Nonlinear Poroelastoplastic Behavior of Geothermal Rocks

Transcription

1 Procding World Gothrmal Congr 15 Mlbourn, Autralia, 19-5 April 15 Nonlinar Porolatoplatic havior o Gothrmal Rock Mario-Céar Suárz A., Frnando Samanigo V., Joé-Eduardo Ramírz L.M., Omar A. Vicncio F. and Máximo E. Frnándz M. UMSNH - Ediicio, Cd. Univritaria, 586 Morlia, Mich., México mca5@gmail.com, pxamanigov1@pmx.com, jdramirzlm@gmail.com, maximo.rnandz@giz.d Kyword: Gomchanic, platic rock, porolatoplaticity, porolaticity, latic poroity, platic poroity. ASTRACT At prnt tim, advancd gothrmal rrvoir nginring rquir on to quantiy and prdict non-iothrmal multipha luid low within dormabl porou rock. Frquntly, th gomchanical bhavior o th rock i nonlinar and go into th platic rang. Thi non-linarity compri inlaticity and larg, prmannt dormation. In ordr to undrtand th tru tr-train bhavior o gothrmal rock, thrmoporolato-platicity pciic tchniqu ar ndd. I th rock wr only porolatic thr would b no numrical limit or th valu o th principal componnt o th tr tnor. In uch idalizd thortical ca, gothrmal rock will nvr ail. In thi work, an initial invtigation o th gnral practical apct o th outlind problm i prntd. To idntiy th proc involvd, th low platicity thory i introducd and mathmatical numrical modl ar dvlopd and olvd uing init dirnc and init lmnt. oth modl ar abl to analyz th gomchanical dormation o gothrmal rock ubjctd to a nonlinar bhavior. Th irt modl i radial and includ th luid low, which oby Darcy law, and a porolatic analyi o th rock. Th cond modl i 3D bad on continuum mchanic and low platicity thory with porolatoplatic dormation. In thi ca, th total tr that can b applid to gothrmal rock i phyically limitd by a ailur tr critrion. An xtndd Druckr-Pragr ailur-yilding critrion i ud to rprnt thi limit ralitically. Uing availabl ild data prviouly publihd, th 3D modl wa applid to comput th porolatoplatic dormation o a alt dom locatd in a hybrid oil-thrmal rrvoir locatd in th outhrn part o th Gul o Mxico, which i rlatd to a dp gothrmal aquir producing hot brin at 16 C and 184 bar that invad th oil producing wll. Th modl how that th vrtical dormation o th alt dom i not ngligibl in having an inlunc in th global rrvoir draw-down prur. 1. INTRODUCTION World litratur on gomchanic dcrib land ubidnc in aquir, ptrolum and ga ild, a wll a gothrmal rrvoir caud by thir xploitation (Wang, ; undchuh & Suárz, 1). Thi phnomnon i a dirct conqunc o prmannt and irrvribl rock dormation. In Enhancd Gothrmal Sytm (EGS), artiicial timulation i applid to dorm th rock and to incra poroity and prmability. Porolaticity tudi only th bhavior o porou latic rock containing vicou luid uch a watr, brin, ga and oil. A porolatic rock i charactrizd by it ctiv poroity, it latic moduli and by th phyical proprti o th luid that it contain. Th porolatic rock dormation can b linar or non-linar, iothrmal or non-iothrmal. On th othr hand, any luid containd in a porou rock rduc it trngth. Th cohiv tructur o a rock i waknd by th prnc o liquid. All gomchanical paramtr ar inluncd by thi cohion and dirctly actd by th prur and amount o liquid prnt in both por and ractur. Thi i th por-ractur-watr ct. Rock compaction, racturing, tim dpndnt dormation, a wll a crp and ubidnc mchanim ar ntially producd by volcanic and tctonic activiti, by lithotatic prur and by luid xtraction/injction. In gothrmal aturatd rock, dnity and wav propagation pd ar incrad, whil trngth i rducd. Th main hypothi in linar porolaticity i that th luid low through a dormabl porou rock who olid klton can b dormd latically. Auming that rock ar only ubjctd to mall dormation, Hook law can b applid to rlat train and tr. Fluid xtraction in gothrmal rrvoir cau th rduction o th intrnal por-ractur prur and th ctiv aprtur o por and iur. Many naturally racturd ytm xprincd intn tctonic activity in thir rmot pat, and thir original racturing wa qually intn. Howvr, om o th ytm contain iurd zon whr many ractur appar clod, a in th Lo Humro, México gothrmal ild (Suarz, 1998). Thi phnomnon wa du to th act that th rock dormation wa prmannt, irrvribl and porolatoplatic. Th poroplaticity o gothrmal rock, can xhibit ithr platic dilation producd by poitiv tnional tr, or platic contraction producd by ngativ comprional tr. oth proc ar mchanical and thrmodynamically irrvribl, yilding prmannt platic dormation that could rduc th rrvoir torag capacity. Thr ar othr important go-thrmo-mchanical ct in gothrmal and hydrocarbon rrvoir. High prur and tmpratur incra ductility and lowr th yild point o th rock. Thror, high conining prur and tmpratur ct induc platic low, producing dormation byond th limit o latic train (undchuh & Suárz, 1). Th prnt low platicity thory (Couy, 4; Chin Wu, 5; d Souza t al, 8) aum that a low rul xit that can b ud to comput th platic rock dormation. It i alo aumd that th total train in gothrmal rock can b dcompod into two part, on o thm latic or rvribl and th othr on platic or irrvribl. Thi dcompoition can b additiv or multiplicativ; both ar uul in gothrmal rrvoir nginring. Th additiv dcompoition i ud or mall platic dormation with Hook Law applid to th latic portion, th total train ε i qual to th latic train ε plu th platic train ε p, (ε = ε + ε p ). Th multiplicativ dcompoition can b ud or larg platic dormation, auming that th dormation gradint tnor F i th product o an latic tnor and a platic tnor, or F = F F p. To dtrmin th platic portion o th total train, a low rul, a yild critrion and a hardning modl ar rquird in both dcompoition (L, 1969; Couy, 4; Souza t al, 8; Anandarajah, 1). 1

2 I th proc i non-iothrmal, an xtra thrmal train tnor hould b conidrd and th total train bcom ε = ε + ε p + ε T. Th platic low rul din th volution o th platic train; th hardning law charactriz th volution o th yild limit according to th yild critrion. Th main objctiv o thi papr i to introduc th important apct o platicity thory applid to a coupl o iothrmal porolatoplatic xampl o dormabl rrvoir.. ELASTIC AND PLASTIC POROSITY. STRAINS AND STRESSES In ordr to introduc in a logical way th main variabl o porolatoplaticity, w din irt th dirntial rlationhip btwn poroity and rock volum. Th tructural volum V or bulk volum i th global volum occupid by th rock, with it olid grain, por and ractur. Th dirntial rlationhip btwn th thr volum and th poroity, ar: dv S V = S S 1 dv V + V d V = d V + d V = + S (1) d V d V whr V, V, V S, φ and φ S ar bulk, por and olid volum, raction o th por or poroity and raction o th olid grain, rpctivly. W aum that th por ar all intrconnctd and compltly aturatd with luid, and thror th por volum i qual to th luid volum V. Porolatoplaticity produc irrvribl and prmannt chang in poroity and in th luid ma contnt inid th rock, which i dind a: m M M V M kg 3 V V V V m () whr m, M, ρ ar luid ma contnt, luid ma, and luid dnity, rpctivly. Th two main variabl that charactriz poroplaticity ar th platic poroity φ p and th platic train ε pij. Th undamntal poroplatic thory, aum that platic dormation occur intantanouly in rpon to incrmnt or dcrmnt o om pciic tr σ ij and luid prur p (Couy, 4). Auming that th train dcompoition i additiv, th corrponding dirntial rlationhip btwn th variabl ar: d d d, d d d, i j i j p i j p i j i j p i j p 1 u u i j ui whr: i j, 11 33, ii ; i, j 1,, 3 x j x i xi (3) Whr ε ij, ε ij, ε pij,, φ, φ p, φ ar total, latic and platic train, latic, platic and initial poroiti, rpctivly; u = (u 1, u, u 3 ) i th vctor diplacmnt o th olid particl in a cartian rrnc bai { i, i = 1,, 3} and ε i th volumtric train. Th variabl ε ij and φ ar latic or rvribl, whil ε pij and φ p ar platic and irrvribl variabl in th porolatoplatic non-linar proc. Th platic poroity hould b intrprtd a th irrvribl chang o th porou volum..1 Thrmoporolatic Equation or Hookan Rock In thi papr w ar introducing only iothrmal xampl bad on th iothrmal platicity low thory. Th thrmoporolatic thory can b tratd paratly bcau o th additiv dcompoition o th total train: (ε = ε - ε p ). A ull tratmnt o thrmoporolaticity can b ound in Couy (4) and, or gothrmal proc, in undchuh and Suárz, (1). Th quation o linar non-iothrmal porou rock rlating tr and dormation ar ormd by two part; on or th klton: i j = b p p K T T i j Gi j ; i, j 1,, 3 (4) And anothr on or th luid inid th por and ractur (undchuh & Suárz, 1): p p p p M C M m T T b m T T (5) M whr σ ij, λ, b, K,, T, G, ar tr, Lamé modulu, iot-willi coicint, bulk modulu, bulk thrmal xpanivity, tmpratur and har modulu, rpctivly. Th trm p i th initial prur, and T i th initial tmpratur; ij i th unit tnor. In quation (5) M, C, ζ, m, ar th two iot moduli, th variation o th luid contnt, and th thrmal xpanion o th luid ma contnt, rpctivly. Th right part o quation (5) i th oprational dinition o ζ, th main porolatic variabl in th linar thory o iot (197). Th mathmatical dinition o all th coicint ar a ollow: K 1 1 V 1 m b 1,, C b M,, m K M p V T p m T S k p (6) whr K S i th olid bulk modulu and p k i th conining lithotatic prur. Th coicint M i th invr o th contraind pciic torag, maurd at contant volumtric train; thi paramtr charactriz th latic proprti o th luid bcau it maur how th luid prur chang whn ζ chang. Th coicint C rprnt th coupling o dormation btwn th olid grain and th luid. Th thr paramtr b, M and C ar th xprimntal cor o th porolatic quation.

3 3. THE ELASTOPLASTIC EQUATIONS. A FLOW RULE AND A HARDENING MODEL Uing ull tnor notation or th train, th gnral, additiv latoplatic modl i: p p p t t t, and t Suárz, Samanigo, Ramírz and Vicncio ε ε ε ε ε ε ε ε ε ε ε (7) Th point abov th train man tim drivativ and th lat trm in quation (7) i an initial condition or th total train. It i aumd that th latic train tnor ε i computd with Hook' law and thror th only unknown i th platic train tnor ε p ; it calculation olv th latoplatic problm. Th non-iothrmal latic part can b olvd a indicatd in ubction.1. Lt ψ b th r nrgy potntial, which i aumd to b a unction o th train and o a t o intrnal variabl = { i, i = 1, n} uch a th platic poroity; th i ar calld th hardning variabl. It i alo aumd that thi potntial can b dcompod into th addition o an latic part ψ and a platic part ψ p (d Souza t al., 8):, p, = p ε ε α ε α (8) Th platic part ψ p dcrib th hardning (or otning) o th rock. W aum in thi dcription that th gothrmal rock ar linar and iotropic. Uing thi dcompoition o th r nrgy potntial, th gnral latic law to comput th tr tnor σ and th hardning thrmodynamical orc τ ar: p σ= ; τ= ε α (9) 3.1 Th Platic Domain Th platic low occur whn th tr attain a critical valu. In th gnral ca, thi xprimntal principl can b rprntd by a yild unction or low potntial Y (σ, τ), which can b poitiv, ngativ or zro. Th poibl valu din thr dirnt domain: Y, Y, Y, σ σ τ, σ σ τ, σ σ τ (1) E P Y Ω E i th latic domain o tr or which platic yilding do not occur, Ω P i th domain o platically admiibl tr and Ω Y i th domain o tr or which platic yilding can occur. Th t dind by Y (σ, τ) > ha no phyical maning and it i aid that thi proc i impoibl to occur thrmodynamically. 3. Platic low rul or rock Th nxt condition i th contruction o a platic low rul that din th quation and volution law or th intrnal variabl, which ar aociatd with th diipativ unction o th proc. Th intrnal variabl ar th platic train and th t. Th platic low rul potulatd by d Souza (t al., 8) i dind a ollow: ε = N σ, τ p (11) Th matrix N i calld th low tnor and i an unknown platic multiplir dind in th nxt ction. 3.3 Th Hardning Modl Th hardning modl i (Couy, 4; d Souza t al., 8): τ= H σ, τ (1) Th matrix H i calld th gnralizd hardning modulu which din th volution o th hardning variabl i. Equation (11) and (1) ar volution law that rquir loading/unloading critrion or condition that dtrmin whn th volution o platic train and intrnal variabl may occur. Th condition ar: Y σ, τ,, Y σ, τ (13) Undr platic yilding, a complmntary quation dducd rom (13) impli th ollowing conitncy condition: Y σ, τ Y σ, τ (14) Th matric N and H can b computd rom th low potntial Y (σ, τ): Y σ,, Hσ, τ N σ τ Y τ (15) 3

4 Th analytical ormula to comput th platic multiplir can b obtaind rom dirntial calculu and algbra o tnor applid to prviou quation (or dtail d Souza t al., 8): ND : : ε p N: D : N H H α (16) Whr matrix D i th iotropic laticity tnor dind by th ymmtric part o th gradint o th olid low vlocity v. Th ymbol : rprnt th innr product btwn two tnor o th am ordr: 1 1 T v v i j ND : : ε ni j d i j i j, N: D : N ni j d i j ni j, D v v i j (17) xj x i Th quation hrin introducd corrpond to th gnral rat-indpndnt platicity modl. 3.4 Th Druckr-Pragr Yild Critrion Th Druckr-Pragr (195; rrnc in Couy, 4) yild critrion tat that (d Souza t al., 8): th platic yilding bgin whn th cond invariant I o th dviatoric tr and th hydrotatic prur p, rach a critical combination. Th cond invariant i a unction o th tr dviator : : 1 I, σ pi, p i j i j i j i j i j (18) Th latoplatic numrical imulation dcribd latr in ction 5, adopt th modiid Druckr-Pragr yilding critrion, which includ platicity and platic damag (Shn t al., 1). Th platic potntial Y DP in th Druckr-Pragr orm i: Y t g I p t g (19) DP t Whr ξ i a paramtr that din th ccntricity o th loading urac in th ctiv tr pac; σ t i th thrhold valu o th tnil tr at which platic low initiat; i th dilatancy angl; I i th cond invariant at th compriv mridian, and p i prur. Th modiid Druckr-Pragr yild/ailur critrion i a gnralization o th claical Mohr-Coulomb critrion; it i uul to dcrib th tr-train bhavior o prur dpndnt matrial uch a porou rock. Th ollowing crp law i adoptd: cr n m t A () cr Whr & cr rprnt quivalnt crp train rat; cr rprnt Von Mi quivalnt tr; t i th total tim variabl; A, n, m ar pciic modl paramtr dcribd in ction FIRST EXAMPLE: A GENERAL POROELASTIC ISOTHERMAL COUPLED MODEL Chin, Raghavan and Thoma (), dvlopd an iothrmal coupld modl to analyz oil wll with tr dpndnt prmability. In thi ction w dduc and gnraliz thir modl rom th gnral thory introducd in prviou ction; w alo add a ormula to comput th poroity a a unction o por prur and th latic dormation or mall train. Thi modl i ormd by th ollowing calar and tnor quation: D 4.1) luid ma Dt v D S S 4.) S S S rock ma Dt v K 4.3) Darcy' law rrrd to th rock v vs p g 4.4) σ = I G ε Hook' law or th porou rock u 4.5) Nwton t nd σ p δ ' law or rock dynamic (1) Whr D ( ) / Dt ( ) / t ( ) v rprnt th matrial or total drivativ o ( ), ρ, ρ, v, v, K,, g ar luid and olid dniti and vlociti, abolut prmability tnor (k ij i j ), luid vicoity, and gravity acclration, rpctivly. 4

5 Auming that th olid rock dnity ρ i contant, and that th low vlocity o th olid grain v i ngligibl or much mallr than th poroity chang, quation (4.) can b impliid. Thi i alo th raon why quation in (4.5) i qual to zro: D 1 1 D v v v Dt t t 1 t Dt t () Subcript man olid grain proprti. Not that th hypothi o ngligibl v i quivalnt to aum that th total drivativ D/Dt i th am a th traditional partial drivativ. Combining th luid ma quation (4.1) with Darcy Law (4.3): K K v v p g v v p g K 1 K 1 1 p g p g v t (3) Introducing th iothrmal luid compribility C, and auming that luid dnity i only a unction o prur: 1 p C ; p p p ; p p t p t (4) Combining both quation (3) and (4): K 1 K p g p p g p v v v p p 1 p p p C p v v v C C v p t t t t t t (5) Rplacing thi quation into quation (3) w obtain a practical, coupld porolatic gnral modl: K K D D p p g C p p g C v D t D t D D D p 1 D D p D p C C v C D t 1 D t D t 1 D t D t D t (6) Introducing th dinition o th iothrmal por compribility C p : p p Cp v Cp p 1 p 1 p 1 p t t (7) Thror, th modl givn in quation (6) can b writtn in trm o both compribiliti: K K p p g C p p g p Cp C t (8) I th por compribility C p i aumd contant, thn quation (7) can b intgratd to obtain a uul ormula or φ (p): p 1 d 1 Cp Cp d p Ln Ln C p p p 1 p 1 1 p 1 Cp p p Cp p p p Cp p p C p p p thror: (9) 5

6 φ i any rrnc poroity corrponding to a luid prur p. 4.1 A coupld porolatic non-linar radial modl W conidr a radial gomtry a occur in a lowing wll in an iotropic porou mdium. Th mathmatical modl (1) can b imply writtn in trm o th radiu r i all variabl and unction ar aumd to b unction o th radial coordinat and tim. k r p 1 p p p CT Cp C C CT p r r r r t (3) Whr p = p (r, t). Modl (3) i a non-linar partial dirntial quation and th total compribility C T, th luid vicoity μ and th rock prmability k r could b conidrd contant or dpndnt on prur. For xampl, i w aum that th prmability dpnd only on prur, th modl bcom: k p p k p p p r C CT p r r r r t 1 r r (31) Th prmability a a unction o por prur can b approximatd uing th mpirical Paron ormula (1976): p k p (3) r Th numrical valu o th coicint and xponnt could b xprimntally adaptd to dirnt typ o rock. In radial coordinat th divrgnc o th tr in quation (4.5) σ r (r, t) i a linar dirntial quation in trm o r and can b partially intgratd a ollow: r r p r,t r r r r p pr,t r r, t pr,t E r r,t w w w r r r r r (33) Whr E i th Young latic modulu o th rock in th radial dirction, r w i th wllbor radiu and σ w i th corrponding tr at th am it. Thi olution i valid or ach ixd tim t, and quation (31) mut b numrically olvd irt to obtain p (r, t ). Th latic train ε r (r, t ) can thn b computd dirctly uing Hook law, a hown in th am olution (33). In quation (), th divrgnc o th rock low vlocity v r (r, t) i a linar dirntial quation o th radiu r and can b partially intgratd a in quation (33): 1 v v r,t v r v v r,t r r r r,t r r r w w r r r,t r w r r r r t r r (34) Thi olution i only valid or ach ixd tim t = t, and th latic train ε r (r, t ) mut b computd irt rom quation (33) to obtain a ixd valu in tim. ε w i th latic train at th wllbor radiu. Thi train could alo includ th platic part o th wllbor, it latoplatic dormation and it poibl collap. From Darcy law (4.3), w can comput th luid vlocity uing quation (34) and th numrical olution o (31): v r,t v r,t r r p k p p r,t r (35) Thi olution i only valid or ach ixd tim t = t, quation (31) and (34) mut b olvd irt to obtain p (r, t ) and v r (r, t ). Th radial diplacmnt u r (r, t) can b intgratd uing quation (33): r r u r,t r r r r dr pr,t r r r r,t r w r w w w w w w r r E r E E r E r rw rw r,t p u r,t u d r u r,t u Log p r Log rw w r r,t r r w r r w w E rw E rw (36) Whr Log rprnt th natural logarithm and u w i th radial diplacmnt at th wllbor that can b null or not, dpnding on th purpo o th modl. 6

7 Th ordr o th olution prntd in quation (1) to (36), contitut a practical algorithm to olv th coupld porolatoplatic problm hrin introducd. Th inal olution itl i practical and uul in gothrmal rrvoir nginring oundary and initial condition or th radial modl Th boundary and initial condition o th modl prviouly dvlopd ar a ollow: 1. Initial radial diplacmnt u r (r, ) =. Wllbor diplacmnt u r (r r w, t) = ( i wll dormation i conidrd). 3. Extrnal boundary diplacmnt u r (r r E, t) = ( i dormation o thi boundary i conidrd). 4. Vlociti o th luid and olid grain at r = r E, v = v r = i u r = 5. Initial luid prur in th rrvoir p (r, ) = p i 6. p r w,t q Intrnal boundary at th wll r k r rwh 7. Extrnal boundary o th rrvoir p r,t E r 4. Ect o rock porolaticity on prur and radial dormation in an oil rrvoir Th modl aum a homognou and iotropic rrvoir in it initial tat, with variabl poroity and prmability du to rrvoir radial dormation. W conidr contant oil rat production rom on wll, locatd at th cntr o a radial rrvoir with uniorm thickn, uniorm initial proprti, and ixd outr and innr boundari. W din a porolatic coicint ξ a: b G (37) whr b i th iot-willi coicint, λ i th draind Lamé coicint and G i th draind har modulu. For tr nitiv rock, th olid bulk modulu i largr than th bulk modulu. Th luid do not produc har tr th draind har modulu i th am a th undraind har modulu. Th draind Lamé coicint i dirctly proportional to th bulk modulu, i th bulk modulu i dcrad th porolatic rock will b mor prur dpndnt (incraing it compribility) and mor diplacmnt/ xpanion would b xpctd. Thror or tr nitiv rrvoir th largr th porolatic coicint o a rock, mor diplacmnt i xpctd a provd by th ollowing igur. Uing th rrvoir proprti o Tabl 1, thr computr run wr mad uing th porolatic coicint valu hown in Tabl or th rrvoir rock. Th rult corrpond to 15 day o oil production. Tabl 1 Numrical valu o Rrvoir Proprti Simulator Rrvoir Proprti Rrvoir nod Initial prmability (m ) Wllbor radiu (m).9 Initial poroity.3 Extrnal radiu (m) 15 Porolatic Coicint (Pa -1 ) Initial Prur (MPa) 5 Rat o oil pr day (m 3 /d) 45 Oil Proprti Rrvoir Thickn 18 Vicoity (Pa ) Firt Timtp (cond) 1 Compribility(Pa -1 ) Producing tim (day) 15 Formation Volum Factor 1.5 Tabl. Porolatic coicint ξ (Pa -1 ) lu Lin Rd Lin Grn Lin Figur 1 and illutrat th ct that thi porolatic coicint ha on th prur proil and on th rock radial diplacmnt. 7

8 Figur 1: Ect o th porolatic coicint on prur proil p (r, t) at a ixd tim. Figur : Ect o th porolatic coicint on th radial diplacmnt u r (r, t) at a ixd tim Ect o th porolatic proprti o th al rock on th prur maintnanc and radial diplacmnt Uing th proprti o tabl 1 or th rrvoir rock and tabl or th al rock, thr run wr mad. Figur 3 and 4 how th obtaind rult. Figur 3: Ect o th al porolatic coicint on th prur proil. 8

9 Figur 4: Ect o th al porolatic coicint on radial diplacmnt. 4.. Dicuion o porolatic rult A can b n rom igur 1,, 3 and 4, i th porolatic coicint ξ incra, th radial diplacmnt incra and th prur drop dcra. For th ca whr th rrvoir boundari ar ixd, it can b n that at th outr boundary th prur i largr and th radial diplacmnt i lowr. Thi can b xplaind with th aid o igur : bcau th boundari o th rrvoir rock ar ixd, thr i no compaction o th rock bulk volum, but thr i a rock movmnt within th rrvoir, it go mor compactd nar th wllbor, rducing th poroity and prmability, but at th outr boundary th rock i xpanding, and poroity and prmability ar incrad, thi cau th prur to b largr at th outr boundary. For th ca whr th al rock i r to mov, th prur maintnanc obviouly incra compard to th ca whr th rrvoir outr boundary i ixd. Th prur proil ar dirnt, but nar th wllbor th prur i almot th am in th thr ca, thy chang whn gtting clor to th outr boundary; i th porolatic coicint i largr, th prur maintnanc i improvd and thr i a largr radial diplacmnt. 5. SECOND EXAMPLE: APPLICATION OF THE POROELASTOPLASTIC MODEL IN 3D 5.1 Dormation o a alt dom in an oil rrvoir rlatd to a gothrmal aquir Th abnormally high concntration o diolvd alt and minral ound in ampl rom activ wll in an oil rrvoir, locatd in th outhrn part o th Gul o Mxico at 6 m dpth, indicat th prnc o olid alt dom undrlying th oil rrvoir. Th dom ar inluncing th chmical compoition o both oil and brin accompanying production. Th xtraction o hydrocarbon at thi it i rlatd to typical gothrmal phnomna coupld with th activ gomchanic o th alt dom. Ma migration occur undr non-iothrmal condition at dirnt tmpratur (up to 16 C) and at vry high prur (ovr 184 bar). Th gological ytm o intrt i locatd in 1 m watr dpth. Rrvoir ormation location i at 5 m blow a lvl, location and approximatd gomtry o th alt dom i hown in Figur 5. Th total dpth o th modl i 7 m, th width i 8 m, and th lngth i 8 m. A long a thi xampl i only mad or computational purpo, it i not ncary to includ a gological map o th ral it. Intrtd radr can ound a dtaild dcription o thi problm in a rcntly publihd papr (Suárz, Samanigo & Shn, 14). Figur 5: Poition o th rrvoir and th alt dom within th gological modl. To imulat th alt dom crp dormation caud by oil and brin xtraction, w ud th 3D modl dcribd by Shn, ai & Standiird (1). Th modl includ a vico-lato-platic dormation analyi and th porou luid low rlatd to prur 9

10 dpltion. Th modl wa olvd uing th init lmnt mthod. Th numrical imulation adopt th modiid Druckr-Pragr yilding critrion, which includ platicity and platic damag that wa prviouly dicud in ction 3. In th crp law (Eq. ) cr A( cr ) n t m, th paramtr ar givn by th ollowing valu: A = ; n =.667; m = -.. For th rock ormation, th cohiv trngth and rictional angl o th Druckr-Pragr modl (Shn at al., 1) ar givn by th ollowing valu: d = 1.56 MPa, β = 44, which corrpond to valu in th Mohr-Coulomb modl a c =.5 MPa, φ = 5. Th valu o th trngth paramtr or alt, adoptd hr by th modiid Druckr-Pragr modl, ar: d = 4 MPa, β = 44, which corrpond to valu in th Mohr-Coulomb modl a c = 1.5 MPa, φ = 5. A impliid modl and our kind o matrial hav bn adoptd, including th uppr ormation, th lowr ormation, th ormation urrounding th alt, and th alt body. Th corrponding paramtr ar litd in Tabl 3. Tabl 3. Numrical valu o matrial paramtr Rock tting Dnity Young co. Poion co. (Kg/m 3 ) (Pacal) (ad) Top layr 1,, ottom layr Rrvoir Salt ormation To kp th coupld bhavior proprty o th modl to hav th lat computational burdn, coupld analyi or dormation and porou low wa prormd or only thi rrvoir ormation and th lowr rgion. Othr part o th modl ar aumd to b nonprmabl. Por prur variation in th rrvoir du to oil xtraction go rom it original valu o 8 MPa down to 7 MPa, which i a rgular valu o prur drawdown. Load applid to th modl at ild cal includ: 1) awatr prur and ) l-gravity o ormation and alt, which i balancd with th initial gotr. Mud-wight prur will not appar in th ild cal modl. Zro-diplacmnt contraint ar applid to th our latral id and th bottom (Fig. 5). Oil production includd in th imulation covr a priod btwn January 199 (5 bpd = barrl pr day), July 4 (8, bpd) and Dcmbr 1 (15, bpd), ditributd among 19 producing wll. 5. Numrical rult o th alt dom dormation and o th por prur Ditribution o ubidnc caud by oil xtraction i hown in Figur 6. A multi-cut viw i ud in viualization o th vrtical diplacmnt/ubidnc U3. It i n that th maximum ubidnc i.348 m, which occur at th top o th rrvoir. Figur 6: Contour o th vrtical diplacmnt/ubidnc (U3) o th alt dom du to oil production. 1

11 Ditribution o por prur corrponding to oil production i hown in Figur 7. A multi-cut viw i ud in viualization o th vrtical diplacmnt/ubidnc U3. Prur drawdown i limitd to th rrvoir rgion dind in Figur 5. Valu o por prur ar t unchangd or rgion byond th rrvoir by tting vry low prmability. Figur 7: Contour o th por prur variation du to oil xtraction btwn 199 and CONCLUSIONS In thi rarch papr w prntd th gnral problm o latoplaticity o gothrmal rock. A bri dcription o th gnral low platic thory wa introducd and two xampl o application wr dvlopd and olvd. Th irt xampl covr a gnral porolatic modl and a impliid particular ca whn th ytm ha radial gomtry a occur in a lowing wll in an iotropic porou mdium. Numrical rult wr graphically prntd, howing th ct that porolaticity ha on th prur proil and on th rock radial diplacmnt. Th rult can b xtndd to th numrical tudy o wll collap. Th cond xampl olv a thr-dimnional porolatoplatic problm, corrponding to th prmannt dormation o a alt dom locatd in th Gul o Mxico, which i rlatd to an oil rrvoir undr xploitation. In thi ca, hot brin invaion occur that involv typical gothrmal phnomna coupld to th xtraction o oil rom th rrvoir rlatd to th alt dom. Th olid alt body xprinc a continuou vico-lato-platic dormation originatd by th oil xtraction and brin. Th gomchanic o thi dormation wa modld uing th modiid Druckr-Pragr yild/ailur critrion and init lmnt. Th maximum ubidnc calculatd i.348 m, which occur at th top o th rrvoir. Thi dormation act th por prur proil and th prur drawdown producd by th oil and brin xtraction. REFERENCES Anandarajah, A.: Computational Mthod in Elaticity and Platicity, Springr, (1), 633 pp. iot, M.A.: Thory o init dormation o porou olid. Indiana Univ. Math. J. 1:7, 597 6, (197). undchuh, J. and Suárz-Arriaga, M.C. Introduction to th Numrical Modling o Groundwatr and Gothrmal Sytm - Fundamntal o Ma, Enrgy and Solut Tranport in Porolatic Rock. Taylor & Franci Group, (1), 479 pp. Chin, L.Y., Raghavan, R., and Thoma, L.K.: Fully Coupld Gomchanic and Fluid-Flow Analyi o Wll with Str - Dpndnt Prmability, SPE Journal, 5:1, (), Couy, O.: Poromchanic, Ed. John Wily & Son, Ltd (4). D Souza, E.A.N., Pric, D. and Own, D.R.J.: Computational Mthod or Platicity, Wily, (8), 791 pp. L, E. H.: Elatic-Platic Dormation at Finit Strain.,Journal o Applid Mchanic 36:1, (1969), 1-6. Shn, X., ai, M. and Standiird, W.: Drilling and Compltion in Ptrolum Enginring Thory and numrical application. Volum 3, Multiphyic Modling ri, Taylor & Franci Group, (1), 33 pp. Suárz, M.C., Samanigo, V.F., and Shn, X.: Salt dom dormation coupld to th low o gothrmal brin and oil, 14IACMAG Procding, 14th Intrnational Conrnc o th Intrnational Aociation or Computr Mthod and Advanc in Gomchanic, Kyoto, Japan, Sptmbr -5, (14). 11