PRESSURE AND PRESSURE DERIVATIVE ANALYSIS FOR PSEUDOPLASTIC FLUIDS IN VERTICAL FRACTURED WELLS

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1 VOL. 7, NO. 8, AUGUST 0 ISSN ARN Joual of Egieeig ad Applied Scieces Asia Reseach ublishig Neo (ARN). All ighs eseved..apjouals.com RESSURE AN RESSURE ERIVATIVE ANALYSIS FOR SEUOLASTIC FLUIS IN VERTICAL FRACTURE WELLS Feddy Humbeo Escoba, iego Feado Boilla ad Yuslly Yeih Cicey Uivesidad Sucolombiaa, Aveida asaa Ca, Neiva, Huila, Colombia fescoba@usco.edu.co ABSTRACT eoleum egiees ofe deal ih No-Neoia fluids i may aciviies of he oil idusy. Some of hese fluids ae used as facuig, EOR ad dillig mud fluids. If oe of hese fluids is used o facue a ell ad a posfacue es is u, afeads, he iepeaio cao be coduced ih he coveioal models. A pseudoplasic model has o be used. The oil lieaue peses oly oe o o ell es aalysis fo facued ells ih o- Neoia fluids. The applicaio as specific fo fall-off esig ad he iepeaio fo deemiaio of he halffacued legh is coduced usig boh he coveioal saigh-lie mehod ad ype-cuve machig. Hoeve, applicaios of he pessue deivaive fo his ype of sysems have o bee pefomed ye. This pape peses a mehodology usig he pessue ad pessue deivaive log-log plo fo iepeaio of pessue asie ess i ifiiecoduciviy veically facued ells ih a pseudoplasic fluid. The iepeaio as exeded o esimae he halffacue legh ad he ell daiage aea. I as successfully esed o field ad syheic daa. Keyods: No-Neoia, dilaa, pseudoplasic, esevoi aea, facues legh.. INTROUCTION seudoplasic fluids obey he poe la. The flo idex behavio,, fo hem is smalle ha he uiy. Mos of he fluids i he oil idusy possess a pseudoplasic behavio. The pessue deivaive duig adial flo egime of a pseudoplasic fluid is a saigh lie hich iceases is slope as he flo idex behavio deceases. Iou has bee he mos ousadig eseache o he field of ell es aalysis of o-neoia fluids. His coibuios compise may aalyical soluios fo flo of foam ad ohe o-neoia fluids hough poous maeials as ca be see i Iou (978), Iou (979), Iou ad Ramey (979a) ad Iou ad Ramey (979b). Applicaio of he pessue deivaive o o- Neoia fluids as fis used by Vogvuhipochai ad Raghava (987a) ad exesio of he TS echique fo such fluids as peseed iiially by Kaime-Meidl ad Tiab (00). Escoba, Maiez ad Moealege (00) peseed a TS echique, Tiab (993), o adial composie esevois ih a No-Neoia/Neoia ieface fo pseudoplasic ad dilaas sysems, especively. Vogvuhipochai ad Raghava (987b) peseed he oly o o facued ell fo poe-la fluids. They used a dimesioless pessue soluio povided by Odeh ad Yag (979) o geeae a ypecuve machig ad coveioal mehodology fo ell es iepeaio. The soluio give Odeh ad Yag (979) is also used i his o o geeae pessue ad pessue deivaive behavio fo facue ells ih o- Neoia fluids. As fo he case of Neoia fluids he pessue ad pessue deivaive also displays a half-slope saigh lie i boh cuves. Exesio of he TS echique as used o deemie half-facue legh ad ell-daiage aea.. MATHEMATICAL FORMULATION The dimesioless quaiies ioduced by Iou ad Ramey (979) ae: NN () µ 4.( ) h NN xf A () G 3 (3) Gx G 3 f 3 ( π e ) (4) (5) I hich; φ cµ h G ad, H 3 µ φ ( ( ) )/. (6) (7) 975

2 VOL. 7, NO. 8, AUGUST 0 ISSN ARN Joual of Egieeig ad Applied Scieces Asia Reseach ublishig Neo (ARN). All ighs eseved..apjouals.com Whee suffix N idicaes Neoia ad suffix NN idicaes o-neoia. Escoba e al. (00) peseed moe pacical expessios fo he deemiaio of boh pemeabiliy ad si faco: α α α ( ) ( )( ) ( ) α µ φc h ( * ' ) (8) s α h φµ c ( * ) h l φµ c (9) Whee α is he slope of he pessue deivaive cuve ad is defied by: α (0) 3 beig he flo behavio idex hich may be foud fom he slope of he pessue deivaive cuve duig adial flo egime. The dimesioless pessue deivaive duig adial-flo egime is goveed by: ( * ') 05 NN NN. α () Odeh ad Yag (979) lieaized he paialdiffeeial equaio fo he poblem of a ell ieceped by a veical facue. Thei dimesioless pessue soluio is give belo: ( ) v v (3 ) Γ( ν) ( ) Whee ν (-)/ (3-) () Vogvuhipochai ad Raghava (987b) peseed o iepeaio mehodologies: ype-cuve machig ad coveioal saigh-lie fo chaaceizaio of fall-off ess i veically hydaulic ells ih a pseudoplasic fluid. They idicaed ha a ealy imes, a ell-defied saigh lie ih slope equal o 0.5 o loglog coodiaes ill be evide, he, xf π f xf (3) (4) φ cµ x Equaio () is used hee o develop pessue ad pessue deivaive cuves fo veically ifiiecoduciviy facued ells. A example of a cuve fo 0.5 is give i Figue-. Noice ha he slope of he pessue deivaive duig adial flo egime is geae ha zeo. Zeo value of he pessue deivaive coespods he case fo (Neoia). I is impoa o obseve ha duig he fomaio liea flo o he facue he slope of he pessue ad pessue deivaive is ye 0.5 as fo he Neoia case. The pessue deivaive goveig equaio is: 976

3 VOL. 7, NO. 8, AUGUST 0 ISSN ARN Joual of Egieeig ad Applied Scieces Asia Reseach ublishig Neo (ARN). All ighs eseved..apjouals.com.e+0.e+0 Liea flo egime Radial flo egime * ' xf ad.e+00.e-0 LRi essue deivaive cuve.e-0.e-03.e-05.e-03.e-0.e+0.e+03.e+05.e+0 xf Figue-. imesioless pessue ad pessue deivaive behavio fo a veical ifiie-coduciviy facued ell ih a o-neoia pseudoplasic fluid ih 0.5. * ' 0.5 π xf xf (5) By usig he iesec poi of he pessue deivaive duig liea flo egime, Equaio (5), ih he adial flo egime goveig equaio, Equaio (), called LRi, a expessio o obai he half-facue legh is peseed: x f ( ) LRi α φ cµ LRi φ cµ 3 + (6) Escoba, Vega ad Boilla (0) shoed ha he lae-ime pessue deivaive plo duig pseudoseadysae he expessio goveig he lae-ime pseudoseadysae flo egime is: * ' π A (7) The poi of iesecio of he pessue deivaives duig liea flo ad pseudoseady-sae (o sho hee), called, allos o obai he ell daiage aea. I ohe ods, equaig Equaios (5) ad (7) ad solvig fo he ell-daiage aea leads o he folloig expessio: psi ( * ') 5.06 psi, * ', psi h LRi 0.77 h , hs Figue-. essue ad pessue deivaive fo field example. 977

4 VOL. 7, NO. 8, AUGUST 0 ISSN ARN Joual of Egieeig ad Applied Scieces Asia Reseach ublishig Neo (ARN). All ighs eseved..apjouals.com A π G 3. EXAMLES /(3 ) (8) 3.. Field example Fa (998) peseed a pessue es of a es coduced i a hydaulic facued ell ih he ifomaio give belo. essue ad pessue deivaive daa fo his es is epoed i Figue-. Soluio The folloig ifomaio as ead fom he pessue ad pessue deivaive plo, Figue-, Usig Equaio (0), a value of 0.3 is foud foα. Resevoi pemeabiliy, si faco, half-facue legh ee esimaed ih Equaios (8), (9) ad (6). Thei especive values ae 0.65 md,.4 ad 77 f. Resevoi pemeabiliy ad half-facue legh ae eesimaed by simulaig he es povidig values of 0.65 md ad 776 f, especively; heefoe, he absolue eos fo hese calculaios ae less 0.06 % ad 0.5 %. 3.. Syheic example Figue-3 peses he pessue ad pessue deivaive daa fo a simulaed es of a 50-f log hydaulic facue. Ohe ifomaio is give belo: Soluio Fom Figue-3, he folloig ifomaio as ead: Equaio (0) povided a α value of 0.. The, a si faco of 3.73 as esimaed ih Equaio (9) ad a half-facue legh of 50.0 as foud ih Equaio (6). I his case he absolue eo as of 0.08 %. 4. COMMENTS ON THE RESULTS The esuls obaied fom he oed examples povide vey lo deviaio eos hich cofim ha he developed mehodology os vey ell fo chaaceizig hydaulic facued ells i esevois coaiig o Neoia poe-la fluids. CONCLUSIONS A esimaio of he half-facue legh fom pessue asie aalysis usig he pessue ad pessue deivaive log-log plo fo esevois beaig poe-la o-neoia fluids is peseed fo he fis ime ad successfully esed ih syheic ad field examples. ACKNOWLEGMENTS The auhos gaefully ha he Mos Holy Tiiy ad he Vigi May mohe of God fo all he blessig eceived duig hei lives. Nomeclaue B c C C f h Volumeic faco, RB/STB Sysem oal compessibiliy, /psi Wellboe soage, bbl/psi imesioless facue coduciviy Fomaio hicess, f H Cosisecy (oe-la paamee), cp*s - G Goup defied by equaio 3 G G m q * * Gees φ Miimum pessue gadie, si/f imesioless pessue gadie emeabiliy, md Flo cosisecy paamee Slope Flo behavio idex (poe-la paamee) essue, psi Flo/ijecio ae, STB/ Time, h Radius, f essue deivaive, psi imesioless pessue deivaive Chage, dop oosiy, facio µ Viscosiy, cp Effecive viscosiy fo poe-la fluids, µ cp*(s/f) - Suffices app Appae imesioless 978

5 VOL. 7, NO. 8, AUGUST 0 ISSN ARN Joual of Egieeig ad Applied Scieces Asia Reseach ublishig Neo (ARN). All ighs eseved..apjouals.com A i NN R W imesioless based o aea Effecive Iiial Iesec of liea ad pseudoseady-sae lies Iesec of liea ad liea lies No-Neoia Radial (ay poi o adial flo) Wellboe Tiab Aalysis of essue ad essue eivaive ihou Type-Cuve Machig: - Si ad Wellboe Soage. Joual of eoleum Sciece ad Egieeig, : 7-8. Also ape SE 543, oducio Opeaios Symposium held i Olahoma Ciy, OK, USA. pp Vogvuhipochai S. ad Raghava R. 987a. Well Tes Aalysis of aa omiaed by Soage ad Si: No- Neoia oe-la Fluids. SE Fomaio Evaluaio, ecembe. pp REFERENCES Escoba F.H., Maiez J.A. ad Moealege-M. Mailde. 00. essue ad essue eivaive Aalysis fo a Well i a Radial Composie Resevoi ih a No- Neoia/Neoia Ieface. CT ad F. 4(): ecembe. Vogvuhipochai S. ad Raghava R. 987b. essue Falloff Behavio i Veically Facued Wells: No- Neoia oe-la Fluids. SE Fomaio Evaluaio, ecembe. pp Escoba F.H., Vega L.J. ad Boilla L.F. 0. eemiaio of Well-aiage Aea fo oe-la Fluids by Tasie essue Aalysis. Se o CT ad F o eques fo publicaio. Fa Y A Ne Iepeaio Model fo Facue- Calibaio Teames. SE Joual. pp Jue. Iou C.U Tasie Flo of No-Neoia oe-la Fluids i oous Media. h.. disseaio. Safod U., Safod, CA. Iou C.U acical Applicaio of No-Neoia Tasie Flo Aalysis. ape SE 835 peseed a he SE 64 h Aual Techical Cofeece ad Exhibiio, Las Vegas, NV, Sep Iou C.U. ad Ramey H.J. J. 979a. Tasie Flo of No-Neoia oe-la fluids Though i oous Media. Soc. e. Eg. Joual. pp Jue. Iou C.U. ad Ramey H.J. J. 979b. Wellboe Soage ad Si Effecs duig he Tasie Flo of No- Neoia oe-la fluids Though i oous Media. Soc. e. Eg. Joual. pp Jue. Kaime-Meidl I. ad Tiab. 00. Aalysis of essue Tasie Tes of No-Neoia Fluids i Ifiie Resevoi ad i he esece of a Sigle Liea Bouday by he iec Syhesis Techique. ape SE 7587 pepaed fo peseaio a he 00 SE Aual Techical Cofeece ad Exhibiio held i Ne Oleas, Louisiaa, 30 h Sepembe-3 d Ocobe. Odeh A.S. ad Yag H.T Flo of o-neoia oe-la Fluids Though i oous Media. Soc. e. Eg. Joual. pp Jue. 979

Supplementary Information

Supplementary Information Supplemeay Ifomaio No-ivasive, asie deemiaio of he coe empeaue of a hea-geeaig solid body Dea Ahoy, Daipaya Saka, Aku Jai * Mechaical ad Aeospace Egieeig Depame Uivesiy of Texas a Aligo, Aligo, TX, USA.