NUMERICAL AND ANALYTICAL MODELING OF SANDING ONSET PREDICTION

Transcription

1 UMEICAL AD AALYTICAL MODELIG OF SADIG OSET PEDICTIO A Disstatio y XIAJIE YI Sumittd to th Oic o Gaduat Studis o Txas A&M Uivsity i patial ulillmt o th quimts o th dg o DOCTO OF PHILOSOPHY August 3 Majo Sujct: Ptolum Egiig

2 UMEICAL AD AALYTICAL MODELIG OF SADIG OSET PEDICTIO A Disstatio y XIAJIE YI Sumittd to Txas A&M Uivsity i patial ulillmt o th quimts o th dg o DOCTO OF PHILOSOPHY Appovd as to styl ad cott y: Pt P. Valkó (Co-Chai o Committ) Jams E. ussll (Co-Chai o Committ) W. Joh L Dimitis C. Lagoudas (Mm) (Mm) Has C. Juvkam-Wold (Had o Dpatmt) August 3 Majo Sujct: Ptolum Egiig

3 iii ABSTACT umical ad Aalytical Modlig o Sadig Ost Pdictio. (August 3) Xiaji Yi, B.S.; M.S., Southst Ptolum Istitut Chai o Advisoy Committ: D. Pt P. Valkó D. Jams E. ussll To povid tchical suppot o sad cotol dcisio-makig, it is cssay to pdict th poductio coditio at hich sad poductio occus. Sadig ost pdictio ivolvs simulatig th stss stat o th suac o a oil/gas poducig cavity (.g. ohol, poatio tul) ad applyig appopiat sad poductio citio to pdict th luid pssu o pssu gadit at hich sad poductio occus. I this ok, pst umical ad aalytical poolastoplastic stss modls dsciig stss aoud poducig cavity ad viy thos modls agaist ach oth. Usig thos modls, valuat th stss stat o th cavity suac ad div sadig ost pdictio modls i tms o luid pssu o pssu gadit asd o th giv sad poductio citio. W th u ild cas studis ad validat th sadig ost pdictio modls. ock stgth citio plays impotat ols i sadig ost pdictio. W ivstigat ho th sadig ost pdictio sults vay ith th slctio o o o aoth ock stgth citio. I this ok, pst ou commoly usd ock stgth citia i sadig ost pdictio ad llo staility studis: Moh-Coulom, Hok-Bo, Duck-Pag, ad Modiid Lad citia. I ach o th citio, th a to o mo paamts ivolvd. I th litatu, a to-stp pocdu is applid to dtmi th paamts i th ock stgth citio. Fist, th Moh-Coulom paamts lik cohsio S o ad ital ictio agl φ a gssd om th laoatoy tst data. Th, th paamts i oth citia a calculatd usig th gssd Moh-Coulom

4 iv paamts. W popos that th st ay to valuat th paamts i a spciic ock stgth citio is to pom dict gssio o th laoatoy tst data usig that citio. Usig this mthodology, dmostat that th ct o vaious ock stgth citia o sadig ost pdictio is lss damatic tha usig th commoly usd mthod. With this mthodology, th uctaitis o th ct o ock stgth citio o sadig ost pdictio a also ducd.

5 v ACKOWLEDGEMETS I ould lik to thak my advisos D. Pt P. Valkó ad D. Jams E. ussll o dictig my Ph.D. study at Txas A&M Uivsity. Thi scitiic appoach toad sach taught m to a sious schola. I ould lik to xpss my gatitud to D. Jams E. ussll o gously povidig iacial suppot o my study. Without that, my study h ould hav impossil. I ould lik to thak D. W. Joh L o ig my committ mm. I also itd a lot om may couss taught y him. I thak him o tachig m to thik i physics i additio to mathmatics. I thak also D. Dimitis C. Lagoudas o svig o my committ. His plasticity cous lvatd my udstadig o solid mchaics ad is ully usd i my disstatio. As a itatioal gaduat studt i th Dpatmt o Ptolum Egiig, I appciat D. Thomas A. Blasigam ad Ms. Elao Schul o ogaiig lots o xtacuicula activitis o gaduat studts. Thy gav us a ss o amily. I appciat M. Josph A. Ayou o Schlumg o povidig m a itship oppotuity. His tust i m ad his illigss to tach m koldg ill missd. I thak my oicmats Javi Faqut, Olado Zamao ad Sgui Joui o shaig thi koldg ith m ad couagig ou mutual itsts i ock mchaics. I appciat my i ad my pats o thi costat ad ucoditioal suppot, ithout hich, it ould hav tak much log o v impossil to pusu ad iish my doctoal study.

6 vi TABLE OF COTETS Pag ABSTACT... iii ACKOWLEDGEMETS...v TABLE OF COTETS... vi LIST OF FIGUES... ix LIST OF TABLES... xii CHAPTE I ITODUCTIO.... Statmt o th polm.... Ojctivs o this sach....3 Summay o sults... II LITEATUE EVIEW...4. Sad ach staility...4. Poatio tul staility Op hol staility Aalytical axisymmtical pla stai modl Aalytical D pla stai modl umical modl....4 Sad poductio citio Tsil ailu Sha ailu ad ock stgth citio Po collaps...3 III AALYTICAL MODELS FO SADIG OSET PEDICTIO Aalytical modl o llo/poatio tul ailu iducd sad poductio-isotopic i-situ stss cas Poolasticity solutio ad sha ailu sadig citio Poolastoplasticity solutio ad tsil ailu sadig citio Aalytical modl o llo/poatio tul ailu iducd

7 vii CHAPTE Pag sad poductio-aisotopic i-situ stss cas Aalytical modl o poatio tip ailu iducd sad poductio Poolasticity solutio ad sha ailu sadig citio Poolastoplasticity solutio ad tsil ailu sadig citio...7 IV UMEICAL MODEL FO SADIG OSET PEDICTIO D stss modl D tasit luid lo modl Moh-Coulom plasticity Moh-Coulom i tms o stss ivaiats Flo ul Elastoplastic stss-stai matix Fiit lmt solutio to th 3D stss modl Fiit lmt solutio to th 3D luid lo modl Viicatio o th umical modls Viicatio o th 3D luid lo modl Viicatio o th 3D stss modl Compaiso o umical modl ad aalytical modl Uctaity assssmt o sadig ost pdictio modl...5 V EFFECT OF OCK STEGTH CITEIO O SADIG OSET PEDICTIO ock stgth citio Moh-Coulom citio Hok-Bo citio Duck-Pag citio Modiid Lad citio a llo poolastic stss modl Dtmiatio o ock stgth citio paamts Applicatio o ock stgth citio i sad poductio pdictio...68 VI SADIG OSET PEDICTIO CASE STUDIES Sad poductio i gas ilds i th oth ad Ctal Adiatic Sa Sad poductio i Wll # VII SUMMAY AD COCLUSIOS...9 OMECLATUE...9

8 viii Pag EFEECES...98 APPEDIX A...7 APPEDIX B...8 APPEDIX C... APPEDIX D...5 APPEDIX E...8 VITA...3

9 ix LIST OF FIGUES FIGUE Pag. Sad ach a poatio...5. Gid systm usd y Moita t al omalid ailu vlop o sadsto Complt Moh-Coulom ailu vlop icludig po collaps Duck-Pag lliptic cap modl Dimsiolss citical dado plot assumig Poisso s atio Compaiso o sha ailu iducd sadig citio ad tsil ailu iducd sadig citio Aisotopic i-situ stss aoud a ohol Dimsiolss citical dado plot o dit stss aisotopy atio-assumig Poisso s atio.3, h (t)c o Compaiso o citical dado valus o sha ailu iducd sadig om poatio tul ad poatio tip Fiit lmt gid usd i simulatio o sigl ll poductio om a closd axisymmtical svoi Compaiso o pssu ad pssu divativ cuvs o sigl ll poductio om a closd axisymmtical svoi Fiit lmt gid usd i simulatio o sigl ll poductio om a closd ctagula svoi Compaiso o pssu ad pssu divativ cuvs o sigl ll poductio om a closd ctagula svoi Compaiso o 3D iit lmt solutio ith aalytical solutio o stss stat a llo duig poductio Picipal ctiv stsss alog adial dictio o to x coodiat...45

10 x FIGUE Pag 4.7 Picipal ctiv stsss aoud llo o aisotopic i-situ stss cas ith luid lo Bouday coditio ad gid usd i iit lmt simulatio o a llo stss Compaiso o stss stat simulatio sults at 5 days o poductio usig aalytical ad umical modls lativ o o stsss usig aalytical modl Histogam o th pdictd citical ottom hol loig P avg 5 psi Cumulativ qucy distiutio o th pdictd citical ottom hol loig P avg 5 psi Compaiso schm i Chapt V Moh-Coulom stgth citio Failu vlops pojctd o th π-pla o Moh-Coulom citio, cicumscid Duck-Pag citio, iscid Duck-Pag citio ad Duck-Pag middl cicl Pssu chag ith tim duig gas poductio Wllo suac picipal ctiv stsss chag ith poductio tim asumig lia lasticity Pdictd sadig ost sults usig dit ock stgth citia o ock data i Cas A & B, ith ock stgth paamts om Tals 5. ad 5.4 spctivly Pdictd sadig ost sults usig dit ock stgth citia o tst data i Cas A, ith ock stgth paamts om Tal Pdictd sadig ost sults usig dit ock stgth citia o ock tst data i Cas B, ith ock stgth paamts om Tal Th oth Adiatic Basi...76

11 xi FIGUE Pag 6. Compaiso o pdictd ad ild masud BHFP at sadig ost assumig poatio tul o poatio tip sha ailu iducd sadig Gas dsity ad pssu colatio i a id pssu ag Gas dsity ad pssu colatio i a ao pssu ag Pdictd ad ild masud BHFP at sadig ost assumig poatio tip tsil stssd iducd sadig Compaiso o pdictd citical BHFP ad ild masud BHFP o 3 lls ith sad poductio Compaiso o pdictd citical BHFP ad ild masud BHFP o 8 lls ithout sad poductio Loggig data o ll # Pdictd ad ild masud BHFP at sadig ost assumig poatio tip tsil stssd iducd sadig Compaiso o pdictd ad ild masud BHFP at sadig ost assumig poatio tul o poatio tip sha ailu iducd sadig...88

12 xii LIST OF TABLES TABLE Pag 4. Wll, svoi ad poductio paamts - Cas Wll, svoi ad poductio paamts - Cas ock mchaics paamts usd i compaiso study Miimum, ma ad maximum valus o Poisso s atio, UCS ad i-situ hoiotal stss Assumd ock stgth tst data poits Stgth citio paamts o Cas A-Divd om gssd Moh-Coulom S o ad φ Stgth citio paamts o Cas A- Calculatd dictly om th to assumd tst data poits Stgth citio paamts o Cas B-Divd om gssd Moh-Coulom S o ad φ Stgth citio paamts o Cas B- Calculatd dictly om th to assumd tst data poits Wll, svoi ad poductio paamts usd i studyig th ct o ock stgth citio o sad poductio oth Adiatic gas lls h sad poductio as osvd ud th giv coditio oth Adiatic gas lls h sad poductio as ot osvd ud th giv coditio...78

13 CHAPTE I ITODUCTIO. Statmt o th polm Sad poductio is th poductio o sad paticls duig th poducig o oil/gas. It poss som disadvatags such as osio to suac ad dohol tuulas, i haad to gas lls. Gally sad poductio is ot dsial ad sad cotol acilitis a istalld to pvt sad poductio. Sic sad cotol is gally a xpsiv ivstmt o a oil/gas opato, it is o gat itsts o th opato to stimat i sad cotol is dd o poductio, o h sad cotol is dd at som tim o sad- poductio. To povid tchical suppot o sad cotol dcisio-makig, it is cssay to pdict th poductio coditio at hich sad poductio occus. This oms th sach topics i this disstatio.. Ojctivs o this sach Th ojctivs o this sach a: To dvlop umical ad aalytical modls hich ca usd to pdict th ost o sad poductio o oil ad gas lls. To study th ct o ock stgth citio o sadig ost pdictio. To us thos modls to xplai som ild sadig polms such as poductio o sad at som tim o sad- poductio. This disstatio ollos th styl ad omat o th SPE Joual.

14 .3 Summay o sults Though this sach, accomplish th olloig: W div a aalytical poolastoplastic stss modl o thick-alld hollo cylid modl. Basd o th thick-alld hollo cylid modl, div simpl sadig ost pdictio modls assumig sha ailu iducd sadig ad tsil stss iducd sadig at sha ailu. Ths modls may usd to study sadig om op-hol ll o poatio tul o casd ll. W div a aalytical poolastoplastic stss modl o thick-alld hollo sph modl. Basd o th thick-alld hollo sph modl, div simpl sadig ost pdictio modls assumig sha ailu iducd sadig ad tsil stss iducd sadig at sha ailu. Ths modls may usd to study sadig om poatio tip o casd ll. W div a aalytical poolastic stss modl o thick-alld hollo cylid modl ud aisotopic i-situ stss coditio. Basd o this modl, div sadig ost pdictio modl assumig sha ailu iducd sadig. This modl may usd to study sadig om op-hol lls h th i-situ hoiotal stsss a dit. W implmt a 3D gal iit lmt cod o stss stat simulatio ad sigl-phas luid lo simulatio. Its tchical pomac is chckd agaist availal aalytical ad umical solutios. W study th ct o ock stgth citio o sadig ost pdictio. W popos that th st ay to valuat th ock stgth citio paamts is to pom dict gssio o th laoatoy tst data. Usig th gssd Moh- Coulom paamts to calculat th ock stgth paamts i th oth citia is ot commdd. Usig this mthodology, th uctaitis o th ct o ock stgth citio o sad poductio pdictio a ducd.

15 3 W study th oth Adiatic Basi lls sadig cass. Though this study, coclud that sad poductio i oth Adiatic Basi lls might caus o iducd tsil stss at poatio tip at sha ailu. W validat this coclusio y lls ithout sad poductio polm ud th giv coditios. Cas study o sadig polm i Wll#3 idicats that sadig might causd y sha ailu at poatio tip. Usig this assumptio, achd pct agmt t pdictd ad ild masud ottom hol loig pssu at sadig ost.

16 4 CHAPTE II LITEATUE EVIEW Sadig ost pdictio is gally asd o sad ach istaility -, poatio tul istaility -5, o vtical, hoiotal o dviatd ohol istaility 6-3. Gally, a stss modl is stalishd to otai th stss stat a th sad ach, poatio tul o ohol, ad th a sad poductio citio is applid to pdict th stss stat o luid lo coditio at hich sad poductio occus. Post-sadig havio has also studid to quatiy th amout o sad poducd umous actos such as ock mchaical poptis, i-situ stss stat, llo/poatio gomty, pssu dado, pssu dpltio, ad at cut may iluc sad poductio. May ots hav mad to study th ct o thos paamts. Th ial goal o ths ots is to ko h sad poductio occus ad ho much sad ill poducd. Th olloig paagaphs summai thos mthods goupd y th udlyig assumptios.. Sad ach staility Th ol o achig i sad staility as ist tatd y Taghi i his tap doo xpimt, hich dmostatd that achig as a al ad stal phomo. Hall ad Haisg iitiatd th study o sad ach staility i th oil idusty. Thi pap dscis that a ach is a cuvd stuctu spaig a opig, svig to suppot a load y solvig th vtical stss ito hoiotal stsss. Sad ach is visualid as i Fig... Thi xpimts dsigd to dtmi hth luid lo o chag i load acts th staility o sad ach. Ects o sad oudss, gai cush, luid lo ad ttaility o sad ach omatio ad staility studid. It as osvd that agula sads a mo likly to om sad ach tha oud sads. Iad luid lo may hlp to staili sad ach omd y oud sads. Slo outad

17 5 luid lo dos ot disupt sad ach hil ast lo dos. Wat cut tds to dstoy th sad ach. Sti ad cooks 3-4 dscid a applicatio o sad ach staility thoy, hich assums that th maximum sad- at a ach ca tolat is popotioal to th sha modulus o th sad. Sad Ach Figu. Sad ach a poatio Lat Tippi ad Kohlhaas 5 xpimtally ivstigatd uth th ct o luid lo at o sad ach omatio ad staility. Thy cocludd om thi xpimts that sustatial sad- poducig ats ca maitaid though stal sad achs i ucosolidatd sads. Ach goth is a uctio o poductio at ad iitial ach si.

18 6 A ach may dstoyd ad a ach omd though gadual icas o lo at. Clay t al. 6 xpimtally studid th ct o stss ad luid poptis o sad ach staility i ucosolidatd sads. Thy potd that th ach si dcass ith icasig coiig stss. Thy oud also that a mo stal ach occus h th hoiotal stss is th maximum picipal stss ad th vtical stss is th miimum picipal stss. Batli ad iss 7 studid i laoatoy th sad ach phomo du to stsss imposd y loig luid. A lastoplastic stss solutio as otaid y simpliyig th ach gomty as a hmisph. Thy pstd a staility citio dsciig th total ailu o sad, as ll as th ailu o a idividual ach. Good agmt t thoy ad xpimtal data as sho. Lat Polillo t al. 8 studid th sam polm ith lastoplastic iit lmt mthod. Yim t al. 9 s xpimtal study shod that th atio o sad gai si to outlt hol si ad gai si distiutio a impotat actos. Mo ctly, Biaco ad Hallck xtdd Hall ad Haisg s ok to study th cts o chag i ttig phas satuatio o ach havio, mophology ad staility. Thi mai coclusio is that ithi a did ag, a pogssiv icas i ttig phas satuatio dos ot impact sad ach staility. As th ttig phas satuatio xcds a citical valu, sad ach istaility occus. I th aov paps, it is assumd that sad ach is omd aoud a poatio ad th poatio just ptats th ll casig ad cmt shath. This thoy may also usd assumig th is a sad ach at th poatio tip i th poatio tul is log ough. Hov, o modl cosidd th itactio o multipl achs h th shot dsity is high.

19 7. Poatio tul staility Athuis t al. simulatd th poatio collaps y loadig a um o thick-all hollo cylids to ailu. Th thick-alld hollo cylid xpimts umically aalyd ith th aid o lastoplastic thoy. It as oud that yildig ad ultimat ailu is ot quivalt. It as cocludd om thi xpimts that th ailu citio could xpssd i tms o a limitig valu o th quivalt plastic stai. Moita t al., y spaatig th ct o ll pssu ad local pssu gadit aoud th cavity, poposd a aalytical appoach to study th cts o may paamts o sad poductio. It as poitd out that th olloig paamts may act sad poductio: () ouday loads such as ll pssu ad i-situ stsss, () luid lo iducd oc hich is dpdt o such actos as lo at, pmaility, viscositis o luids, lativ pmaility o multiphas lo ad luid satuatio, (3) ock domatio chaact, (4) ock stgth chaact, (5) poatio cavity gomty ad shot dsity, (6) cyclic loadig histoy. A cavity ailu vlop, composd o a tsil ailu vlop ad sha ailu vlop, is gatd ith thi modl. I th ll pssu is too lo, sha ailu ill occu, i th pssu gadit is too high, tsil ailu ill occu. Thy dmostatd that th svoi pssu dpltio icass th ctiv i-situ stss, spcially th ctiv vtical stss compot, hich sults i th shit o th cavity ailu vlop. Tho, dpltio-tiggd tsil stss iducd sad istaility is lss likly ut sha iducd sadig polm may com domiat. Th authos ot that th sults otaid should usd o a qualitativ asis caus o som dg o simpliicatio. Moita t al. 3 povidd a aalysis o poatio tul staility usig a st o 3D tasit luid lo ad gomchaical iit lmt cods. Th sults a said to quatitativly accuat ad ca applid to ild dsig. Th gid systm usd i this study is sho i Fig...

20 8 Figu. Gid systm usd y Moita t al. 3 Satalli t al. 4 usd a 3D iit lmt cod to study th stss coctatio aoud poatios ad to lat it to sad poductio isk. Iluc o poatio gomtic atus such as poatio lgth, diamt, shot dsity, phasig ad cmtatio quality studid. sults sho that: () Sad poductio isk is idpdt o poatio lgth; () Lag shot dsity ill icas th mchaical itactio t adjact poatios ad thus sad poductio isk; (3) Cavity lagmt ill uth os poatio itactio; (4) I pooly cmtd lls, poatios ca oitd to dcas sad poductio isk ad th optimal agl oly dpds o th isitu stss stat. Tovoll t al. 5 pomd comphsiv laoatoy ad umical studis o poatio cavity ad sad poductio om a poatio tul. Jacktd cos ith a cavity simulatig a poatio tul a loadd i a high-pssu vssl ad luid lo is applid. Cavity domatio ad ailu a moitod. A 3D olia iit lmt modl as mployd to study cavity staility. It as sho, om oth umical modlig ad xpimts, that th ost o sad poductio is maily cotolld y th

21 9 omatio stgth ad th i-situ stss stat. Th sults o thi study xplai quatitativly som ild osvatios. I thi study, th ailu poit is chos as th poit o th cavity displacmt cuv cospodig to a citical plastic stai o th matial i th viciity o th cavity all. W mak that i th aov modls, oly o poatio is cosidd, i., o itactios t multipl tuls a cosidd..3 Op hol staility This typ o study ocuss o th a ohol, vtical, hoiotal o dviatd, stss stat simulatio ad its applicatio to sadig pdictio. Th stss-stai latio o th matial, th stgth citio, th post ailu havio, ad th ct o luid lo, a vy impotat o accuat simulatio o th havio o th matial. Dtaild iit lmt simulatio igs mo accuat sults ut a aalytical solutio may tt mphasi th ct o a sigl paamt. Laoatoy hollo cylid xpimts may cssay to validat th aalytical o umical solutio o it is usd i ild sadig pdictio. Th a ohol stss modls pulishd i th litatu a catgoid as: () aalytical axisymmtical pla stai modl, this typ o modl may usd to study sadig phomo i vtical ad hoiotal lls; () aalytical D pla stai modl cosidig th ct o i-situ stss aisotopy, this typ o modl may usd to cosid th ct o aisotopic hoiotal stsss o hoiotal, vtical ad dviatd ll sad poductio; (3) umical modl, this typ o modl may tak ito accout th ct o i-situ stss aisotopy, matial oliaity ad dit ouday coditios..3. Aalytical axisymmtical pla stai modl Paslay ad Chatham 6 studid th a ll stss iducd y luid lo ito th ohol usig a axisymmtical pla stai modl. Th omatio matial is

22 cosidd as lia lastic, luid is icompssil ad lo is i stady stat. Pmaility vaiatio ad its ct o ock ailu studid. Gtsma 7 povidd a gal poolastic pla stai solutio ithout assumig a paticula luid lo gim. iss ad Batli 8 oud a poolastoplastic solutio o icompssil luid lo ud stady stat coditio. Th simpliicatio, hich assums o luid lo h th maximum picipal stss is th vtical stss, may d to impovd o pactical usag. It is oth mtioig that a iitial vtical stai is cosidd, hich may lad to mo accuat dsciptio o th i-situ stss coditio. Wag t al. 9 povidd a complt axisymmtical pla stai poolastoplastic solutio, hich as usd to study poatio tul staility ad gas ll sad poductio. Badod ad Cook poposd a smi-aalytical modl to study llo staility ad sad poductio. Sigl luid lo is assumd tasit lo i iiit actig svoi. Mo ctly, Va d Hok t al. studid th a ll stss stat y cosidig th chag o matial cohsio i th plastic o..3. Aalytical D pla stai modl Ud aisotopic i-situ stss coditio h th maximum ad miimum hoiotal stsss a dit, th Kisch solutio ca usd. Hov, it is assumd that th is o luid lo om th llo to th omatio o om th omatio to llo, tho, it may ot appopiat to us this solutio to sad poductio pdictio o a loig ll. Dspit that, a coupl o applicatios o this modl had potd to pdict maximum sad- poductio at o dictioal ll 3 ad hoiotal ll 4 y assumig uiom po pssu i th svoi ad a dit pssu i th llo. Fo a spciic avag svoi pssu, a citical llo pssu is calculatd ad th it is usd to pdict th citical sad- poductio at. o aalytical modl cosidig a llo pssu gadit ud aisotopic isitu stss coditio has oud.

23 .3.3 umical modl Gally, umical modls povid mo dtaild ad accuat dsciptio o th stss stat. Aouslima t al. 5-6 povidd a poolastic modl to study llo staility. Som oth cts such as thmal, chmical ad physical also icludd i thi modl. Vadoulakis t al. 7 studid th hydo-mchaical aspcts o th sad poductio polm. A mathmatical modl is stalishd to dsci th taspot o sad paticls i omatio. As a sult, th amout o sad poducd as a uctio o tim is calculatd om thi modl. Matial alac quatios a stalishd o th loig sad paticl, th omatio matix ad th loig luid. Th amout o odd sad ad dpositd sad is latd to lo at, poosity, sad coctatio, ad a sad poductio coicit, hich is dtmid though xpimt ad caliatd though ild simulatio. Th ct o poosity icas o pmaility is tak ito accout though th Cama-Koy quatio. Lat, Stavopoulou t al. 8 coupld th osio modl ith a ll stss modl. Th ct o poosity chag o lasticity ad cohsio is also icludd i thi modl. Papamichos t al. 9-3 usd simila mthod to study volumtic sad poductio. ctly, Cooml t al. 3 coupld th osio modl ith mo complicatd a llo stss dliatio ad luid lo modl to dsci th cold poductio pocss o havy oil. Wag t al usd a coupld logomchaical modl to simulat th sad poductio pocss ad llo staility. Sad poductio is assumd to occu i ith a citical stss stat o a citical stai stat is achd. Th amout o sad poducd is assumd to popotioal to th sadig o si..4 Sad poductio citio Sval mchaisms a cogid as sposil o sad poductio, such as tsil ailu, sha ailu ad po collaps.

24 .4. Tsil ailu Batli ad iss 7 poposd a tsil ailu citio o poatio tul i shll collaps. iss ad Batli 8 usd th sam citio to hollo cylid. Tsil ailu may occu h th ctiv adial stss is qual to th tsil stgth o th omatio ock. Basd o iss ad Batli 8 s ok, Vaii mployd a ully coupld luid lo ad domatio modl to cosid th ct o tasit luid lo, olia soil ad luid havio o sad poductio. It as oud that a cavity, tsil o, plastic o ad olia lastic o may om aoud llo. Pkis ad Wigat 38 studid th coditios cssay o staility o ailu o a sphical cavity i ucosolidatd o akly cosolidatd ock. Wigat ad Pkis 39 divd a quatio dsciig tsil ailu coditio i tms o pssu dado, llo pssu, omatio ock cohsio ad ictioal agl. I thi pap, dimsiolss cuvs a povidd o dtmig th pssu dado at a spciid llo pssu. Og t al. 4 xtdd Pkis ad Wigat 38 ad Wigat ad Pkis s ok 39 y cosidig th ct o o-dacy lo. A oot-y-oot aalysis o sad poductio causd y tsil ailu is mad possil though thi ok..4. Sha ailu ad ock stgth citio I sad poductio is causd y sha ailu, ock stgth citio plays a impotat ol i sad poductio. Sval ock stgth citia hav mployd to pdict llo staility ad sad poductio i th litatu. Amog hich a Vo Miss 3-4, Duck-Pag 4, Moh-Coulom 38-39, Hok-Bo 4, Modiid Lad ad Modiid Wiols & Cook 45. Laoatoy tsts may cssay to ko hich stgth citio st dscis th havio o th ock studid. Amog thos stgth citia, th Vo Miss citio is usd mo i mtal tha i poous mdia, th Moh-

25 3 Coulom ad Hok-Bo citia cosid oly th ct o maximum ad miimum picipal stsss hil th Duck-Pag, Modiid Lad ad Modiid Wilols & Cook citia ivolv also itmdiat picipal stss. Th systmatic compaiso o th us o all thos citia has ot mad. Fo ock havs i th ittl gim, th sad poductio citio may th sam as th ock stgth citio. Hov, o ock havs i th ductil gim, it may cssay to simulat th post yild havio (hadig o sotig) ad to popos som oth sad poductio citio..4.3 Po collaps With th dpltio o th svoi pssu, th ctiv stss actig o th omatio ock icass. At a ctai stss lvl, po collaps may occu ad this may lad to sad poductio. Th pvious listd citia a just good to dsci ailu i th ittl gim ad caot usd to dsci ailu y po collaps. As a sult, it is cssay to u oth tiaxial ad hydostatic tsts to costuct a complt ailu vlop. Som ots hav mad to costuct a complt ailu vlop ad apply it i sad poductio pdictio. Though tiaxial ad hydostatic tsts o a vaity o sadstos, a sigl omalid ailu vlop as stalishd y Zhag t al. 46, s Fig..3. Th oly paamt appas i th omalid vlop is citical pssu, hich is colatd to th compssio av vlocity. It is ot cla i this omalid vlop ca applid uivsally. Aal t al. 47 dmostatd th applicatio o this omalid cuv i sadig ost pdictio ut th applicaility is ot viid.

26 4. omalid Dviatoic Stss, Ds/Pc Ba Salt ash B B B3 cuv-it B4 A A A3 A omalid Ma Ectiv Pssu, m /P c Figu.3 omalid ailu vlop o sadsto (at Zhag t al. 46 ) Th po collaps potio is sho i th Moh-Coulom citio, s Fig..4 ad i th Duck-Pag citio, s Fig..5. I summay, i this chapt, sadig ost pdictio mthods usd i litatu a vid. Th limitatios i som o th mthods a poitd out. I th olloig chapts i this ok, impovd ad mo gal poolastoplastic stss modls o thick-alld hollo cylid ad hollo sph modls a divd. Simpl sadig ost pdictio modls a divd asd o thos stss modls ad usd i ild sadig cas studis. Th ct o ock stgth citio o sadig ost pdictio is studid.

27 5 Sha Stss Sha Failu Tsil Failu Po Collaps Ectiv omal Stss Figu.4 Complt Moh-Coulom ailu vlop icludig po collaps (at Aass t al. 48 ) Ditial Stss Duck-Pag Failu Evlop Hadig Cap Ectiv Ma Stss Figu.5 Duck-Pag lliptic cap modl (at Ch ad Miuo 49 )

28 6 CHAPTE III AALYTICAL MODELS FO SADIG OSET PEDICTIO Sadig ost pdictio ivolvs stss calculatio at cavity (icludig llo, poatio tul i ou cass) suac. Ev though a umical modl, such as Fiit Elmt modl, is mo gal, aalytical o smi-aalytical modls may mo covit ad asi to us ud spcial coditios. Bsids, a aalytical modl is alays usul to viy umical modls. This chapt povids som impovd aalytical modls o sadig ost pdictio. 3. Aalytical modl o llo/poatio tul ailu iducd sad poductio-isotopic i-situ stss cas I ptolum giig, th vtical/hoiotal llo, poatio tul ad thi adjact omatio a ot appoximatd as thick-alld hollo cylid. Usig this appoximatio, a al to otai a aalytical o smi-aalytical solutio o th a llo/poatio tul stss stat ad us it i sad poductio pdictio. As idicatd i th pvious litatu vi, iss t al. 8 studid th a llo stss stat cosidig icompssil, stady stat luid lo ito llo i oudd lastoplastic svoi. Iitial vtical stai is tak ito accout i thi modl. Badod ad Cook studid th o-stady stat oil lo ito llo i iiit ouday lastoplastic svoi. Sailippo t al. 5 poposd a mo gal poolastoplastic modl o iiit ouday svoi y takig ito accout th iitial stsss o a ll is dilld ad poducd. Hov, o simplicity, th pssu i th ll daiag aa is assumd to uiom. I vi o th limitatios i th aomtiod modls, i this chapt, popos a mo gal modl suital o sad poductio pdictio though comiig th mits o th pvious modls. I this modl, assum

29 7 () Th llo/poatio tul-omatio stuctu is axisymmtical. () Th omatio ock mchaics poptis a homogous ad isotopic. Extal stsss act axisymmtically. (3) Th domatio o omatio ock satisis th pla stai coditio. (4) Th omatio ock oys th Moh-Coulom ailu citio ad havs as lia lastic pct plastic matial. With th aov assumptios, div a poolastoplastic solutio, s Appdix A o dtaild divatios. 3.. Poolasticity solutio ad sha ailu sadig citio I th matial is i lastic stat, th stsss distiutio is B α G ' (λ G)A G p(, t)d α p(, t) ' h (t ) λ G (3.) B α G λ ' (λ G)A G θ p(, t)d α p(, t) ' h (t ) λ G λ G (3.) Fo sadig ost pdictio, a oly itstd i stsss at llo/poatio tul suac. Th stsss at th suac a ' ( α )P (t) (3.3) ' θ P (t) ( h (t) P (t)) - ν ν α P(t) α P (t) (3.4) ν ν I ou cas, >>, ad th aov quatio simpliis to

30 8 ' θ - ν ν h (t) α P(t) ( α )P (t) (3.5) ν ν I assum that th ctiv tagtial stss is th maximum picipal stss ad th ctiv adial stss is th miimum picipal stss, ad i assum sadig occus at sha ailu coditio, usig Moh-Coulom citio, ' ' θ ' ' θ siφ S cosφ o (3.6) Sustitutig (3.3) ad (3.5) ito (3.6), otai th olloig sad poductio citio siφ ( α ) ( α siφ ν ) P ν (t) (t) α h - ν Socosφ P(t) - ν siφ Sic (3.7) C o S cosφ o siφ (3.8) quatio (3.7) ca itt i a dimsiolss om, hich is ( α siφ ) siφ ( α ν P (t) ) ν Co h (t) α C o - ν ν P(t) C o - (3.9) I α is tak to o, th aov quatio simpliis to P (t) h (t) - ν P(t) (- ν) - (3.) Co Co - ν Co Sadig occus h th LHS is lss tha th HS.

31 9 Fig. 3. povids a gaphics pstatio o dimsiolss citical dado assumig Poisso s atio quals to Sigmh(t)/Co3. Sigmh(t)/Co. Sigmh(t)/Co.. P (t)/co.5 P (t)p avg (t) Citical Dado P av g (t)/c o Figu 3. Dimsiolss citical dado plot assumig Poisso s atio Poolastoplasticity solutio ad tsil ailu sadig citio Th matial coms plastic i th ailu citio is violatd. Usig thoy o plasticity, otaid th olloig solutio. I th plastic o,

32 ' m α p(,t) m d C hc m S cotφ o (3.) ' m θ αq p(,t) m d C hc q m S cotφ o (3.) h m siφ siφ (3.3) q siφ siφ (3.4) C hc Socotφ ( α m )P (t) (3.5) I th lastic o, th solutios a th sam as thos i quatios (3.) ad (3.) xcpt that th itgatio coicits a dit. S Appdix A o dtails. Ev though th matial a cavity may ail ud sha stsss, it dos ot cssaily ma sad poductio occus. It is possil that sad paticls still stay i plac ad sustai loads util th lo at is lag ough to dislodg th aild sad paticls. Wigat ad Pkis 39 poposd this sad poductio citio. I assum that th tsil stgth o th matial is gligil, th th sad poductio citio is d ' d (3.6) Sustitutig (3.) ito (3.6), yilds α p(, t) siφ siφ S cotφ ( α )P o (t) (3.7)

33 I α is tak to o, th aov quatio simpliis to th olloig p(, t) C o (3.8) Sad poductio occus h th LHS is lag tha th HS. I assum luid lo i svoi achs stady stat o psudo-stady stat, th LHS is latd to avag svoi pssu ad ottom hol loig pssu. A citical dado o sadig ost ca divd. Og t al. 4 divd such solutio o stady stat luid lo coditio cosidig o-dacy lo. Fo slightly compssil oil lo i psudo-stady stat, Equatio (3.8) coms P (t) P(t) 3 [l( ) ] (3.9) C C 4 o o Th compaiso t th to citia is sho i Fig. 3.. W s that th tsil stss iducd sadig citio pdicts high dado tha th sha ailu iducd sadig citio.

34 5 Citical P Assumig Sha Filu Iducd Sadig-Sigmh(t)/Co, Poisso atio.3 Citical P Assumig Tsil Failu Iducd Sadig-/8 P (t)p avg (t) Citical dado assumig sha ailu iducd sadig P (t)/co 5 Citical dado assumig tsil ailu iducd sadig 5 5 P avg (t)/co Figu 3. Compaiso o sha ailu iducd sadig citio ad tsil ailu iducd sadig citio 3. Aalytical modl o llo/poatio tul ailu iducd sad poductio-aisotopic i-situ stss cas I th aov divatios, assum that th xtal stsss actig aoud th llo o poatio tul a axisymmtical. I som cass this may ot tu. Fo xampl, o vtical ll, th hoiotal i-situ stsss may dit i dit dictio. Fo hoiotal ll, v i th hoiotal stss is isotopic, it is ot that th vtical stss is ot qual to th hoiotal stss. So, it is cssaily to dvlop a solutio o this cas. I th olloig divatios, assum that th i-situ stss i x

35 3 dictio is th maximum hoiotal stss H ad i y dictio is th miimum hoiotal stss h, s Fig Th poolastic solutio is otaid though suppositio o th stss causd y i-situ stss (Kisch solutio) ad th stss iducd y luid lo. Th stss solutio is Figu 3.3 Aisotopic i-situ stss aoud a ohol ' B α G (λ G)A G λ G H (t) h (t) H (t) ( ) p(, t)d h (t) ( 4 α p(, t) )cos ( θ ) (3.)

36 4 ' θ B α G (λ G)A G λ G H (t) h (t) H (t) ( ) p(, t)d h α (t) ( λ λ p(, t) G )cos ( θ ) (3.) 4 H (t) h (t) ( 3 )si ( θ ) (3.) τ θ 4 Th itgatio costats A ad B a giv i Appdix B. At th llo suac, th stsss a ' ( α )P (t) (3.3) ' θ H (α (t)(- cos ( θ )) (t)( cos ( θ )) α ν - ν )P (t) h - ν - ν P(t) (3.4) Th maximum tagtial stss is - ν ν 3 H (t) h (t) α P(t) (α )P (t) (3.5) - ν - ν ' θ Usig Moh-Coulom citio, th sha ailu coditio is siφ ν (- α ) (α ) P siφ - ν - ν So cosφ α P(t) - - ν siφ I dimsiolss om, it coms (t) 3 H (t) h (t) (3.6) (- α siφ ) siφ (α ν - P (t) ) ν Co H (t) h (t) 3 α C C o o - ν - ν P(t) C o - (3.7) I α is o, th

37 5 P (t) H (t) h (t) - ν P(t) (- ν) 3 - (3.8) Co Co Co - ν Co Sadig occus h th HS is lss tha th LHS. Similaly, otai th citical dado gaph o dit hoiotal stss aisotopy atios, s Fig To th autho s koldg, th tsil stss iducd sad poductio citio caot divd aalytically. Tho, umical mthods may dd. 3. SigmH(t)/Sigmh(t)..5 SigmH(t)/Sigmh(t).5 SigmH(t)/Sigmh(t).. P (t)/co P av g (t)/c o Figu 3.4 Dimsiolss citical dado plot o dit stss aisotopy atio- Assumig Poisso s atio.3, h (t) C o

38 6 3.3 Aalytical modl o poatio tip ailu iducd sad poductio Bsids poatio tul staility, poatio tip staility may also caus sad poductio. At poatio ad a piod o oil/gas poductio, th poatio tip may volv ad com a hmisph-lik stuctu. It is possil to study th staility o th hmisph ad us it i sad poductio. I th litatu, Batli ad iss 7 studid th staility o sad ach. A poolastoplasticity modl is povidd assumig stady stat lo coditio. Pkis ad Wigat 38 divd th sha staility ad tsil staility citia. Wigat ad Pkis 39 ad Og t al. 4 usd this modl i ild studis. Usig simila assumptios as thos listd i sctio 3., div a impovd ad mo gal stss solutio ad sadig ost citio Poolasticity solutio ad sha ailu sadig citio I th matial is i lastic stat, th stss solutio is A B 4G (3λ G) 4G α p(, t) α p(, t) d ' h (t ) (λ G) (3.9) ' A B λ G ' θ (3λ G) G α p(, t) α p(, t) d ' h (t ) (λ G) (λ G) (3.3) Th stsss at th i suac o th sph a ' ( α )P (t) (3.3) ' θ 3 h (t) v v α P(t) (α ) P (t) (3.3) v v I assum sad poductio is causd y sha ailu, usig Moh-Coulom citio, yilds

39 7 siφ ( α ) siφ ( α ν ) P ν 3 h (t) (t) α v Socosφ P(t) v siφ (3.33) I dimsiolss om, th aov quatio coms siφ ( α ) siφ ( α ν P (t) ) ν Co 3 h (t) α C o v P(t) v C o (3.34) I α is tak to o, th aov quatio simpliis to th olloig ( ν ) ν P (t) C o 3 h (t) v P(t) C v C o o (3.35) Sadig occus h th LHS is lss tha th HS. Fig. 3.5 povids a compaiso o th citical dado valus o sha ailu iducd sadig om poatio tul ad poatio tip Poolastoplasticity solutio ad tsil ailu sadig citio Th matial coms plastic i th ailu citio is violatd. Usig thoy o plasticity, otaid th olloig solutio. ' m α p(,t) m d C hs m S cotφ o (3.36) ' m θ αq p(,t) m d C hs q m S cotφ o (3.37) h th m, q a th sam as i (3.3) ad (3.4), ut th C hs valu is giv y th olloig xpssio.

40 Sigmh(t)/Co3.-Poatio Tul Sadig Sigmh(t)/Co.-Poatio Tul Sadig Sigmh(t)/Co.-Poatio Tul Sadig Sigmh(t)/Co3.-Poatio Tip Sadig Sigmh(t)/Co.-Poatio Tip Sadig Sigmh(t)/Co.-Poatio Tip Sadig P (t)/co P av g (t)/c o Figu 3.5 Compaiso o citical dado valus o sha ailu iducd sadig om poatio tul ad poatio tip C hs Socotφ ( α m )P (t) (3.38) I sad poductio dos ot occu util tsil stss is iducd at sha ailu, usig (3.6), otai th olloig sad poductio citio α p(, t) siφ siφ S cotφ ( α )P o (t) (3.39) I α is o, th

41 9 p(, t) C o (3.4) Sad poductio occus h th LHS is lag tha th HS. I assum th luid lo i svoi achs stady stat o psudo-stady stat, th LHS is latd to avag svoi pssu ad ottom hol loig pssu. A citical dado o sadig ost ca divd. Wigat ad Pkis 39 divd such solutio usig stady stat lo assumptio. Lat Og t al. 4 xtdd thi solutio y cosidig o-dacy lo. I summay, i this chapt, () W div a poolastoplastic stss modl o thick-alld hollo cylid modl o aitay luid pssu distiutio. Basd o th thick-alld hollo cylid modl, div simpl sadig ost pdictio modls assumig sha ailu iducd sadig ad tsil stss iducd sadig at sha ailu. Ths modls may usd to study sadig om op hol ll o poatio tul o casd ll. () W div a poolastoplastic stss modl o thick-alld hollo sph modl o aitay luid pssu distiutio. Basd o th thick-alld hollo sph modl, div simpl sadig ost pdictio modls assumig sha ailu iducd sadig ad tsil stss iducd sadig at sha ailu. Ths modls may usd to study sadig om poatio tip o casd ll. (3) W div a poolastic stss modl o thick-alld hollo cylid modl ud aisotopic i-situ stsss coditio. W div a sadig ost citio assumig sha ailu iducd sadig. This modl may usd to study sadig om op hol ll h th i-situ stsss a dit.

42 3 CHAPTE IV UMEICAL MODEL FO SADIG OSET PEDICTIO Th us o pla stai modls dvlopd i th pvious chapt is limitd caus o th udlyig pla stai assumptio. To dsci th stss stat mo alistically, a 3D stss modl may quid. This chapt dscis th 3D poolastoplastic modl o a lastic-pct plastic matial. 4. 3D stss modl Usig th thoy o poolasticity 5, otai th momtum quiliium quatio o a poous mdia ith luid lo as x p α τ y τ x x x xy x ' (4.) y p α x τ τ y y xy y y ' (4.) p α y τ x τ y x ' (4.3) Assumig th matial havs lia lastically o yild, x y xy y x x y xy ' v ' ' h ' y ' H ' x γ γ γ ε ε ε τ τ τ ) (,t ) (,t ) (,t D (4.4) h

43 3 ( )( ) ν ν ν ν ν ν ν ν ν ν ν ν - ν ν E D (4.5) Th latio t displacmt ad stai is did as v u x y x y y x γ γ γ ε ε ε x y xy y x (4.6) Bouday coditios may vay ith th spciic polm ig solvd. Most commoly, th displacmt at a spciic locatio may spciid, o th oc applid at a spciic locatio spciid. 4. 3D tasit luid lo modl Assumig sigl-phas lo accodig to Dacy s la, stalish a matial alac quatio 5. t p B ) C (C )) H γ p ( B µ k ( )) y H γ y p ( B µ k ( y )) x H γ x p ( B µ k ( x y x φ (4.7)

44 3 Th iitial coditio is p(x, y,, t) p i (x, y,) (4.8) t h p i is th iitial pssu dpdig o locatio. Th ouday coditios may vay ith th polm ig solvd. Gally, a pssu at a spciic locatio may spciid, o th lo at acoss a spciic suac spciid. 4.3 Moh-Coulom plasticity 4.3. Moh-Coulom citio i tms o stss ivaiats I th pvious chapt, it as assumd that th tagtial stss aoud a cavity is th maximum picipal stss ad th adial stss is th miimum picipal stss. I this chapt, povid a mo gal om o th Moh-Coulom citio so that it ca usd mo covitly i plastic domatio simulatio. I is th maximum picipal stss ad 3 is th miimum picipal stss, th Moh- Coulom ailu citio is itt as 3 3 si φ S o cosφ (4.9) W ca also xpss quatio (4.9) i tms o stss ivaiats usig th olloig quatios 53 I J si θ L π (4.) I J siθ L (4.) 3 3

45 33 I 3 J si θ L π (4.) I quatios (4.)-(4.), th stss ivaiats a did i th olloig quatios 53 I (4.3) ' x ' y ' [(' ' ) (' ' ) (' ' ) ] τ τ J x y y x xy y τ x (4.4) J3 π π acsi( ) ( θ L ) (4.5) 3 J 6 6 θ L 3/ J 3 I ' x τ xy τ x 3 I τ xy ' y τ y 3 (4.6) I τ x τ y ' 3 Sustituts (4.)-(4.) ito (4.9), it coms I siφ J (cosθ L siθ L siφ ) Socosφ 3 3 (4.7) Th ailu uctio F is did as I F siφ cos J (cosθ L siθ L siφ ) So φ (4.8) 3 3 Failu occus h F. Fo o-associatd plasticity, th plastic pottial uctio Q ca itt as

46 d34 I Q siψ J (cosθ L siθ L siψ ) Socosψ 3 3 (4.9) h ψ is calld dilatio agl Flo ul I od to div th latioship t plastic stai compot ad stss icmt, a uth assumptio o matial havio must mad. I paticula it is assumd that th plastic stai icmt is popotioal to th stss gadit o plastic pottial uctio Q, that is dq ε pd λ ' (4.) h th plastic multipli, d λ, is dtmid Elastoplastic stss-stai matix Fo lastic-pct plastic matial, th is o hadig ad th yild uctio is oly a uctio o th stss. It dos ot mov duig loadig ad uloadig, hich mas F F( ' ) (4.) By ditiatig th yild uctio, yild df T d' d' (4.) I it is assumd that th stss is iducd oly y lastic stai, th )(ddp' D ε ε (4.3) Sustitutig quatio (4.) ito th aov quatio,

47 35)(d)d(ddp' Q d λ ε D ε ε D '(4.4) Sustitutig quatio (4.4) ito quatio (4.), )(d ' Q d d df d d df λ T T ε D ' ' ' (4.5) aagig th aov quatio, yilds ' Q ' d df d ' d df d T T λ D ε D (4.6) Tho ε D D D D D ε D ε D ε D d ' Q ' d df ' d df ' Q ' Q ' Q ' d df d ' d df ' Q d T T T T λ )()(d)(dd '(4.7) Th aov quatio is simpliid as ε D pd 'd(4.8) h p D p D D D D D D ' Q ' d df ' d df ' Q T T (4.9) D p is th lastoplastic stss-stai matix. Expssio o D p ca oud Appdix i c 53. W ot that it is a uctio o th stss stat. Tho it is cssay to solv th lastoplastic polm itativly.

48 Fiit lmt solutio to th 3D stss modl Accodig to th picipl o vitual ok, th Fiit Elmt omulatio o th polm is T B D p BdV u F (4.3) Ω Dtaild omula o th aov matics a povidd i Appdix C. Usig (4.9), Equatio (4.3) coms T B (D - D p )B dv u F (4.3) Ω Sic [D p ] is a uctio o stss stat, solvig th aov quatio ivolvs itatio tchiqus. Hov, caus calculatig ad ivtig th stiss matix i Fiit Elmt Mthod is tim-cosumig, th olia pat ca movd to th ight sid duig itatio, hich sults i T B DB u F T dv B D pεdv Ω Ω (4.3) Th scod tm o th ight had sid o (4.3) acts lik a ody oc, it is updatd duig ach itatio hil th lt sid mais uchagd. This algoithm is calld iitial stss algoithm 54. Aoth ay o gatig th ody loads as poposd y Zikiic ad Comau 55. This algoithm is calld visco-plasticity algoithm, h th ody load is gatd though itatio i th olloig ay, at th i th itatio, th ody load p is p i p i Ω B T vp i ( Dε ) dv (4.33) h

49 37 ε vp Q tf ' (4.34) t 4( ν)( - ν) E( - ν si ) φ (4.35) 4.5 Fiit lmt solutio to th 3D luid lo modl Th luid lo modl ca solvd ith oth iit dic mthod ad Fiit Elmt Mthod. Fo complicatd gomtis, th iit dic mthod may diicult to apply ad th iit lmt mthod is pd. I this ok, th 3D luid lo modl is solvd y th iit lmt mthod. Th iit lmt omulatio o 3D Fluid Flo Modl is 53 K p t P p PM q (4.36) Dtaild omula o th aov matics a dscid i Appdix D. 4.6 Viicatio o th umical modls 4.6. Viicatio o th 3D luid lo modl W u sval cass i this sctio to viy th 3D luid lo modl, sho th sults o to cass. Th ist cas is sigl ll poductio at costat at om a closd axisymmtical svoi, ith ll ad svoi data pstd i Tal 4.. I th iit lmt modl, th gid usd is sho i Fig. 4.. W cosid oly ¼ o th svoi. Compaiso o th ottom hol loig pssu ith th o otaid om commcial svoi lo simulato is sho i Fig. 4.. W s good agmt t iit lmt modl simulatio ad iit dic simulatio.

50 38 Tal 4. Wll, svoi ad poductio paamts-cas Wllo adius (t).5 Daiag Aa (ac) 4 svoi Thickss (t) Gas Spciic Gavity (actio).7 svoi Tmpatu ( F) 8 Iitial svoi Pssu (psi),8 Pmaility (md) Poosity (actio). Fomatio Compssiility (/psi) -6 Iitial Poductio at (Msc/Day),

51 39 (a) Gid o ¼ o a svoi () Gid i a llo gio Figu 4. Fiit lmt gid usd i simulatio o sigl ll poductio om a closd axisymmtical svoi

52 4.E3 Eclips FEM p,d p/d(l(t)).e.e.e.e-3.e-.e-.e.e.e.e3.e4 t(hs) Figu 4. Compaiso o pssu ad pssu divativ cuvs o sigl ll poductio om a closd axisymmtical svoi Th scod cas is sigl ll poductio at cost at om a closd ctagula svoi, ith th ll ad svoi data pstd i Tal 4.. Gid usd i iit lmt simulatio is sho i Fig Compaiso o th ottom hol loig pssu ith th o otaid om commcial svoi lo simulato is sho i Fig As s om this igu, good agmt is achd t th iit lmt modl simulatio ad th iit dic simulatio. This agai viis th luid lo simulato dvlopd i this ok. Oth cass lik loig at costat ottom hol loig pssu a u ad xcllt agmt is achd t sults om commcial simulato ad th modl implmtd i this ok.

53 4 Tal 4. Wll, svoi ad poductio paamts-cas Wllo adius (t).5 Daiag Aa (ac) 4 Aspctio atio ½ svoi Thickss (t) Gas Spciic Gavity (actio).7 svoi Tmpatu ( F) 8 Iitial svoi Pssu (psi),8 X Dictio Pmaility (md) 5 Y Dictio Pmaility (md) Z Dictio Pmaility (md). Poosity (actio). Fomatio Compssiility (/psi) -6 Iitial Poductio at (Msc/Day),

54 4 (a) Gid o ¼ o a svoi () Gid i a llo gio Figu 4.3 Fiit lmt gid usd i simulatio o sigl ll poductio om a closd ctagula svoi

55 43.E3 Eclips FEM p,d p/d(l(t)).e.e.e.e-3.e-.e-.e.e.e.e3.e4 t(hs) Figu 4.4 Compaiso o pssu ad pssu divativ cuvs o sigl ll poductio om a closd ctagula svoi 4.6. Viicatio o th 3D stss modl Th tchical pomac o this 3D iit lmt pogam is chckd agaist th aalytical solutios otaid i th pvious chapt. W u sval cass i this compaiso study. To cass a pstd i this ok. Th ist cas is to compa th sults ith th os otaid usig th poolastoplastic solutio o th thick-alld hollo cylid modl ad this iit lmt cod. I this compaiso, th olloig ock mchaics data a usd i additio to th ll ad svoi paamts listd i th Tal 4., s Tal 4.3. Fig. 4.5 psts th compaiso o umical ad aalytical solutios. Fom Fig. 4.5, s xcllt agmt t umical ad aalytical solutios.

56 44 Tal 4.3 ock mchaics paamts usd i compaiso study Youg Modulus (psi).4e6 Poisso atio (actio).3 Cohsiv Stgth (psi) 5 Fictio Agl (Dg) 3 Ovud Stss (psi) 34 Hoiotal Stss (psi) 36 Picipal Ectiv Stsss Ectiv adial Stss-Aalytical Solutio Ectiv adial Stss-FE Solutio Ectiv Vtical Stss-Aalytical Solutio Ectiv Vtical Stss-FE Solutio Ectiv Tagtial Stss-Aalytical Solutio Ectiv Tagtial Stss-FE Solutio / Figu 4.5 Compaiso o 3D iit lmt solutio ith aalytical solutio o stss stat a llo duig poductio

57 45 W u th scod cas to compa th iit lmt solutio ith th xtdd Kisch solutio o aisotopic i-situ stss cas. Th data usd is th sam as th isotopic isitu stss cas xcpt that th hoiotal stss i x dictio is icasd to 459 psi (hich mas H / h.5). With ths data, compa th picipal ctiv stsss alog a spciic adial dictio, s Fig Th agmt t th iit lmt modl ad th aalytical modl is satisactoy. W also gt th picipal ctiv stss distiutio dsity plot aoud th llo, s Fig Picipal Ectiv Stss (psi) Max. Picipal Ectiv Stss-FE Solutio Max. Picipal Ectiv Stss-Aalytical Solutio Mi. Picipal Ectiv Stss-FE Solutio Mi. Picipal Ectiv Stss-Aalytical Solutio / Figu 4.6 Picipal ctiv stsss alog adial dictio o to x coodiat

58 46 (a) Maximum picipal ctiv stss aoud llo ith luid lo () Miimum picipal ctiv stss aoud llo ith luid lo Figu 4.7 Picipal ctiv stsss aoud llo o aisotopic i-situ stss cas ith luid lo.

59 Compaiso o umical modl ad aalytical modl It is oth mtioig that i sctio 4.6, th compaiso t umical ad aalytical modls is pomd to viy th umical modl. I th compaiso, sam ouday coditio ad physical gomty a usd i th to modls. Th ouday coditio usd i th compaiso is pstativ to a al cas i th cap ock ad udlig omatio hav high Youg modulus. I this situatio, oth modls giv accuat stss dsciptio. Usig oth modls ill yild idtical stss simulatio sults ad sadig pdictio sults. Hov, th lxiility o th Fiit Elmt modl lis i that it ca dal ith dit ouday coditios ad gomtis oth tha thos usd i th aalytical modl. Fo xampl, i th cap ock has simila ock mchaics poptis as th poductiv o ut th udlig omatio has sigiicatly high Youg modulus tha th poductiv o, th umical modl ca dsci th stss stat mo alistically tha th aalytical modl. I this sctio, usig th umical ad aalytical stss modls, compa th dics o stss simulatio sults ad sadig ost pdictio sults. Usig data pstd i Tal 4. ad Tal 4.3, pomd a llo stss simulatio ith th aalytical modl ad Fiit Elmt modl. Fig. 4.8 psts th ouday coditios ad gid usd i Fiit Elmt simulatio. Fig. 4.9 psts a compaiso o th stss stat at th ct o th omatio at 5 days o poductio usig th aalytical ad umical modls. W s that th mai dic t umical ad aalytical simulatio sults is th vtical stss. This is du to th pla stai assumptio usd i th aalytical modl, hich assums that th is o domatio i vtical dictio duig poductio ad lads to a lss vtical stss tha th al o. Fig. 4. psts th lativ o o th stsss otaid om th aalytical modl. W s that usig th aalytical modl o this cas causs som o o stss stat simulatio. Hov, it is oticd that, at th llo, adial stss o is oly 3%, tagtial stss o is oly.4% ad vtical stss o is %.

60 48 Figu 4.8 Bouday coditio ad gid usd i iit lmt simulatio o a llo stss 5 Stss (psi) / Ectiv adial Stss-Aalytical Modl Ectiv Vtical Stss-Aalytical Modl Ectiv Tagtial Stss-umical Modl Po pssu Ectiv Tagtial Stss-Aalytical Modl Ectiv adial Stss-umical Modl Ectiv Vtical Stss-umical Modl Figu 4.9 Compaiso o stss stat simulatio sults at 5 days o poductio usig aalytical ad umical modls

61 lativ Eo-Ectiv adial Stss lativ Eo-Ectiv Tagtial Stss lativ Eo-Ectiv Vtical Stss Eo (%) / Figu 4. lativ o o stsss usig aalytical modl I assum sadig is causd y llo sha ailu, usig th sadig modl divd om th aalytical stss modl, quatio (3.), id that sadig occus at 7 days o poductio at P psi ad P avg 54.7 psi. Usig th umical modl, id that sadig occus at 65 days o poductio at P psi ad P avg 7.5 psi. W s that, o th cas studid, sadig ost pdictio sults a ot vy dit y usig umical ad aalytical stss modls. Ev though s lag vtical stss dic i th compaiso, sic th vtical stss is th itmdiat picipal stss hich is ot tak ito accout i Moh- Coulom ailu citio, oly th tagtial stss ad adial stss ill act th sadig ost pdictio sults. Du to th act that th dics o th tagtial stss ad adial stss at llo om th umical modl ad aalytical modl a miimum, th sadig ost pdictio sults di miimally.

62 5 4.8 Uctaity assssmt o sadig ost pdictio modl Mchaistic modlig o physical systms is ot complicatd y th psc o uctaitis. This also applis to sadig ost pdictio modlig. Ev though sigiicat ot may dd to icopoat uctaitis ito th modlig pocss, this could pottially sult i povidig usul iomatio that ca hlp sad cotol dcisio-makig. A systmatic uctaity aalysis povids isight ito th lvl o coidc i modl stimats, ad ca hlp assssig ho vaious possil modl stimats should ighd. Futh, it ca lad to th idtiicatio o th ky soucs o uctaity, as ll as th soucs o uctaity that a ot impotat ith spct to a giv spos. Th olloig stags a ivolvd i th uctaity aalysis o a modl: (a) stimatio o uctaitis i modl iputs ad paamt (i.. chaactiatio o iput uctaitis), () stimatio o th uctaity i modl outputs sultig om th uctaity i modl iputs ad modl paamts (i.. uctaity popagatio), (c) chaactiatio o uctaitis associatd ith dit modl stuctus ad modl omulatios (i.. chaactiatio o modl uctaity), ad (d) chaactiatio o th uctaitis i modl pdictios sultig om uctaitis i th valuatio data. Vaious mthods a availal o uctaitis aalysis. Amog hich, Mot Calo mthods a th most idly usd os. Ths mthods ivolv adom samplig om th distiutio o iputs ad succssiv modl us util a statistically sigiicat distiutio o outputs is otaid. Thy ca usd to solv polms ith physical poailistic stuctus, such as uctaity popagatio i modls o solutio o stochastic quatios, o ca usd to solv o-poailistic polms. Commcial dvlopd y Palisad Copoatio (.palisad.com) is o o th availal tools to pom uctaitis aalysis. I this sctio, study ho iput data uctaitis popagat i th sadig ost pdictio modl ad act modl pdictio uctaity. W igo modl uctaitis