Iranian Journal of Chmical Enginring Vol. 9, No. 4 (Auumn), 22, IAChE Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl A.R. Basami, M. Dlshad 2, P. Pourafshary 3 - Gradua Sudn, Insiu of Prolum Enginring, Univrsiy of Thran 2- Gradua Sudn, Insiu of Prolum Enginring, Univrsiy of Thran 3- Assisan Profssor, Insiu of Prolum Enginring, Univrsiy of Thran Absrac War flooding, h olds and mos common EOR mhod, incrass h displacmn fficincy in a rsrvoir and also mainains h rsrvoir prssur for a long priod of im. In Iran, war incion is widly usd as a mhod o nhanc rcovry from oil rsrvoirs. Dfining h opimizd incion ras and incion parns, dpndn on h gological srucur of h rsrvoir, is ssnial in opraional and conomical dcisions for rsrvoir managmn. In his papr, h Capacianc-Rsisiv Modl is usd o find inrwll connciviy, and opimizd incion ras in a synhic fild. In his approach, h rsrvoir rcivs incor ra variaions as an inpu signal, whil h producr rsponss drmin h incor/producr pair connciviy quaniaivly. This modl is usd o prdic oil producion for a spcific rsrvoir, if h producion/incion ra and boomhol prssur daa ar availabl. Th rsuls show ha h Capacianc- Rsisiv modl has h capabiliy o b usd for h producion hisory maching and o opimiz h incion ra in diffrn wlls of a rsrvoir during h immiscibl flooding o maximiz h oil producion. Morovr, hy show ha any chang in oil and war prics can significanly influnc h opimizd war incion ras. Kywords: Warflooding, War Incion, Producion Forcas, Opimizaion, Capacianc Rsisiv Modl. Inroducion Oil rcovry opraions hav radiionally bn subdividd ino hr sags: primary, scondary, and riary. Hisorically, hs sags dscribd h producion from a rsrvoir in a chronological sns. Primary producion, h iniial producion sag, rsuld from h displacmn nrgy naurally xising in a rsrvoir. Scondary rcovry, h scond sag of opraions, was usually implmnd afr primary producion dclind. Tradiional scondary rcovry procsss ar warflooding, prssur mainnanc, and gas incion, alhough h rm scondary rcovry is now almos synonymous wih warflooding. Triary rcovry, h hird sag of producion, was ha obaind afr Corrsponding auhor: pourafshari@u.ac.ir 3
Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl warflooding (or whavr scondary procss was usd). Triary procsss usd miscibl gass, chmicals, and/or hrmal nrgy o displac addiional oil afr h scondary rcovry procss bcam unconomical [, 2]. 2. Capacianc rsisiv modl Th ida of his modl is similar o h lcrical currn in h lcrical circuis, including a nwork of capaciors and rsisors. Hnc, by considring flow sram, sorag capaciy in porous mdia, prssur diffrnc and prmabiliy similar o lcrical currn, capacianc, ponial diffrnc and rsisanc, rspcivly, h mahmaical rlaions in lcrical circuis can b usd in h rsrvoir. In his modl, a rsrvoir is supposd o rciv a signal (incion) and dlivr a racion signal (producion) o wha i rcivs. Hisorical incion and producion ras ar h main inpu daa for his modl. Analyzing hs daa could provid a gra dal of usful informaion abou h rsrvoir and rduc h uncrainis in rsrvoir modling. Th final purpos is h conomical and opraional opimizaion in warflooding procs o maximiz h oil rcovry [4]. Albroni al. usd a simpl modl o find h inrwll connciviy. H showd ha vn h incion ras of incors which ar far away from producrs can affc hir producion ras. H simad h inrwll connciviy by a linar modl and simad cofficins by using MLR [5]. Yousf al. addd a nw paramr and dvlopd h modl o considr boh capacianc and. Mulipl Linar Rgrssion rsisanc ffcs by using comprssibiliy and ransmissibiliy concps, rspcivly [6, 7]. Sayarpour al. dfind hr diffrn simplifid modls and prsnd analyical soluions for ach and validad hs soluions by applying hm for som ral filds [8, 9]. Wbr al. rviwd h problms of using his modl in larg scal filds wih a larg numbr of wlls and suggsd som soluions for minimizing h rror causd by hs problms []. Dlshad al. usd his modl o dc h prsnc of fracurs in a rsrvoir and calcula h fracur prmabiliy []. 3. Mahmaical dvlopmns Th marial balanc for a simpl rsrvoir including on incor/producr pair as shown in Fig., is as follows: Figur. A conrol volum including on incor/ producr pair dp cv p i( ) q( ) () d To find an quaion basd on only incion and producion ras, linar produciviy indx can b usd. q J P P ) (2) ( wf By rplacing avrag prssur form quaion 4 Iranian Journal of Chmical Enginring, Vol. 9, No. 4
Basami, Dlshad, Pourafshary (2) in quaion (), dq dpwf q( ) i( ) J (3) d d whr is im consan. Th soluion of his firs ordr diffrnial quaion is as follows: ( ) q( ) q( ) i()d J P( ) P( ) P()d wf ( ) wf wf (4) This is h basic formulaion of his modl. Now, assum ha h rsrvoir consiss of diffrn conrol volums wih a producr and incor around i. [2]. A schmaic rprsnaion of his approach is illusrad in Fig. 2. ach producr and a wigh cofficin for ach pair of incors and producrs. Hnc, h quaions which dscrib h rlaions bwn incors and producrs ar diffrn in his approach: dq ( ) q ( ) d I i. i ( ) J whr h im consan is dfind as: c V J i i dp wf d (5) p (6) Th soluion of h abov quaion is as follows: q( ) q( ) ( ) I dp wf, i i i d i()d J d (7) Ingraion of h abov quaion lads o: Incor q( ) q( ) i( ) i( ) ( )( ) I i i i i I di() i dpwf, i J d i d d (8) Producr Figur 2. A schmaic viw of a rsrvoir wih 2 incors and 2 producrs which dscrib h paramrs of CRM in his modifid approach. In his approach, hr is a im consan for By assuming ha during h producion inrvals, h boomhol prssur is varying linarly and h incion ras ar consan, h dscribing quaion would b as follows: Iranian Journal of Chmical Enginring, Vol.9, No. 4 5
Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl q( ) q( ) ( ) ( ) n k I ( k) p ( k) wf, ( ) I J i i k i k (,,..., 2 I) (9) 4. Oil producion modl To sima h oil fracional flow in h producion sram, an oil fracional flow modl should b usd in associaion wih Capacianc Rsisiv Modl. On of h suggsd modls is a powr law rlaion bwn insananous war/oil raio, F, and cumulaiv war incion ra, wo W i [3]. Hnc, h simad war/oil raio can b calculad from q.. F W () wo i For a balancd sysm, oal incion and oal producion ar qual; hnc, Wi is qual o h oal liquid producion. This quaion can b usd for ach wll sparaly. By using q. as h oil fracional flow modl, w hav: q o f o ( ), q q o qw o Fwo, Wi, q w () whr is h connciviy bwn an incor/producr wll pair or wigh cofficin. On h ohr hand, oil producion ra from producr is qual o h oil fracion, f o, muliplid by oal producion, q () as follows: q o ( ) f ( ) q ( ) (3) o Th combinaion of q., q. 2 & q. 3 lads o h oil producion ra from ach producr: q( ) o N in ( k i( k s).ds) (4) k (, 2,..., N) pro q( ) Hnc, afr simaing h CRM paramrs (im consans & wigh cofficins), h oil fracional flow paramrs (, ) should b simad by minimizing h diffrnc bwn ral and simad valus during hisory maching. Sinc h dirc applicaion of q. in his form for finding modl paramrs is vry difficul, a logarihmic form of his quaion is suggsd hr. log( Fwo, ) log( ) log( Wi, ) (5) In his quaion, cumulaiv war incion is as follows: W Nin N in i, [ k k k k. k k i ( s)]. ds i ( s) ds (2) By using a linar rgrssion and minimizing h diffrnc bwn ral and simad valus, log( ) and can b calculad. This linar form of powr law rlaion shows h limiaions of using his modl in 6 Iranian Journal of Chmical Enginring, Vol. 9, No. 4
Basami, Dlshad, Pourafshary prdicing oil producion ra. In ohr words, his modl can b applid if h logarihmic plo of war/oil raio vrsus cumulaiv war incion is linar. 5. Opimizaion algorihm Th mos imporan par of an opimizaion problm is drmining h obciv funcion. Dpnding on h purpos of h opimizaion, diffrn obciv funcions can b dfind. Som of hs obciv funcions ar as follows:. Maximizing cumulaiv oil producion during a dfini im inrval 2. Minimizing cumulaiv war producion during a dfini im inrval 3. Maximizing n prsn valu of h proc by considring incion coss 4. Mainaining h oil producion ra from a spcific fild Hnc, h purpos of opimizaion can b o maximiz ulima oil producion ra or o maximiz h profi of h warflooding in a rsrvoir. To maximiz h profi of a warflooding proc during a spcific im inrval [, ], h suggsd obciv funcion is as follows: N pro N in R p. q ( s). ds p. i ( s). ds (6) whr, o po and o w k k pw ar oil and war pric pr barrl, rspcivly. Furhrmor, dcision variabls ar producion and incion ras. An uppr and a lowr boundary for incion hrough incors can b dfind according o h condiions of h rsrvoir. Th ovrall procdur of his opimizaion by using capacianc rsisiv modl and oil fracional flow modl is illusrad in Fig. 3. In his papr, h Microsof Excl Solvr is usd o drmin h CRM and fracional flow paramrs. Morovr, i is usd o solv h nonlinar mahmaical quaion o drmin incion ras. Microsof Excl Solvr uss h Gnralizd Rducd Gradin (GRG2) Algorihm for opimizing nonlinar problms. Figur 3. Workflow for h CRM applicaion in hisory-maching and opimizaion [4] 6. Cas sudy 6-. * Modl In his scion, Capacianc-Rsisiv Modl for a synhic fild wih incor- producr is applid. Ths hisorical incion and producion ras ar for a ral fild and his wll pair is supposd o b sparad from h ohr par of h fild. In Fig. 4, h incion ra vrsus im is illusrad. By applying CRM, h inrwll paramrs ar calculad as i is shown in Tabl. Iranian Journal of Chmical Enginring, Vol.9, No. 4 7
Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl Ra (bbl/day) 9 8 7 6 5 4 3 2 Incion Ra Incion 5 5 2 25 3 35 Tim (day) Fig. 4. Th incion ra vrsus im obciv funcion in his sudy is o maximiz h profi by considring h war incion cos qual o -3 $/bbl and diffrn oil prics. In addiion, h minimum and maximum limis of incion ra ar supposd o b, bbl/day, rspcivly. Tabl. CRM paramrs P τ.5 9 λ q, RB/T q Error, RB/T q Corrlaion raio 5657 267.97 In Fig. 5, h ral producion ra is compard o h simad producion ra which is calculad by CRM. Hnc, CRM is abl o sima liquid (war + oil) producion ra accuraly. In addiion, by using CRM paramrs w ar abl o forcas h producion ra by accouning any chang in h incion ra. In Fig. 6, h calculaion procdur o find fracional flow modl paramrs (powr low modl) is illusrad. Ra (bbl/day) 9 8 7 6 5 4 3 2 Producion Ra Ral Esimad 5 5 2 25 3 35 Tim (day) Figur 5. Th ral producion ra in comparison wih simad producion ra Using powr low modl paramrs and slcing h dsird obciv funcion which is discussd in h prvious scion, incion ra opimizaion is possibl. Th Figur 6. calculaion Th graph of fracional flow modl Tabl 2. Powr low paramrs α β q o Corrlaion raio 6.42E-3.857.9 According o hs rsuls (Tabl 3, 4 & 5), if war incion coss $/bbl, for oil prics lss han 8 $/bbl, war incion is no fasibl. Also, if ach oil barrl is worh $9, opimizd war incion ra is qual o 5 bbl/day. Finally, if ach oil barrl is worh $2, opimizd war incion ra is qual o bbl/day. Ths calculaions for diffrn war incion coss and oil prics ar prsnd in h abls blow. Tabl 3. Opimizd incion ra dpndn on oil pric, and war incion cos qual o $/bbl Oil Pric ($) War Pric ($) Opimizd Incion ra (RB/D) 8 9 5 2 8 Iranian Journal of Chmical Enginring, Vol. 9, No. 4
Basami, Dlshad, Pourafshary Tabl 4. Opimizd incion ra dpndn on oil pric, and war incion cos qual o 2 $/bbl Oil Pric ($) War Pric ($) Opimizd Incion ra (RB/D) 36 2 37 2 774 38 2 5 39 2 9329 4 2 Tabl 5. Opimizd incion ra dpndn on oil pric, and war incion cos qual o 3 $/bbl Oil Pric ($) War Pric ($) Opimizd Incion ra (RB/D) 55 3 56 3 2227 57 3 5 58 3 793 59 3 6-2. 5*4 Modl 6-2-. Inroducion In his cas, a synfild wih 5 incors, 4 producrs and wo high prmabl sraks is considrd. I is assumd ha all h wlls ar vrical and hy ar prforad in all layrs. Th mos imporan rock and fluid propris ar prsnd in Tabl 6 and a schmaic viw of his rsrvoir is illusrad in Fig. 7. This modl consiss of 3*3*5 grids in X, Y and Z dircion, rspcivly. Figur 7. A schmaic viw of a synfild wih 5 incors, 4 producrs and wo high prmabl sraks Tabl 6. Rock and fluid propris of 5*4 Modl Paramr X dircion Valu Grid Siz, f Y dircion 8 Porosiy, fracion Prmabiliy, md Viscosiy, cp Z dircion Oil War Oil 8 4.8 5 2..5 5-6 Comprssibiliy, psi - War -6 Rock -6 6-2-2. Hisory maching In his sudy, Capacianc Rsisiv Modl paramrs ar usd o forcas h producion ra and o sima h warflooding prformanc quickly. In his cas, 35 days of opraion in h form of 2 incion/producion daa ar pu ino h modl, and h modl paramrs, as prsnd in Tabl 7, ar obaind afr hisory maching. Tabl 7. CRM paramrs for 5*4 Modl Paramrs P P2 P3 P4 τ.53 5.324 2.57.566 λ λ 2 λ 3 λ 4 λ 5.87.69.24..522.34.98.246.277.74.88.56.27.22.62.52.96.43.94.567 Sum Iranian Journal of Chmical Enginring, Vol.9, No. 4 9
Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl Error % 6.2.9.6 9.96 6-2-3. Producion forcas By using CRM paramrs which ar obaind hrough hisory maching, producion has bn prdicd for 7 days. Hnc, i is possibl o sudy h capabiliy of capacianc rsisiv modl in prdicing fuur producion of h rsrvoir by comparing h prdicd producion ras wih ras calculad by fin grid simulaors (Eclips). Fig. 8 shows h comparison bwn h valus obaind by CRM and Eclips for ach producr. In his figur, Ral rprsns h Eclips rsuls and h CRM rsuls ar shown as Esimad. Th rsuls show h simad valus of CRM ar in agrmn wih h calculad valus by Eclips. Hnc, CRM is capabl of forcasing h fuur producion vn for rsrvoirs wih mor complxiy and hrogniis. 6-2-4. Opimizaion of h incion ras In his scion, h calculaion of incion ra opimizaion by using CRM is prsnd. Fig. 9 shows h oil fracional flow modl (Powr law) graphs in which paramrs of modls can b calculad for ach producr and h inrval ha his modl can b applid. Minimizing h rror bwn simad valus and ral valus (lik hisory maching) rsuls in oil fracional flow modl paramrs, as i is prsnd in Tabl 8 for ach producr. Tabl 8. Esimad valus for oil fracional flow modl paramrs Wlls P P2 P3 P4 9.65E-2.9E-22 8.59E-5.78E-5.77 3.77 2.397 2.377 Iranian Journal of Chmical Enginring, Vol. 9, No. 4
Basami, Dlshad, Pourafshary Figur 8. Comparison bwn simad valus of CRM and h calculad valus by Eclips for ach producr Iranian Journal of Chmical Enginring, Vol.9, No. 4
Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl Figur 9. Th graphs basd on powr law modl for simaing h fracional flow modl paramrs To opimiz h incion ra, i is assumd ha h ra of all incors is consan and qual o h las valu of h hisory for 3 monhs. Morovr, i is assumd ha h minimum and maximum incion ras ar and 4 STB/D, rspcivly. Thus, using CRM and fracional flow modl paramrs, wih an assumpion of 2 and 7 dollars cos pr barrl for war and oil, rspcivly, h opimizaion basd on maximizing h ulima profi can b accomplishd. Tabl 9 compars h ras of incors bfor and afr opimizaion. Using fin grid simulaors provids us wih a masuring ool o compar h diffrnc bwn producion ras ihr by h iniial incion ras or h opimizd ras. This comparison for oal producion ra and profi is illusrad in Fig. and Fig., rspcivly. Tabl 9. Comparison bwn h ras of incors (bbl/day) bfor and afr opimizaion Iniial ras Opimizd ras I I2 I3 I4 I5 Toal Incion 3 54 29 963 97 7892 3862 4 7892 2 Iranian Journal of Chmical Enginring, Vol. 9, No. 4
Basami, Dlshad, Pourafshary Figur. Comparison bwn h oal producion bfor and afr opimizaion Figur. Comparison bwn h profi bfor and afr opimizaion 7. Conclusions Nowadays, warflooding is known as on of h mos common EOR mhods and also as a mhod o mainain h rsrvoir prssur in ordr o incras h ulima oil rcovry in h oil indusry. Thrfor, opimizd war incion ra as an opraional facor and also as an conomical facor is vry imporan. I has a considrabl ffc on h ulima prformanc of h nhancd oil rcovry proc. In his papr, a nw mhod for a rapid and coninuous opraional and conomical calculaion in an incion proc is prsnd. In addiion, i has bn shown Iranian Journal of Chmical Enginring, Vol.9, No. 4 3
Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl ha i is possibl o calcula opimizd incion ra by using his modl for diffrn rsrvoir and wll gomris. Nomnclaur Modl cofficin Modl cofficin Wigh cofficin Ingraing variabl Tim consan c L 2 /F Toal comprssibiliy Fwo War/oil raio i() L 3 /day Toal incion ra J L 5 /F- Produciviy indx P F/L 2 Avrag prssur in h por volum Pwf F/L 2 Boomhol prssur q() L 3 /day Toal producion ra Vp L 3 Por volum Wi L 3 Cumulaiv war incion Appndix To driv h analyical soluion of q. 3, w firs xprss h diffrnial quaion in a gnral form, or: dq d q whr, i r r dp J d wf A- A-2 This firs-ordr diffrnial quaion can b gnrally solvd by using h ingraing facor chniqu. For his quaion, h ingraing facor is givn by: f d A-3 Thn, by muliplying q. A- by h ingraing facor, or: dq q r A-4 d i immdialy follows ha d d q r A-5 W now spara and ingra his quaion wih rspc o, q c r d A-6 whr c is h consan of h ingraion. By dividing boh sids by, q c r d A-7 Th consan of h ingraion c can b simad by using h iniial condiion q q. Thn Eq. A-7 bcoms, q q By subsiuing for quaion, w g q q or r r d dp d i J wf d A-8 in h abov 4 Iranian Journal of Chmical Enginring, Vol. 9, No. 4
Basami, Dlshad, Pourafshary q q d dp wf i d J d A-9 whr, i is a variabl of ingraion. Th hird rm on h righ of Eq. A-9 can b simplifid by using ingraion by pars. Th final analyical soluion for Eq. 3 is, q q i d J pwf pwf pwf d Rfrncs [] Lail, M., Enhancd oil rcovry, s diion, Tchnip, Paris, Franc, p. -2 (98). [2] Grn, D. W. and Willhi, G. P., Enhancd oil rcovry, SPE Txbook Sris VOL. 6, 2 nd diion, Txas, USA, p. (23). [3] Bruc, W. A., "An lcrical dvics for analyzing oil rsrvoir bhavior", Trans. AIME, p. 3-24, (943). [4] Liang, X., Wbr, D. B., Edgar, T. F., Lak, L. W., Sayarpour, M., and Yousf, A. A., "Opimizaion of oil producion basd on a capacianc modl of producion and incion ras", SPE 773, Hydrocarbon Economics and Evaluaion Symposium, Dallas, Txas, U.S.A (27). [5] Albroni, A., and Lak L. W., "Infrring connciviy only from wllra flucuaions in warfloods", SPE Rsrvoir Evaluaion and Enginring Journal, 6(), 6 (23). [6] Yousf, A. A., Gnil, P., Jnsn, J. L., and Lak, L. W., "A Capacianc modl o infr inrwll connciviy from producion and incion ra flucuaions", SPE 95322, SPE Annual Tchnical Confrnc and Exhibiion, Dallas, Txas, U.S.A. (25). [7] Yousf, A. A., Jnsn, J. L., and Lak, L. W., "Analysis and inrpraion of inrwll connciviy from producion and incion ra flucuaions using a capacianc modl", SPE 99998, SPE/DOE Symposium on Improvd Oil Rcovry, Tulsa, Oklahoma, U.S.A. (26). [8] Sayarpour, M., Zuluaga, E., Kabir, C. S. and Lak, L. W., "Th us of capacianc-rsisiv modls for rapid simaion of warflood prformanc", SPE 8, SPE Annual Tchnical Confrnc and Exhibiion, Anahim, California, U.S.A. (27). [9] Sayarpour, M., Kabir, C. S., and Lak, L. W. "Fild applicaions of capacianc-rsisiv modls in warfloods", SPE 4983, SPE Annual Tchnical Confrnc and Exhibiion, Dnvr, Colorado, U.S.A (28). [] Wbr, D. B., Edgar, T. F., Lak, L. W., Lasdon, L., Kawas, S., and Sayarpour, M., "Improvmns in capaciancrsisiv modling and opimizaion of larg scal rsrvoirs", SPE 2299, SPE Wsrn Rgional Ming, San Jos, California, U.S.A (29). [] Dlshad, M., Basami, A. and Pourafshary, P., "Th us of Iranian Journal of Chmical Enginring, Vol.9, No. 4 5
Rapid Esimaion of War Flooding Prformanc and Opimizaion in EOR by Using Capacianc Rsisiv Modl capacianc-rsisiv modl for simaion of fracur disribuion in h hydrocarbon rsrvoir", SPE 2676, SPE Saudi Arabia Scion Tchnical Symposium, Khobar, Saudi Arabia, (29). [2] Basami, A., Th us of capaciancrsisiv mhod for rapid simaion of immiscibl flooding in nhancd oil rcovry, M.Sc. hsis in prolum nginring, Univrsiy of Thran, Iran, (29). [3] Gnil, P. H., Th us of mulilinar rgrssion modls in parnd warfloods: Physical maning of h rgrssion cofficins, M.Sc. hsis in prolum nginring, Th Univrsiy of Txas a Ausin, Txas, (27). [4] Sayarpour, M., Dvlopmn and applicaion of capacianc-rsisiv modls o war/co 2 floods, Ph.D. dissraion in prolum nginring, Th Univrsiy of Txas a Ausin, Txas, (28). 6 Iranian Journal of Chmical Enginring, Vol. 9, No. 4