WELL DELIVERABILITY PREDICTIONS OF GAS FLOW IN GAS-CONDENSATE RESERVOIRS, MODELLING NEAR-CRITICAL WELLBORE PROBLEM OF TWO PHASE FLOW IN 1 -DIMENSION

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1 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN WELL ELIVERABILITY PREICTIONS OF GAS FLOW IN GAS-CONENSATE RESERVOIRS, MOELLING NEAR-CRITICAL WELLBORE PROBLEM OF TWO PHASE FLOW IN -IMENSION a Ordu, O.. ; a Ak, C. T.; a Makinde, F. A.; a Owarume, M. O. a Cvenan Universiy, eparmen f Perleum Enineerin, Oun Sae, Nieria ABSTRACT Prducin f ases frm as-cndensae reservirs are knwn bear cerain challenes larely due he frmain f rerrade cndensaes ha hinder as flw. The drp u f his liquid creaes flw reins ha are characerized by he liquid saurain as i affecs he mbiliy f he w phase flw, hereby prevenin he effecive mdelin f well prduciviy. In his sudy, a predicive mdel based n an analyical apprach is develped predic as flw in as cndensae reservirs. This sudy cmpares he esimaed as flw frm he develped mdel fr as-cndensae reservirs he flw f an exisin mdel fr as reservirs. This sudy bserves he effecs f liquid drp-u n prduciviy a lw pressures and he cndensae unladin pressure, which is cmparable ha f cmmercial sfware. KEYWORS well deliverabiliy mdelin; wellbre prblems; w- phase flw; -imensin T whm all crrespndence shuld be addressed. Address: eparmen f Perleum Enineerin, Schl f Enineerin, Cllee f Science & Technly, Cvenan Universiy, Km. 0 Idirk Rad, Oa, Oun Sae, Nieria Telephne: +34 (0) david.rdu@cvenanuniversiy.edu.n, preye.d.rdu@mail.cm di:0.549/bjp

2 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN INTROUCTION Reservirs bearin as cndensaes are becmin mre cmmn as develpmens are encunerin reaer dephs and hiher pressures and emperaures. Accuracy in enineerin cmpuains fr as-cndensae sysems such as precisin in well esin, in esimain reserves, in sizin surface faciliies, and in predicin prduciviy rends depends upn basic undersandin f phase and flw relainships. When cmparin dry as reservirs wih ascndensae reservirs, ne can bserve he exisence f many special facrs ha affec he perfrmance f as-cndensae reservirs durin he expliain prcess. A he ime f discvery, a ypical ascndensae reservir pressure mih be abve r clse he criical pressure. A his ime, here exiss nly sinle-phase as. Hwever, as he prducin is carried u, decline in ishermal pressure ccurs. Als, and as he bm-hle pressure in a flwin well falls belw he dew pin f he fluid, a liquid-hydrcarbn phase is frmed. The frmain f a rerrade cndensae resuls in a buildup f a liquid phase arund he wellbre, leadin a decrease in he effecive permeabiliy as in he wellbre. The prduciviy lss assciaed wih cndensae buildup can be subsanial. Afidick e al. (994) and Barnum e al. (995) lised several insances in which well prduciviies have been repred decline by a facr f w fur as a resul f cndensae accumulain. Barnum e al. (995) fund ha prducin lss is severe fr lw prduciviy reservirs, such as hse wih a K h less han 000md-f. They repred ha he criical cndensae saurain rane frm 0-30% and can decrease he prduciviy by a facr up five due cndensae accumulain near he well bre. Fevan (995), Ali e al. (997), and Grinaren and Al-Lamki (000) shwed ha, when reservir pressure arund a well drps belw he dew pin pressure, rerrade cndensain ccurs and hree reins are creaed wih differen liquid saurains. Varius mdels have been develped sudy deliverabiliy. Amn hese mdels is he seadysae flw cncep by O ell and Miller (965), wih pessimisic prducin raes a averae reservir pressure belw saurain pressure f fluid-in-place. Laer, Fussell (973) mdified he versin f - radial mdel develped by Rebuck e al. (969) sudy ln erm well perfrmance. espie he mdificains, he cndensae accumulains in he prducin rein remained reaer han he nes bained experimenally fr he cnsan vlume deplein prcess. Cable e al. (000) adped he use f special cre analysis daa fr near-well relaive permeabiliy mdel prduciviy in a full-field mdel. Their sudy cnsiders he imprance f liquid recvery and chane in yield, cmpsiin radiens, and reducin in well deliverabiliy caused by cndensae blckae. Because he ms impran and cmplex phenmena assciaed wih cndensae bankin and prduciviy reducin is relaive permeabiliy, here have been many invesiains f as cndensae relaive permeabiliy. Hinchman and Barree (985) shwed ha he prduciviy abve he dew pin pressure is cnrlled by he reservir s permeabiliy and hickness, in addiin he viscsiy f he as. Belw he dew pin, he deree f prduciviy reducin is cnrlled by he criical cndensae saurain and he shape f he as and cndensae relaive permeabiliy curves, as well as he chice beween imbibiins and drainae relaive permeabiliy curves. Whisn e al. (999) shwed ha relaive permeabiliy effecs in as-cndensae reservirs can be classified in hree caeries: near well seadysae as/il flw; bulk f reservir, beynd well viciniy, where liquid mbiliy is zer/ neliible; and waer encrachmen, where as and/r rerrade cndensae are rapped. Baue e al. (005) develped a nvel apprach fr calculain represenaive field relaive permeabiliy. This mehd is based n a physical mdel ha akes in accun he varius mechanisms f he prcess: bubble nucleain (pre-exisin bubbles mdel), phase ransfer (vlumeric ransfer funcin), and as displacemen (bubble flw). In he mdel, he researchers idenified a few invarian parameers which were n sensiive deplein rae and were specific he rck/fluid sysem. These invarian parameers were deermined by hisry machin ne experimen a a iven deplein rae. Jamilahmady e al. (006) used a lare daa 60

3 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN bank f as/cndensae relaive permeabiliy develp a eneral crrelain accunin fr he cmbined effec f cuplin and ineria as a funcin f fracinal flw. The parameers f he new crrelain were eiher universal, applicable all ypes f rcks, r culd be deermined frm cmmnly measured perphysical daa. And, Bzrzadeh and Grinaren (007) shwed ha well deliverabiliy depends mainly n he as relaive permeabiliies a bh he end pin and near wellbre saurain, as well as n he reservir permeabiliy. This wrk sudies well deliverabiliy (prduciviy) predicins f as flw in a ascndensae reservir mdelin near criical wellbre prblems in -. Since as cndensae is a mixure f as and liquid, his wrk will fcus n findin he pimum way imprve as flw (reducin cndensae build up) arund he well bre. number f reservir and well parameers. Cndensae blckae is deemed impran if he pressure drp frm he reservir he wellbre represens a sinifican percenae f he al pressure drp frm reservir delivery pin (e.. a surface separar) a he ime and afer a well es n decline. Reservirs wih lw--mderae permeabiliy (<0 50md) are fen cnsidered prblem wells and i becmes criical address he cndensae blckae issue prperly. Wells wih hih k h prducs (>5 0,000 md-f) are ypically n affeced by reservir pressure drp because he well s deliverabiliy is cnsrained alms enirely by he ubin. In his case, cndensae blckae is a nn-issue. In erms f reservir well perfrmance, he near-well behavir, deermined by he nearwell relaive permeabiliy funcins, is he dminan facr. The diffusiviy equain (Eq. ) is slved based n dimensinless rups wih respec cerain cndiins and assumpins which are elabraed and saed belw.. METHOOLOGY The ypical chemical cmpsiin f a ascndensae mixure is dminaed by vlaile cmpnens such as mehane, and a raher small amun f heavy hydrcarbn cmpnens (<5ml-%). espie is cmpsiin, hese heavier cmpnens make up a cnsiderably larer percenae f he liquid phase, rerrade cndensae, frmed durin pressure decrease belw an upper dew pin (McCain Jr., 989). This sudy is based n a fluid characerized by C7+ f 3.35ml-% and mlecular weih f 44.77/ml.; GOR, scf/sepp bbl; 5. O API, and reservir fluid mlecular weih f 40./ml. Fr pracically any rerrade cndensae reservir, he cndensae saurain is, hruhu ms f he reservir, s lw ha is mbiliy is much less han as mbiliy. Fr pracical purpses i can be cnsidered immbile. Neverheless, his as-dminaed flw behavir is n a all crrec in he viciniy f as-cndensae wells, where cndensae saurains fen reach hih values (>50%), and il permeabiliy may exceed as permeabiliy (kr/kr < ). Cndensae blckae near he wellbre may reduce as well deliverabiliy appreciably, huh he severiy f he bsrucin depends n a C p r r r r k p () An expressin fr al cmpressibiliy (Eq. ) wih respec he pressure derivaive f il frmain vlume facr and il saurain was prpsed. I was derived based n he cmbinain cninuiy equain fr he rae f in and u a cnrl vlume made up f a prus medium fr il and as respecively (see Appendix). This is incrpraed in he diffusiviy equain. SB S C B where B p B and S S p () Where r represens he radius; p, pressure; φ, prsiy; µ, viscsiy; k, permeabiliy; C, al cmpressibiliy;, ime; S, il saurain; B, il frmain vlume facr; λ, al mbiliy, and λ, il mbiliy. The reservir is caerized pseud-seady in naure havin a cnsan prducin. The wellbre 6

4 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN is classified as finie. The flw in he reservir is als cnsidered wih a n-flw uer bundary. Physically, a n-flw uer bundary culd be sealin fauls r pinch-us. Fr he mahemaical inerpreain f hese cndiins, cerain mahemaical relains such as Laplace and Bessel funcins are pu in use. Als, make he simplificain easier, dimensinless rups are used. Frm he diffusiviy equain, he parial differenial equain is iven by: p p r r r r (3) where r, p and are dimensinless radius, dimensinless pressure and dimensinless ime, respecively. A iniial cndiins, p ( r, 0) 0 (4) The inner bundary cndiin is cnsan rae prducin, r p r ( r ) (5) The uer bundary cndiin is fr a n-flw bundary, 8: Subsiuin he iniial cndiins, Eq. 4 in Eq. dp d r r dr dr UP (0) Where Eq. 0 has he frm f he mdified Bessel equain, d dv x v 0 x dx dx This represens he eneral sluin fr: () V AI ( x) BK ( x) () Therefre, he eneral sluin f Eq. 0 ives: P AI ( Ur ) BK ( Ur ) (3) Where A and B are arbirary cnsans deermined by he bundary cndiins. Takin he Laplace ransfrm f he bundary cndiin, Eq. 6 ives: dp dr r e 0 iffereniain Eq. 3, ne can bain: (4) p r re 0 (6) dp A U I( U r) B U K( U r) dr (5) Takin he Laplace ransfrm f he parial differenial equain, Eq. 3; p p L L (7) r r r This cnvers he parial differenial equain in rdinary differenial equains, d r dr dp r UP P ( r, 0) dr Where; (8) p L UP P( 0) (9) P is he Laplace ransfrm f p. A r = r e, his ives, 0 A U I ( Ur ) B U K ( Ur ) (6) A e Wriin A in Eq. 6 in erms f B, ives: B U K ( U r ) K ( U r ) e e B (7) A U I ( U r ) I ( U r ) e Usin he her cndiins slve fr he cnsans A and B. The inner bundary in Laplace space is: dp r dr U r e Subsiuin Eq. 8 in Eq. 5: e (8) 6

5 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN A U I( U re ) B U K( U re ) (9) U k k k ; C S C S C C f (6a,b) Subsiuin Eq. 7 in Eq. 9 and slvin fr B, B U I U re e e K U r I ( U ) I U r K ( U ) 3 (0) Therefre, k k r w SC SC C f (7) One can nw slve fr A, by subsiuin he relainship fr he cnsan B iven in Eq. 0 in Eq. 7: A U K U re e e K U r I ( U ) I U r K ( U ) 3 Subsiuin Eq. and Eq. 0 in Eq. 3: e e K U re I ( U r ) I U re K ( U r ) P 3 U K U r I( U ) I U r K( U ) () () This is he dimensinless pressure sluin in erms f U and r. Usin he Cauchey s inversin herem (O Neil, 987), he sluin becmes: r re ln r P r, r r 4 e re ln e ln r re re re r e re 4 re 4 3re (3) Fr he pressure a wellbre, P w, r e >>, Eq. 3 becmes: r ln 3r 4r ln r Pw re re 4 4 e e re re 4re re 4 4 e e e 4 4 re re 4r ln r 3 ln r 4 e (4) In rder cnver field unis, he dimensinless rups have be prperly expressed k r C (5) w Since his experimen cnsiders w phases (as and il): r r e e (8) rw Where r w is well radius; r e, reservir radius; k, relaive permeabiliy il; k, relaive permeabiliy as; µ, il viscsiy; µ, as viscsiy; S, il saurain; S, as saurain; C f, frmain cmpressibiliy; C, il cmpressibiliy; C, as cmpressibiliy and hers are as defined fr Equains. h k k pw pi pwf 4.qB Subsiuin Eq.9, 8 and 5 in Eq. 4: (30) (9) Where p wf is bm hle flwin pressure; p i, iniial reservir pressure; q, flw rae; h, reservir heih, and hers are as previusly defined. This equain (Eq. 30) shall be used predic as prducin perfrmance in cmparisn wih exisin sluin. 3. RESULTS AN ISCUSSION The bjecive f his sudy is buil arund he accurae predicin f he deliverabiliy f as in a as cndensae reservir. I akes in cnsiderain he varius limiin facrs he prper prducin frm such reservirs. In his cnex, cndensae blckae represens he main prblem experienced in hese reservirs, and is enerally knwn aler he deliverabiliy as a 63

6 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN Table. Reservir and well parameers. Reservir Pressure 4903 psia Reservir Temperaure 00 F Prsiy 0.0 Saurain Pressure 4440 psia Reservir Radius 800 f Reservir Wellbre Radius f Reservir eph 0000 f eph f Mid Perfrains 8000 f Well head Pressure 000 psia Well head Temperaure 40 F Table. Reservir fluid PVT daa. Pressure psia Gas ensiy z-facr Gas FVF /cm 3 cu f/scf Relaive Vl. Gas Viscsiy cp Gas Cmpr. psia - Gas Saurain Oil Saurain E E E E E E E E resul f reduced permeabiliy. The mehd emplyed invlves an analyical apprach. I may n prve be as deailed r as hruh as he mehd f numerical apprximains which, hwever, is mre cmplex in naure. The apprach used validae his research is as fllws: bainin accurae field daa (preferably a variey f hem), usin he bained daa make cmpuains wih he develpin crrelains, usin he bained daa make cmpuains wih esablished crrelains eiher similar r clsely relaed in naure, makin cmparisns wih bained resuls, and makin suiable arumens back hem up. The field daa used is shwn in Table. Table is bained frm a PVT analysis, hese daa have aided he analysis f his wrk. 3. Cmpuain arcyan and Nn- arcyan (Mdels) Usin he derived/develped crrelain belw fr a simplified -phase Nn-arcy unseady sae (frm Eq. 30), cmpuains where made bain he flw rae f he as phase, as seen in Table 3. Likewise, he esablished crrelain fr arcy seady sae flw f naural as is used predic prduced as. 44qZT r p ln e e pwf s kh rw (3) arcy s relainship fr seady sae flw f as (Eq. 3) is used predic his resul. arcy s mdel cann be lked a as an exac cmparisn he develped crrelain fr he -phase unseady sae used in his sudy. Hwever, cnsiderin he 64

7 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN Table 3. Prediced as flw rae. P (iniial) P wf Nn-arcyan arcyan psia psia q(mscf/) q(mscf/) deliverabiliy f as, his relainship can be used wih he allwance f cerain discrepancies. The applicain f Eq. 30 uilizes PVT daa (Table ) and special cre analysis daa carry u he fllwin cmparisns. Crey s relainship permeabiliy mdel is used fi measured cre daa, k /k rai is bained frm k / k ( / ) V where µ and V rcce rcce are exraced frm PVT daa and V rcce is a cnsan cmpsiin expansin il relaive vlume. The uer reservir radius is a he pin where cndensae saurain is abve criical cndensae saurain. The pressure a his pin is he dew pin pressure; his is acually he near wellbre rein, as explained in he nex sub secin. All her parameers required cmpue flw rae in Eqn. 30 are as lised in Tables and. 3. Cmparisn f crrelain (mdels) Cmparisn f he w IPR resuls can be bserved in Fiure. The enire sudy is cenered n Reins and. Rein is characerized by cndensae saurain abve criical cndensae saurain (CCS) and, hence, bh as and liquid phases are mbile. Rein is he main surce f deliverabiliy lss in a as-cndensae well. Gas relaive permeabiliy is reduced drasically in his rein due cndensae buildup. The reducin in relaive permeabiliy as reaches is hihes levels in his rein. Even huh cndensae buildup sars frm rein, he liquid phase is immbile. The w-phase flw in rein is he main cause f as relaive permeabiliy reducin. Rein is he inermediae zne where cndensae drpu beins and defines a rein f ne accumulain f cndensae. The cndensae saurain is belw he criical value (Scc) and effecively nly as is flwin in his rein because il mbiliy is reduced r zer. One can assume ha he resuls bained frm he arcy crrelain are applicable as reservirs ha have very similar cndiins ha Fiure. IPR f he develped crrelain fr Nn-arcyan and arcyan mdels. 65

8 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN f he cndensae reservir. urin he early phase f prducin frm rein, ne can bserve ha he flw rae shwn by he mdel develped in he presen sudy is hiher han he ne shwn by arcy s crrelain (Fiure ). This can be aribued he fac ha a his sae, he as phase sill has cnsiderable mbiliy and he flw rae is sill hiher han ha f he liquid phase. The difference in flw rae akes place because cndensaes are sill buildin up. Bu, as he prducin pressure declines, he mre he cndensaes build and leadin a reducin in he as mbiliy. This leads a siuain where he flw rae f he as shwn under he develped crrelain is lwer han he flw rae shwn by he arcy crrelain. Tha cnfiures he phenmenn bserved arund rein. The pressure pin where he flw rae f as sars reduce represens he cndensae unladin pressure. The cndensae unladin pressure fr he abve scenari is abu,50 psi. The presen sudy may cnain shrcmins. Amn he pssible inaccuracies, he majr pins cnsider are: The develped crrelain is mdeled wih he assumpin ha he fluid flw is arcy in naure, where as in he acual sense, i is a nn- arcy flw; An analyical apprach was cnsidered in he develpmen f his crrelain. This cann be said be a very deailed analysis because his apprach deals wih many assumpins and des n vary sufficien parameers; This mehd des n suppr a case scenari where he cmpsiin f he mixure is cnsanly chanin wih ime. This can be seen, fr example, when as injecin is carried u; The mdel sricly cnsiders a w-phase flw (excludin waer), which is bserved in very rare cases. A beer mdel wuld include waer effecs. Frm he resuls bained, i is safe say ha prducin mus be carried u a a hih flwin pressure (far abve he dew pin), irrespecive f he crrespndin flw rae. This is preven r prln he frmain f cndensaes ha wuld furher hinder prducin. NOMENCLATURE µ = viscsiy µ = as viscsiy µ = il viscsiy B = il frmain vlume facr CCS = criical cndensae saurain C f = frmain cmpressibiliy C = as cmpressibiliy C = il cmpressibiliy C = al cmpressibiliy h = reservir heih k = abslue permeabiliy k = relaive permeabiliy as k = relaive permeabiliy il p = pressure p = dimensinless pressure p i = iniial reservir pressure p w = dimensinless pressure a wellbre p wf = bm hle flwin pressure q = flw rae r = radius r = dimensinless radius r e = reservir radius r w = well radius S = as saurain S = il saurain = ime V rcce = cnsan cmpsiin expansin il relaive vlume = dimensinless ime λ = il mbiliy λ = al mbiliy φ = prsiy 4. CONCLUSIONS This sudy aimed develpin a mehd suiable fr analyzin he delivery paerns f as in a as-cndensae field usin an empirical apprach. I prvides a simple deducin n a pssible sluin a as cndensae challene. 5. REFERENCES Afidick,. ; Kaczrwski, N. J.; Srivinas, B. Prducin perfrmance f a rerrade as reservir: A case sudy f he Arun field. SPE paper 8749-MS, 994. hp://dx.di.r/0.8/8749-ms 66

9 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN Ali, J. K.; Mc Gauley, P. J.;Wilsn, C. J. Experimenal sudies and mdelin f as cndensae flw near he wellbre. SPE paper MS, 997. hp://dx.di.r/0.8/39053-ms Barnum, R. S.; Brinkman, F. P.; Richardsn, T. W.; Spillee, A. G. Gas cndensae reservir behavir: Prduciviy and recvery reducin due cndensain. SPE paper MS, 995. hp://dx.di.r/0.8/30767-ms Baue, F.; Eermann, P.; Lenrmand, R. A new mdel bain represenaive field relaive permeabiliy fr reservirs prduced under sluin-as drive. SPE Jurnal f Reservir Evaluain and Enineerin, v.8(4), p , 005. hp://dx.di.r/0.8/84543-pa Bzrzadeh, M.; Grinaren, A.C. Esimain prduciviy cnrllin parameers in as/cndensae wells frm ransien pressure daa, SPE Reservir Evaluain & Enineerin, v.0, p. 00-, 007. hp://dx.di.r/0.8/9408-pa Cable A. S.; M R. E.; Mike S. X-Ray in-siu saurain in as cndensae relaive permeabiliy sudies. SCA AEA Technly PLC, Winfrih Technly Cener, rcheser, rse, T 8ZE, UK, 000. Fevan, O.; Whisn, C. H.; Trndheim, U. Mdelin as cndensae deliverabiliy. SPE paper 3074-PA, SPE Reservir Enineerin, v. (4), p. - 6, 995. Fussell,.. Sinle-well perfrmance fr as cndensae reservirs. Jurnal f Perleum Technly, v. 5(7), p , 973. hp://dx.di.r/0.8/407-pa Grinaren, A. C.; Al-Lamki, A.; aunkaew, S.; M, R.; While, T. M. Well es Analysis in as Ccndensae reservirs. SPE paper 690, 000. Hinchman, S. B.; Barree, R.. Prduciviy lss in as cndensae reservirs. SPE paper 403-MS, 985. hp://dx.di.r/0.8/403-ms Jamilahmady, M; anesh, A.; Tehrani,. H.; Shrabi, M. Variains f as/cndensae relaive permeabiliy and prducin rae a near-wellbre cndiins: A eneral crrelain. SPE Jurnal f Reservir Evaluain and Enineerin, v. 9(6), p , 006. hp://dx.di.r/0.8/83960-pa Mc Cain Jr, W.. The Prperies f Perleum Fluids, nd Ed., PennWell Publishin Cmpany, p. 55, 989. O ell, H. G.; Miller, R. N. Successfully cyclin a lw-permeabiliy, hih-yield as cndensae reservir. SPE paper 495-PA, p.4-47, 967. hp://dx.di.r/0.8/495-pa O Neil, P.V. Advanced Enineerin Mahemaics. nd Ed., Wadwrh Publishin C., Belmn, Califrnia, 987. Rebuck Jr., I. F.; Frd, W. T.; Hendersn, G. E.; ulas Jr., J. The cmpsiinal reservir simular: case iii The radial emery. Unpublished paper, available frm Cre Labraries, Inc., allas, Texas, 969. Whisn, C. H.; Fevan, O.; Saevareid, A. Gas Cndensae Relaive Permeabiliy fr Well Calculains. SPE paper 56476, 999. APPENIX In deerminin he as deliverabiliy in a as cndensae reservir, sme equains have be derived, akin in cnsiderain fundamenal principles f il and as flw. This is a mre cmplex apprach, cnsiderin ha i invlves w phases which d n presen a cnsan cmpsiinal vlume hruhu he reservir. Sarin frm he basic diffusiviy equain, which cnsiders he principle f cnservain f mass, an equain f min, and an equain f sae, ne can aain he diffusiviy equain. Fr a sinle-phase flw, he diffusiviy equain is iven by: C p r r r r k p (A) Fr he muliphase flw, ne mus cnsider he cninuiy equain fr each phase. The equain includes a uni cnrl vlume cnainin il and as saurains S and S. The il mass balance f a sysem is represened by: 67

10 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN m M (A) frmain vlume facrs and slubiliy are funcins f pressure: Radial flw crdinaes leads : M m r r r ; M u Fr il flw in a radial sysem, u S r s r r B k P u r ; ; m s (A3) (A4) k P s k P u r B r (A5) Based n he n arcy s equain f min in a cnrl vlume fr he il phase, k P S r r r B r B (A6) The equain abve is simplified by assumin cnsan il permeabiliy, viscsiy and frmain vlume facr. B S B r r r r k p B k p p p B S p p S k B p p B Le, B p B S S p B p B B p Therefre, B B S S B p r r r k B B p r (A9) (A0) B S r r r r k B P (A7) Where m is mass f il per vlume; m, mass f as per uni vlume and ρ s, represens as densiy a sandard cndiin; M, mass flux; ρ, as densiy; ρ s, il densiy a sandard cndiin; ρ, il densiy; u, flw velciy; µ, il viscsiy; µ, as viscsiy; k, il permeabiliy; k, as permeabiliy; B, il frmain vlume facr; B, il frmain vlume facr; S, il saurain; S, as saurain; φ, prsiy; p, pressure; r, radius, and, ime. The cninuiy equain can be wrien fr each phase cnsiderin ha he rae f mass f he cnrl vlume less he rae f mass u is iven by he fllwin similariy f as equain: B S r r r r k B p (A8) Expandin he parial derivaives, wih respec ime, usin chain rule and bservin ha he SBp r r r k B p r S Expandin Eq. A8: B S p r r r r k p B p B S S B p r r r k B B p r Oil and as mbiliy are iven by, k (A) (A) (A3) k (A4) Subsiuin Eq. A3 and A4 in A and A: 68

11 BRAZILIAN JOURNAL OF PETROLEUM AN GAS v. 6 n. 4 p ISSN p SB p r S r r r B B S S B p r r r B B p r (A5) (A6) efinin al cmpressibiliy, C, and bainin expressins fr il and as cmpressibiliy in erms f frmain vlume facr: C SC SC (Aa) C B B (Ab) Nw equain il and as relainships derive a sinle equain describe muliphase flw: SB p B S SB p S B B B SB SB S S B B S (A7) (A8) S (A9a) iffereniain Eq. A9a wih respec pressure ives: S S 0 (A9b) Tal mbiliy in his case is λ (A0) Subsiuin Eq. A9a and A0 in A4 C C B (Ac) B Subsiuin Eq. Ab and Ac in Eq. Aa: SB SB (A3) B B Therefre, SB S C B (A4) The abve equain, Eq. A4, may be subsiued in Eq., ivin he diffusiviy equain calculae cmpressibiliy. C p r r r r p (A5) SB SB SB S B B B (A) 69