRACTThis paper presents a new methodto determine in-situstress directions and in-situ formation rock properties using anewly developed downhole extensometer. The extensometerisused to measure the wellbore deformation, pressure, andtemperature during a microfrac stress test. This paper wilfpresent the method used in analyzing the data collected by thisnew downhole extensometer. Someof the parametersthat canbe determined are (1) principal stress directions, which aredetermined from the induced fracture orientation; (2) minimum

in-situ stress magnitude, which is determined from the fracturewidth calculations in addition m the conventional microfracanalysis; and (3) in-situ static rock properties, which aredetermined from the prefracture wellbore deformation. Theemphasis of this paper will lie primarily on the theoreticalmethodologyused in the data analyses. A tleld case example isalso included.

INTRODUCTIONOne of the problems in drilling highly deviated orhorizontal wells is the instability of the wellbore. Beforeaccomplishinga boreholefailure analysi: the stressesaround thewallof the wellbore and the strengthof the rock mustbe known.Knowledge of the in-situ state of stress (orientation andmagnitude) of the formationwill thusbe beneficial in designingthe drilling direction of horizontal and deviated wells to obtainmaximum stability. This knowledge will also improve thefracture design for stimulation treatments of oil and gas wells.Someconventionalways of determiningthe fracturelin-situstressorientationinvolveeitheranelasticstrain relaxationtests(ASR)ormicrofrac tests for which an orientated core is retrieved. Awidely accepted method of determining the closure pressure(minimum in-situ stress) is through a microfrac analysis. Thistype of analysis relies solely upon the pressure decline data to

match a particular model,Formation properties such as Youngs modulus andPoissons ratio also play important roles in hydraulic fracturedesign. Several techniques, covering both static and dynamicmethods, existfor determining these parameters. Staticmethodsusually involve laboratory tests that require carefully preparedsamplesandare oftentime-consuming.Retrievedsamplesare also,1olonger in the in-situ state, and the size of the samples areusually very small comparedto the formation being studied. Onthe other hand, dynamic methods, such as sonic logs, are

essentiallybased upon rapidly applied nondestructiveloads, andare usually performed under in-situ conditions, However,parameters determined statically are preferred in the design ofhydraulic fracturing becausethey are more representativeof theactual loadingconditions.A downholeextensometerwas recentlydevelopedl23omeasurewellboredisplacement,pressure, and temperatureduringmicrofracjobs, The tool differs from a conventionalcaliper inthat it can measure displacement, pressure, and temperaturecontinuously during a microfrac job. Parameters that arecurrently being obtained about the formation from the datacollected by this downhole tool are in-situ shear modulus,hydraulic fracture orientation (yielding the in-siN stressdirections), fracture closure pressure (yieldingthe minimum in-situ stress), breakdown pressure, the fracture width at thewellborewall.The extensometercollects the data during the creationof the fractureunderin-situconditions;thus, thecalculatedin-situstressdirectionsare of greater quality than those calculatedfromother methodssuch as ASR. This tool also provides a way ofobtaining the in-situ shear modulus under static loadingconditions. Most fracture design models would prefer using ashear modulusthat was calculated this way.

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2 A NEWMETHODTODETERMINE IN-SITU STRESSDIRECTIONSAND IN-SITU SPE 26600FORMATIONROCKPROPERTIESDURINGA MICROFRACTESTWith the extensometer, closure pressure is determinedfrom calculatingthe time inwhich the fractureclosesusingdirectborehole displacementmemuremerrts. Becausetheextensometercan see when the fracture really closes while the microfrac

pressure analysis cannot, the extensometerdata shouldallow forgreater accuracy in determining the closure pressure.Charlez et al.45proposeda methodto analyzewellboredeformation data. He assumed that the displacement at thewellbore is a functionof horizontalprincipalstresses,pressure inthe wellbore, radiusof the wellbore, formationelastic properties(E and v), and the fracture geometryand azimuth. Bygivinganinitial value, he used an inverse method to obtain his solution.UnlikeCharlezsmethod, themethodproposedin this paper doesnot couple the prefracture analysiswith the postfractureanalysis,This gives greater stability in the calculationswhendealingwithreal data (imperfect data).

EXTENSOMETERFEATURESThis tool has two sets of six extensometer armsseparated vertically along the tool axis by 18in. (Fig. 1). Theangle between two adjacentarms is 60. The top set of caliperarms is offset by 30 from the bottom set of caliper arms.allowing for the measurement of the radial displacementof thewellbore in 12different directions.

W * I M to7 , iwinacmwulw

- lhbhlg

&...... ..*.. .,.,,..... ,: :;:~.::: Ehcwnhx

;...,.-i.,.. .. :., E&uom9m

!!!!+7 nLds0tr0mPI

F qpc L - ~xtensometer r e a nue s 8.

Each arm is equipped with a potentiometer and anLVDT(linear variable differential transformer), which measurethe total arm movement(providing the wellbore radius) and themovementof the arms after making contact with the wellborewalls (providing the differentialdisplacementsof the wellbore),respectively. The orientationof the tool from the magneticnorthand inclination from the vertical is determined by multi-axesmagnetometers. The bottomhole injection pressure andtemperatureare measuredby a highlyaccuratequartz transducer,

MATHEMATICALMODELIn addition to temperature and pressure, theextensometer measures wellbore radial deformation in 12directions. Themeasuredboreholedeflectionsbeforebreakdowncan be used to determine shear modulus. Youngs moduluscanthen be determined if Poissons ratio is known. The wellboredeformationsafter breakdowncan be usedto calculate the widthand orientation (at the borehole) of the induced fracture. Thefracture closure pressure can also be determined by correlatingthe post-shut-inpressure readingswith the displacementdata.

BasicA~sThe assumptionsmade are summarizedbelow:1, The forma!ion is elastic, isotropicand homogeneous.2. The formationhas three mutuallyindependentprinciplestress directions (i.e. a hydrostatic state of stressdoesnot exist in any plane), with the maximum principle

stressbeingparallel to thevertical axis of the wellbore.3. The cross section of the wellbore is circular. Duringdrilling, the nonhydrostaticin-situ stresses may distortthe shapeof the wellbore, Assumingthat the wellboreundergoeselasticdeformationthat is instantaneous,thedistortionwill beminimized, This isdue to the drill bitactingas a reamer upon the initial elastic response ofthe wellbore walls to the cut-out hole. Thus thedistortedwellborecross section is immediatelyreamedinto a circular wellborewith the stress field around theborehole in equilibriumwith the in-situ stresses.These assumptionsare used in developingthe theoreticalmodel;however, deviation from one or more of these assumptionsmaynot result in any substantial error in the parameters calculated.A lattersectionof thispaper clearly showsthis to be the case fornon-circularwellborecross sections (see assumption 3 above).

WellboreDisdacernentsThe displacements around the circumference of thewellbore caused by a change in pressure within the wellbore,must be studied before any theoreticalmodel using the boreholedisplacementmeasurementscan be applied. Using the theoryoflinear elasticity and conformal mapping techniques the


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4 Lq

SPE 26600 PENG LIN ANDdisplacement of an elliptically shaped hole is described asshown:

-APR m(l + X) COS28 - (1 + Xm2) (1)UP = 2G ~1 - 2m cos2Et + mf

where

U8 = -AP R (1 - X)m sin20 (2)JI - 2m COS21 + mfm= a-b a+bR. (3)~b 2

~=3-4v (4]

a = major axesof the ellipseb = minor axes of the ellipse0 = anglebetween oz and the a axisU. = displacementsin the P - direction

yeb upqz

eo al

igure 2- EllipticalShapedBoreholeu~ = displacementsinAP = pressure changeG = shear modulusv = Poissonsratio

For a circular wellbore crossdisplacements resulting from a change in

TOMMYG. RAY 3Shear ModulusDetermination

Theextensometerprovidesa uniqueway to measuretheformation shear modulus under in-situ conditions. Itcontinuouslymeasures the radial deformations of the wellborewallduringthe microfractest. Thewellboredeformationsbeforebreakdownareusedfor determiningthe formationshearmodulus.As the pressure withinthe wellbore increases, the wellborewallexpands uniformly in all directions, assuming a linear elasticformationand a circular cross section. Equation5 can be solvedfor the shear modulus (G) resultingin the followingequation:D APG=2 AD

where G = shear modulusD = diameterof the wellboreAP = change in pressureAD = change in diameter

(6)

The linearportion of thepressure vs. displacementdatabefore breakdown is used to calculate the shear modulus. Thislinear portion yields a slope of AP/AD, which can be calculatedvia linear regressionanalysisfrom the pressure vs. displacementplot before breakdown.

Yotmgsmodulus(E) and Poissons ratio (u) cannotbedirectly calculatedfrom the extensometerdata, since only radialdeformationsare measured. However, for most formations, agood approximation to Poissons ratio v can be determined(usuallyin the range of 0.15 to 0.3). Youngsmodulus (E) canthen b> calculated from the shear modulus and the estimatedPoissonsratio (v) as6- direction E = 2G(1+v) (7)

In-SituStress Orientationand Width Determination

section, the radialpressure witbin theweilborecan be obtained from Eq. 1tirough4 by setting a=b.Thus, from Eq. 3, it is seen that R becomes the radius of theborehole. This yields the following:AP RUP = -~ (5)

while the tangential displacementsvanish. The equation aboveshows that for any change in pressure, the radial deformationofthe wellbore is the same in all directions (i.e., the radialdeformationof a circular hole is independentof direction). Fora full derivation of the equations shown in this section, seeAppendixA.

The fracture azimuthand widthare calculated from thewellboredeformationsafter breakdown, Assumingthe wellborecross section iscircular before fracture initiation, it canbe shownthat as the wellborepressure changes, the radius of the wellborewill changeequally in all directions (discussedpreviously), Thefracture will developparallelto thewellboreaxis andwill cut thewellbore into two half-circles that move perpendicular to thefracture direction (i.e. The direction of maximum deformationwill be perpendicular to the induced fracture plane and in thedirection of the minimumprincipal in-situ stress, assuming themaximum principal stress is aligned with the wellbore axis).Superpositionof the linearelasticdeformationof thewellboreandthe deformation resultingfrom the growth of the fracture yieldsthe total radialdeformationof the wellbore.

The part of the radialdisplacementcausedby a changein pressure (the elastic deformation) is independentof direction,893


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A A NEWMETHODTO DETERMINEIN-SITU STRESSDIRECTIONSAND IN-SITU SPE 26600FORMATIONROCKPROPERTIESDURINGA MlCROFIL4CTEST

T2B3 (~) B2. .(6) ~ 30 !3~T3 . .. T1 (6) (2) ./

[

I!I

;y2 =30 B4 ~ (1,) B1/I!, /T4 . I1 T61I

B5 I B6T6

(j} [1!1!1:1!.1 ,1 Ii,, rl.ll l.., l.::!1, Illll!:]r,$, ,,! 1{. !!,.,, 1,}, ..[.

l, 1:,. ,11. l:!l 1.111. ,,1 !l. :11. 111 I

T 1.1 !! ,: 1,. 1-.. 1! 1,11,

!_____ * !- r.. . - -- -- -- -- fi.., - -. -. :- -lgt.rrea - I ransuucerurmrauunwhile the part caused by the openingof the fracture will varywith direcfion. The direction of maximumdeformation will beperpendicular to the induced fracture plane. Thus, using themeasured displacements, a relationship between fractureorientation, fracturewidth, andthe changeof wellboreradiuscanbe modeledgeometrically and solvedusingan iterativesolution.Fhally, the direction of the least principal in-situ stress isperpendicular to the direction of the inducedfracture.

From the superpositionprinciplementioned above andthe fact that the direction of maxinarm deformation isperpendicular to the induced fracture plane, the followingrelationshipholds (Fig. 3 ):w lsin(~-yi) I + DR = Dispf (8)i=l,6

whereB = fracture direction with respect to transducerB1w = fracture widthDR = change in wellborediameter

This yieldssix nonlinearalgebraicequations (one for each of thetransducerpairs) for the three unknowns. Theseequationscanbefurther simplifiedby taking a Taylor Series expansion about apoint, The Taylor SeriesExpansionabout a point (~,yo,zO)of a

functionF(x,y,z) is written as follows:

F(x, y, z) = F(xo/ yo, Za) + g Ixojy., ? 2 0X-xo) +dF~y Ixo,yo, 20 (y-yo) + glxo,yo,z.(%)

(9)Thus, Eq. 1 becomes(W-WO)lSiIl(~O-yj) I + (p-p.) ~O~~coS(PO-Yf)(lO)

+ (DR-DRO) = Dispj i=l,6where 6,, = initialguess for the inducedfracture direction

W,, = initial guess for the induced fracture widthDR(, = the initial guess for the change in wellborediameterTI = the transducerlocationw.r.t to B1(indegrees)K, = * 1, dependingon the value for&An iterative method, using the center-correcteddisplacementsandEq. 8, is used to determinethe initialguessfor/3,,. Subtractingthe equationfor the i-l transducer pair from theequation for the i* transducer pair, will eliminate DR from theresultingequation. Now using B(,which is already determined,W. is calculated. When calculating W,,the combinations oftransducers used for this calculation are determined through

statisticaltechniques, DROs nowcalculatedusing Eq, 8. Afterdetermining the initial guess for the three unknowns, Eq. 10 isused inmatrix form to solve for the three unknowns.Upon calculating the induced fracture width ariddirectionfrom a setof measureddisplacements,an idealizeddataset of transducer displacements is generated. This data set isbasedupon the results (i.e. the calculatedfracture azimuth, widthandwellboreexpansion)of the original data, The idealizeddatais then compared to the actual transducer data. With thiscomparison, transducers not operating within acceptableconstraints may be detected and eliminated from any furtherarra!ysis. This anaiysis is made simuhaneously with thecalculationsof fracturewidth and azimuth,

Closure PressureAnalysisThe fracture closure pressure can also be determinedfrom the data collected by the extensometer. Unlike theconventionalmicrofrac pressure analysis, this analysis uses thedisplacementdata to determine when the fracture has closed.Upon shutting in the well after an injection cycle, the fracturewidthbeginsdecreasingbecauseof the decline in pressurewithinthe fracture. The transducerarms maynever return fully to theiroriginalpositionbecauseof the damagedone to the formation andthe physical constraints within the extensometer. Thus, the

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SPE 26600 PENG LIN AND TOMMYG. RAY sfracmre is consideredclosed when the rate of changeof fracturewidth is small. Given the pressure and fracture width vs. timerelation, a correlation betweenbottomholepressureand fracturewidthcan be found. Hence, closurepressurecan be determined.

DISCUSSIONThere were some assumptionsmade in developing thetheoreticalmodel. However,deviationfromoneor moreof theseassumptions may not result in any substantial error in theparameters calculated. This section will clearly show this to bethe case for slightlyellipticalwellborecross sections.In reality, the borehole cross section should be verynearlycircular. This is inherentwithinthedrillingprocess itself.During the drilling process, unequal in-situ stressesmay distortthe shape of the borehole, but the drill bit acts as a reamer andleaves a circular borehole behind, However, the displacementsfor an elliptical shaped cross section has also been studied to

investigatethe accuracy of the circular hole assumption. Here,the displacementson theedge of an ellipticalholeare studiedandcompared to the displacementsof a circular cross section, Thedisplacementson the edge of an ellipticalhole are shown in Eq.1 and 2, The derivationof theseequationsis shown inAppendixA.From the final equationsin AppendixA, it is seenthatthe maximumdisplacementfor the elliptical hole subjectedto anincrease in pressure occurs along the minor axis of the ellipse(i.e. 0 = 900), while theminimumdisplacementoccursalongthemajor axis of the ellipse (i.e. 6 = OO). Thus, subtractingthedisplacements yielded from these two directions results in themaximumpossibledisplacementdifferencein any twodirections.Subtracting Up((l= 0) from U@(f? 90) gives the maximum

possibledifference in the displacements,which is as follows:Aup = -~ m (3 - AV) (11)

For the case of a circular hole (m = O), AuObetweenthe a and b directions is equal to zero. This is because thedisplacementsin the radialdirectionare independentof direction.For the case of a slot (m = 1), u, in the u direction is zero.For the caseof an ellipticalhole (O< m < 1), AuOcanbe estimatedusing Eq. 11, A typical shear modulusG of theformation is on the order of ld psi. Assumea pressurechange,AP, of 1 psi. This results in the term AP/G being on the orderof 104. The term Rxm is one halfof the differencebetweenthemajorandminor axes(i.e. Rxm = l/2[a-b]). Tie factor(3-4P)is approximately equal to 2. Therefore, for a change in pressufeof 1 psi, the maximumdifferenceof the eiastic displacementforan ellipticalhole (Aup)is about six orders of magnitudesmallerthan the differences of the major and minor axes lengths.Assuminga difference between the lengthof the two axes to beabout 100,000 #-in, and a pressure change of 500 psi, themaximum possible difference in the displacement (AuJ isapproximately 50 ~in. For this case, with an 8 in. diameterborehole, the actual radial displacement(u,) is about 1000win.

Thus Au, is about one-twentiethof the actual radialdisplacementin any direction, Therefore, the assumption of a circularborehole is quite adequate for this application.The theory above is basedupon linearelasticitytheory,anddoesnotconsideraneiasticdeformationof the wellborewalls.

However, one of the reasons the cross section of the wellboremay be slightlyelliptical may be due to anelastic deformation,Therefore, the actual wellbore deformations resulting from achange in pressure within the borehole may differ slightly fromthe theory above.If the wellbore wall is ellipticaland the major axis issufficientlylarger than the minor axis, then the shear moduluscalculationsin the six directions may differ significantly,but theaverage overall directions may still approximate that calculatedfrom a circular wellbore. The fracture azimuth and width aremuchless sensitiveto the initial wellbore shape and thuswill beaccurate for even highiy elliptical wellboreshapes.

FIELD EXAMPLESThe fieldcaseselectedhere ismerelyused to showhowthe data collected from this tool can be used to determine theshear modulus of the formation, the fracture direction, and thefracturewidth. From these results we getYoungsModulus, thein-situ stress directions, and the minimum in-situ stressmagnitude, respectively, Additional field applications can befound in an earlier pape~,The shear modulus analysis is performed over theinjectioncyclesbeforethe formationis fractured, Figure 4 showsthe pressure vs. displacementdata used to determine the shear

{IEJMIIIIIIII1

L JFigure 4- DeterminingShear Modulusmodulus from one such cycle. Here we see that all of thetransducer pairs gave approximately the same slope in thepressure vs. displacementplot. This indicatesthat the boreholeis growing approximately the same in all directions. For thisfigure, no common reference point for the displacementswas

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6 A NEWMETHODTODETERMINE IN-SITU STRESS DIRECTIONSAND IN-SITU SPE 26600FORMATIONROCKPROPERTIESDURINGA MICROFFL4CTESTchosen. This is the reason for the offset from one curve to thenext, The curves are plotted in this manner for illustrativepurposesonly (i.e. the result!!do notdependuponwhetheror nota common reference is used).

For the postfracture analysis (i.e. the fracture alreadyexists or is created), one pressure cycle from this samejob ischosen, Figure 5 gives the transducer displacementsfor the time

IJlspheam enisOf Ta nskws PARS Vs TmeFigure 5- Measured Transducer Displacements

interval in which the analysiswas performed, whileFig. 6 showsthe resultingazimuth from this interval. The displacementcurves

S u J n lnucNRFA2wmwTiusFREwEwr

.,,_. < -- ... . . .-.t.lgure o - rracurre wimu tn

shownbegin at the point in time when the fracture is fully open.These curves are referenced to a common point which is justbefore the pressure-up cycle. Thus, unlike the pre-fracturecurves, these curves show the relative difference in themagnitudesof the transducer displacements. This difference isdue to the opening/creationof the fracture. It is this differentialbetween the transducers that allows for the calculation of afracture direction and width. Figure 6 illustrates the fractureazimuthas a circularfrequencyplotwith a 2 bandwidth. Ascanbe anticipated,all of the remahringcyclesshowedthe fracture toopen within2.2 of this direction.The tail end of the data from this cycle is used indeterminingthe closure pressure, This time intervalcorrespondsto that in which the injection has ceased and the well is eitherhut-in or allowed to flow back. Thus, the pressure and

I I i- iE5E4==l. L

igure7- Closure Pressure Analysisdisplacements start to decrease. Figure 7 shows how theprr&sure,width, and derivative of width with respect to time,respond after the well is shut-in. These curves are essential indeterminingfracture closure pressure. The fracturewidthinitiallydecreases rapidly with time, and then levels off to a near-zeroslope. As the width levels off, the rate of fracture closure(DW/DT) approaches zero, The fracture is considered to beclosing in on itselfwhen the DW/DT curve starts to level off toa slope of zero. The pressure related to this time is consideredto be the closure pressure.

CONCLUSIONSA mathematicalmodelbased on linear elasticity theoryhas beendevelopedto analyzepressureandwellboredisplacementdata retrieved during a microfrac test. Someof the informationthat can be obtained from the analysis are (1) formation shearmoduhrs, (2) in-situ principal stress direction, and (3) fractureclosure pressure. Example output using this model wasillustrated. For ibis example, the azimuth obtained through this


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SPE 26600 PENG LIN ANDmethodmatchedto withinone degree, theazimuthsobtainedfroma microfrac core sample retrieved after the microfrac test, anacousticscanningtool, and regionaltrends. All cyclesfrom anyone test have aIways indicated very good consistency in theazimuth angle (within a few degrees of one another). Thecalcrdatedclosurepressures and shearmodulusare alsonormallyconsistentfrom one cycle to the next.

ACKNOWLEDGEMENTSThe authors would like to thank the managementofHalliburtonServices for permission to publish this paper.

REFERENCES1,

2,

3.

4.

5.

6.

7.

8.

Despax, D. M., Martin, J. H., and Gueuret, F.:GeomechanicalProbe for a Drilling Well, UnitedStatesPatent 4,625,795, Dec. 2, 1986.Copland, G. V. and Barrington, B. Q.: Well BoreMeasurement Tool, United States Patent 4,673,890,June 16, 1987.Despax, D. M., Martin, J. H., Gueuret, F.:GeomechanicalProbe for a Drilling Well, UnitedStatesPatent 4,800,753, Jan. 31, 1989.Charlez, Ph., Seleh, K., Despax, D., and Julien, Ph.:ANewWay to Determine the State of Stress and theElasticPropertiesof RockMassive,Proceedingsof theInternational Symposium on Rock Stress and RockStress Measurements, Stockholm, Sep. 1-3, 1986, pp.313-322.Charlez, P., Seleh, K., and Despax, D.: TheFracmeter: A New Numerical method to EvaluatetheState of Stress and the Elastic Properties of Rocks,SPE 15773, Fifth SPE Middle East Oil Show held inManama, Bahrain, Mar. 7-10, 1987,Muskhelishvili, N. L, Some Basic Problems in theMathematicalTheory of Elasticity,PublishingHouseofthe USSRAcademyof Sciences, 1954.Timoshenko, S.P. and Goodier, J.N., Theory ofElasticity, McGraw-Hill Publishing Company, 3rdedition, 1987.Kuhlman, R. D., Heemstra, T. R., Ray, T. G., Lin,P., and Char!ez, P. A.: Field Tests of Downhole13xtensometerUsedTo ObtainFormationIn-SituStressData, SPE 25905, Rocky Mountain Regional/LowPermeabilityReservoirsSymposium,DenverCorolado,Apr. 1214, 1993.

TOMMYG, RAY 7APPENDIXALinear Elastic Disdacement of an Ellimic Hole subiected toUniform Internal Pressure

The material in this appendix gives the derivation forthe displacement field around an elliptical hole subjected touniform internalpressurein an infiniteplate, This is a very closeapproximationto the pressurizingof a wellbore.Elasticity theory and cr-mformalmapping are well-developedanddocumentedwithin the literature, Detailsof thesesubjects can be found in references 6 and 7. Only a briefexplanationis given here,

ConformalMarmingLet z and Jbe two complex variables such that

z = w(f)whereu(~) is a single-vahreda n al yt i c fu n c t io n in some region, E,in the r plane. The above equationrelates every point jof Z tosomedefinite point z withinthe z plane. These latter pointswilloccupysome region S withinthe z plane. Conversely, let it beassumed that each point z within the region S, corresponds tosome definite point of E by the equationabove. It then can besaidthat the aboveequationdeterminesan invertiblesingle-valuedconformal transformationor conformalmapping of the regionSinto the region E (or vise-versa).

Theellipticalhole withinthe z planecan be transformedto a circularholewithinthe t plane, Thus, the area surroundingthe elliptical hole maps to the region E which is Irl >1. Theconformalmappingof the elliptical hole isz = u(() = F?({+ ~) (A-1){

The circle I(1 = 1 correspondsto the ellipse with center at theorigin and semi-axesa = R(I l -m), b = R(l-m)

where a and b are major and minor axes of the ellipserespectively. From the above equations, R and m becomea+bR=- ~. a-b-2 (A-2)a+b

r is a complex functionand can be expressedas( = p# (A-3)

Plane Theorv of ElasticityThe conformalmappingof the basic formulasof linearelasticity is required to solve this problem. After applyingconformalmapping to the formulas from the theoryof elasticity,the displacementat a point f can be expressedas follows:

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8 A NEW METHOD TO DETERMINE IN-S ITU STRESS DIRECTIONS AND IN-S ITU SPE 26600FORMATIONROCKPROPERTIESDURINGA MICROFRACTEST

where

b

$({) . .+2!

and x = 3 -4P (v is Poissonsratio).The derivative of w(~) is

(.!+({) = R(l - +e -26)P

The modulus of w(~)is:R 4 - 2p2m cos26

~Idol = -j [P + m2]2The conjugate of to(t) is

~ = R(l - ~ e*2~P2The conjugate of o(~) is

R)= AP R m ejzeP2The conjugate of #(~) is

*(C)=-APRpe~e (1 + m2)(p2 - me12e)P4 - 2p2mcbs20 + m2

(A-4)

(A-5)

(A-6)

(A-7)

By separatingthe real and imaginarypart of the equationabovethe displacemerm$n the direction of P and d (u, and UO)can bedetermined as follows:~ -- AP R mpz(l + X)COS2L3m 2(l - X) - p2(l + m?P 2G p dP4 - 2p2mcos20 + mi

(A-14)

U. = -APRp (1 - X)msMEi (A-15)~p4 - 2p2mcos20 + m2

where AP = change in pressuree = angle betweenl ine oz and the a axis @lg. 2)

Di5t)laCementSAroundAn Elliotic HoleThe formulas for calculatingthe displacementsaroundan elliptically shaped hole within an infinite plate was derivedabove._ At the boundary of the elliptic hole, p =displacementscan be further simplifiedas follows: 1, the

(A-8)

(A-9)

(A-1O)

(A-n)

(A-12)

LP R m(l + x)cos20 - (1 + xm2)up = -2G ~1 - 2mcos20 + m2

U. = -AP R (1 - x)m sh2Et41 - 2m COS20+ m2

By substituting Eq. A-5 through A-12 into Eq. A-4, thecomponents UPand U8of the displacements in terms of thecurvilinear coordinates areA P Rx2G(uP + fue) = -, P (A-13)

[mxp2e-i26 +m2(l - x) + mpzezo - P2(1 + m?]4P4 - 2p2m cos20 + m2

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00026600_A New Method to Determine in-Situ Formation Rock Properties During a Microfrac Test

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Transcript of 00026600_A New Method to Determine in-Situ Formation Rock Properties During a Microfrac Test