DEVELOPMENT OF A NEW DRAG COEFFICIENT MODEL FOR OIL AND GAS MULTIPHASE FLUID SYSTEMS

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1 Nigrian Journal o Tchnology (NIJOTECH) Vol. No., April 5, pp Copyright Faculty o Enginring, Univrsity o Nigria, Nsukka, ISSN: EVELOPMENT OF A NEW RAG MOEL FOR OIL AN GAS FLUI K. O. Bllo and K. I. Idigb, EPARTMENT OF PETROLEUM ENGINEERING, UNIVERSITY OF BENIN, BENIN CITY, NIGERIA addrsss drsss: blloko@unibn.du, kiidigb@gmail.com ABSTRACT Multiphas lows involving suspnsions o solid particls ar rquntly ncountrd in many industrial procsss including oil & gas production. In ordr to transport solid ntraind multiphas luids, spcially through a piplin, th luids must b capabl o suspnding th ntraind solid particls to prvnt solid dosition and low assuranc challngs. Th particl drag coicint, C, is a ky hydrodynamic paramtr, and its ctiv dtrmination is critical to nsuring ctiv transportation o th particls. In this work, th rsults o laboratory masurmnts o th trminal sttling vlocity o sand particl(s) in multiphas luids ar prsntd. Th rsults allow th propr dtrmination o th particl(s) Rynolds numbr, C. Th dvlopd C modl is an improvmnt on xisting modls, which can b applid to both th turbulnt and laminar low rgions, thus, rasonably prdicting C ovr a wid rang o R in multiphas luid low. It ctivly accounts or th complxitis associatd with multiphas low in pips, and has th potntial to solv th problm o sand dositions, and thus, minimizing low assuranc challngs in multiphas low.. INTROUCTION Multiphas low involving suspnsions o solid particls ar rquntly ncountrd in many industrial procsss including oil & gas production. Th ability o luids in horizontal motion to suspnd solid particls dnds on th balanc o two actions: gravity, which causs th particls to all or sttl in th luid, and an upward diusion o th particls, causd by a concntration gradint o particls which in turn is cratd by gravity [, ]. Th particl movmnt thus, dnds on th proprtis o th solids such as solids dnsity, particl siz and particl shap. Th gravitational orc causing th particl to ris or all can b dind as []: πd F = ( ρ ρ )g () g Th ris or all o th particls in th luid rsults in a lit & drag orc whr this orc may b xprssd as: ρv πd F = C () In (), ρ p is th particl dnsity, ρ is th luid dnsity, d is th particl diamtr, m, g is th coicint o gravity, C is th drag coicint and v is th ris or all vlocity, m/s. Th drag orc ariss rom prssur p and viscous strsss applid to th particl surac and rsist th rlativ luid vlocity v []. Th magnitud o drag is primarily dictatd by th particl s Rynolds numbr, R, dind as: ρ v d p R = () µ In (), v is th luid vlocity (m/s), d p is th particl diamtr (m) and µ is th luid viscosity (cp). Th particl drag coicint is a vry important hydrodynamic paramtr involvd in th modlling and dsign o multiphas procsss spcially whn ntraind with solid particls. From a hydrodynamic viwpoint, th undamntal aspcts o solid-liquid multiphas low ar intr-phas intraction (i.., intraction btwn th luid phas and th particulat phas) and intra-phas intraction (i.., intraction among solid particls making up th particulat phas). Intr-phas intraction btwn th luid phas and th particulat phas is manistd mainly in th drag orc xrtd on th particls by th luid stram and th transr o momntum rom on phas to anothr []. Svral corrlations or drag coicint hav bn proposd ovr a wid rang o Rynolds numbr in th litratur. On o th most * Corrsponding author, Tl:

2 EVELOPMENT OF A NEW RAG MOEL FOR OIL AN GAS FLUI widly usd is th mpirical quation o Schillr and Numann, which is simpl and ctiv in th rang. < R < 8 [5], xprssd as:.87 C = ( +.5R ) () R Chng 9 [], proposd a drag coicint modl which has gratr applicability whn compard with othr modls valuatd. Ths othr modls can only b usd or limitd Rynolds numbrs and vn thos applicabl or widr rang o R, may involv tdious application procdur. Th modl proposd by Chng [] givn blow, dspit its simpl orm, givs th bst approximation o xprimntal data or R, rom Stok s rgim to about ⁵...8 C = ( +.7R ) +.7[ xp(.r )] (5) R or a wid rang o particl Rynolds numbrs, bcaus th low condition during th procss is highly complicatd []. Evn with th advnt o CF, prorming larg scal numrical study o complx multiphas low rquirs som assumptions and also mpirical data dscribing th intractions btwn th luid and th particls []. This rlationship can gnrally b dtrmind xprimntally by obsrving th sttling vlocitis in still luids [, ].. METHOOLOGY FOR EVELOPMENT OF RAG (C ) CORRELATION Th dinition o th drag orc on a particl in a luid low gnrally involvs th undrstanding o th rlationship btwn th drag coicint C and particl s Rynolds numbr, R. Th drag coicint rrsnts th raction o th kintic nrgy o th sttling vlocity that is usd to ovrcom th drag orc on th particl, whil th Rynolds numbr is a ratio btwn th inrtial and viscous orcs o a luid []. As th particl siz or low vlocity incrass or a givn kinmatic viscosity, so dos th Rynolds numbr, and th charactr o low changs as xprssd in Equation. For vry small Rynolds numbrs, Stoks proposd an analytical solution o drag coicint by solving th gnral dirntial quation o Navir Stoks. This drag coicint C is prdictd rom two trms. Th irst trm on th RHS can b considrd as an xtndd Stoks' law, applicabl approximatly or R < ; and th scond trm is an xponntial unction accounting or slight dviations rom th Nwton's law or high R. Th sum o th two trms is usd to prdict drag coicint or any R ovr th ntir rgim. Availabl drag corrlations in th litratur apply to smooth sphrical particls, or rgularly shapd particls lik disks or cylindrs. But particls ncountrd in th oil and gas production usually ar C = () not smooth sphrs but ar irrgularly shapd and do R not hav smooth suracs [7]. In act, thr is An analytical attmpt to xtnd th rang o rmarkabl dirnc btwn sphrical and nonsphrical particls in th contxt o th mthod in approximation or th drag coicint byond Stoks low was proposd by som authors by including th which thy tnd to commnc motion, givn that inrtia trms in th solution o Navir Stoks as sphrical particls tnd to bgin motion via rolling rortd by []. whras non-sphrical particls prrably commnc C + = R (7) motion via dragging [8]. Anothr drawback is that th R corrlations wr dvlopd rom solids transport A numbr o mpirical and smi mpirical C R situations whr th solids loading is vry high. rlationships, can b ound in th litratur [, ]. Typical sand loading in oshor applications is much Four o th C smallr than most industrial solid-liquid slurry corrlations [5, ] ar shown blow or comparisons bcaus thy ar gnrally considrd to transport, on th ordr o 5- lb o sand pr bbl b o high accuracy and with wid rang o o producd liquid [9]. applicability; quations 8 to. For th cas o low around a sphr, crtain hydraulic analyss rquir dtrmining th drag coicint as.57. C = (8) ( + (.7R ) +. 9 wll as lit coicint as a unction o particl R + R Rynolds numbr. This is tru or th prsnt study, in (9) th dtrmination o minimum transport vlocity C = R modls or suspnsion and rolling. A ky paramtr is R R th particl sttling vlocity within th multiphas..8 C () low in piplin. On challng is that th drag = ( +.7 R) +.7( xp(. R ) R coicint cannot b xprssd in an analytical orm Nigrian Journal o Tchnology, Vol., No., April 5 8

3 EVELOPMENT OF A NEW RAG MOEL FOR OIL AN GAS FLUI C = R.87 ( + (.5 R ) (). EXPERIMENTAL SET UP An xprimntal dtrmination o th drag coicint is otn basd on masurmnt o th trminal sttling vlocity o a sand particl in luid mdium. In th prsnt study, data rom xprimnts conductd or particl sttling vlocity has bn usd to dtrmin th R & C rlationship, s Tabl. Th dosit vlocity or solid-liquid mixtur is th vlocity blow which solid particls will sttl out o slurry to orm a moving bd or a stationary bd. Th dosit vlocity was dtrmind xprimntally both at static and dynamic conditions in ordr to valuat th drag and lit coicints. An xprimntal st-up was dsignd and constructd; s th igital pictur o th vrtical tst column in Figur. In ordr to dtrmin dosit vlocity in static condition, th vrtical pip column was illd with tst luid. Th particl sampl was droppd rom th top o th vrtical tub and obsrvd as it travlld through th liquid column. On th sid o th column ar markd dirnt points. Th tim takn or th particl to travl rom th top to dirnt lvl or distant rom th top was notd and rcordd with th aid o stop watch. This procdur was ratd thr tims or ach sampl particl or singl phas watr, oil and two-phas oil-watr, oil-gas, watr-gas and or thr phas oil-watr-gas. In ach cas gas low rats wr varid and sttling vlocity obsrvd at dirnt gas low rats. Schillr Numann C corrlation is commonly usd as drag corrction xprssion in multiphas lows sinc many particls ar constraind to R valus in this rang. Howvr, it was obsrvd that th main limitation o th modl was its limitd capability or prdicting th laminar low rgion. Th modl was thn modiid spcially to improv on this limitation. This ld to th dvlopmnt o a nw C corrlation basd on th xprimntal data or multiphas, watroil-gas low s Tabl. Th dvlopmnt o th nw C corrlation that is a unction o Rynolds numbr was basd on itting th nwly acquird xprimntal data to th bas quation using MS-Excl program. Th gnric orm o th quation is dscribd as: ( ( ) a () ( ) C =. +.5R 87 + d R b + c R In () R is th particl Rynolds numbr and a, b, c and d ar constants dndnt on xprimntal data. Tabl : Sand dosit xprimntal rsults or C modls Singl-Phas Oil Flow Particl Particl Vlocity, R C n Siz m/s Two-Phas Oil-Air Flow Two-Phas Oil-Watr Thr-Phas Oil-Watr-Air Figur : igital pictur o th vrtical tst column Nigrian Journal o Tchnology, Vol., No., April 5 8

4 EVELOPMENT OF A NEW RAG MOEL FOR OIL AN GAS FLUI Exprimnt Exprimnt Schillr & N Nw Modl C R C R Figur : C or two phas oil-gas low Figur : C or singl phas watr showing improvd corrlation with nw modl Exprimnt Exprimnt C C R R Figur : C or two phas oil-watr low Figur 5: C or thr phas oil-watr-gas low Modl Prdiction C 8 R² =.989 Modl Prdiction C 8 R² = Masurd C Masurd C Figur : Comparison o masurd and modl prdiction C or watr-oil-air low Figur 7: Comparison o masurd and modl prdiction C or singl-phas oil low Solving or ths unknowns, th quation that bst its th xprimntal data was dtrmind and can b xprssd as:.87.5 ( ( ) () ( ) C = +.5 R +. 7 R + 5 R Th xprimntal data was dividd into thr parts; on part was usd mainly or modl dvlopmnt, th scond sts o data wr usd or modl tsting and th third sts o data wr ddicatd to modl validation. In total, svn dirnt glass bad sampls wr usd during th xprimnts. Th prdictions with th modl match th masurmnts quit wll. Only a minor dviation was obsrvd as can b sn in Figur with oil-gas xprimnt. Th RSQ corrlation coicint was 79%. Th dvlopd C modl was an improvmnt on Schillr Numann modl, s Figur. It incorporatd an xtndd Stoks law and th dviations rom th Nwton s law. This can b applid to both th turbulnt and laminar low rgions and rasonably prdict C ovr a wid rang o R in multiphas luid low. Nigrian Journal o Tchnology, Vol., No., April 5 8

5 EVELOPMENT OF A NEW RAG MOEL FOR OIL AN GAS FLUI Exprimnt Hattovy & Nicholas data C C R R Figur 8: C or singl phas oil low Figur 9: Nw C modl prdiction with litratur data C Exprimnt Nw Modl T & Lvnspil Schillr & N S & Ojha Modl Prdiction C 5 R² = R Masurd C Figur : Nw C modl compard with slctd modls rom litratur Figur : Comparison o masurd and modl prdiction C or singl-phas watr low. NEW RAG (C ( ) ) MOEL: TESTING, VALIATION AN COMPARISON In ordr to stablish th rliability and accuracy o th nw modl, it was tstd with othr sts o data. Th rsults or two-phas oil-watr low and thr-phas oil-watr-gas low ar prsntd in Figurs & 5. Th nw drag coicint modl prormd wll with th xprimntal data by th high valus o th RSQ corrlation coicint, which wr 98% & 99% rspctivly, s Figurs and 7. Whn th viscosity is rlativly high, such as in singl phas oil low, th rgim is usually laminar and th Rynolds numbr tnds to b vry small, s Figur 8. Th particl in this cas xprincs a highr drag coicint and low particl sttling rat. Th lowing luid proprtis around th solid particl ar an important actor in dtrmining th drag coicint []. Th RSQ corrlation coicint was 99%, s Figur 7. Indndnt C data [] rom litratur was Th nw modl was qually compard with slctd drag coicints rom th litratur, as shown in Figur, giving bttr corrlation with xprimntal data. In gnral, comparison o th xprimntal rsults and that o th C modl prdictions ar ound to b largly in agrmnt. Tabl shows th statistical paramtr btwn th two masurd drag coicints and drag coicint prdictd with th nw modl. Tabl : Statistical Paramtrs or th rag Coicint Modl rag Coicint Modl Prdictions Error margin Twophasphas Thr- Singlhasphas Singl- oilwatoil-air watr- oil watr Avrag prcntag rror (APE) R-Squar valu (R²) usd to validat th modl, s Figur 9. Thr was 5. CONCLUSIONS xcllnt agrmnt btwn th xprimntal data W invstigatd drag coicints or oil and gas and modl prdiction including publishd data. Th multiphas systms. Analysis clarly showd that RSQ corrlation coicint was 97%. Nigrian Journal o Tchnology, Vol., No., April 5 8

6 EVELOPMENT OF A NEW RAG MOEL FOR OIL AN GAS FLUI singl-phas modls ar inadquat or multiphas low prdiction, and also, mrly xtnding C modls usd in hydraulic transport is quit mislading. Th dvlopd modl in this work, has ctivly addrssd ths concrns. It has bn shown to adquatly prdict or both singl-phas and multiphas low whn compard with xprimntal data. This is vidnt in th rsults prsntd. This nw modl ctivly accounts or th complxitis associatd with multiphas low in pips, and has th potntial to solv th problm o sand dositions, and thus, minimizing low assuranc challngs in multiphas low. 5. REFERENCES. GOVIER, G.W. AN AZIZ, K., 97. Th Flow o Complx Mixturs in Pips. th d. Malabar Florida: Robrt E. Krigr Publishing Company.. LIU, H.,. Piplin Enginring. Lwis Publishr, A CRC Prss Company.. LOTH, E., 8. rag o non-sphrical solid particls o rgular and irrgular shap. Powdr Tchnology, 8, pp. -5. OAN, Q. AN GEORGE, A.E., 998. Flow o Oil and Sand in a Horizontal Wll. Journal o Ptrolum Tchnology, 7(), pp TRAN-CONG, S., GAY, M. AN MICHAELIES, E.E.,. rag coicints o irrgularly shapd particls. Powdr Tchnology, 9, pp. -. CHENG, N., 9. Comparison o ormulas or drag coicint and sttling vlocity o sphrical particls. Powdr Tchnology, 89, pp STEVENSON, P. t al.,. Th Transport o Particls at Low Loading in Nar-Horizontal Pips by Intrmittnt Flow. Chmical Enginring Scinc, 5, pp LASKOVSKI,., STEVENSON, P. AN GALVIN, K.P., 9. Lit and drag orcs on an isolatd cubic particl in pip low. Chmical Enginring Rsarch and sign, 87, pp ANIELSON, T.J., 7. Sand Transport Modling in Multiphas Piplins. Oshor Tchnology Conrnc. May. Houston, USA: SPE. ALMEEIJ, J., 8. rag coicint o low around a sphr: Matching asymptotically th wid trnd. Powdr Tchnology, 8, pp. 8-. ZASTAWNY, M., MALLOUPPAS, G., ZHAO, F. AN WACHEM, B.,. rivation o drag and lit orc and torqu coicints or non-sphrical particls in lows. Intrnational Journal o Multiphas Flow, 9, pp CARMICHAEL, G.R., 98. Estimation o th rag Coicint o Rgularly Shapd Particls in Slow Flows rom Morphological scriptors. Amrican Chmical Socity,, pp. -. CHIEN, S., 99. Sttling Vlocity o Irrgularly Shapd Particls. SPE rilling and Compltion,, pp LOTH, E., 8. rag o non-sphrical solid particls o rgular and irrgular shap. Powdr Tchnology, 8, pp TURTON, R., LEVENSPIEL, O. A short not on th drag corrlation o sphrs, Powdr Tchnology. 7 (98) 8 8. HOTTOVY, J.. AN SYVESTER, N.., 979. rag Coicints or Irrgularly Shapd Particls. Amrican Chmical Socity, 8(), pp. - Nomnclatur CF Computational Fluid ynamics C - rag co-icint R ynolds numbr R - Particl Rynolds numbr Nigrian Journal o Tchnology, Vol., No., April 5 85