Zone 1. Pseudopressure and Pseudopressure Derivative behaviors. Mathematical model. Zone 2

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1 INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84) Inepeon o pessue ess n hydulclly cued wells n b-zonl gs esevos Inepecón de puebs de pesón en pozos hdáulcmene cudos pclmene penedos en ycmenos de gs b-zonles F. H. Eob, Y.. Zho nd. H. Zhng 3 ABSTRACT ue o he ecen ncese n he cung o low pemebly omons, mhemcl wellboe d nepeon hs become mpon o genee mhemcl models o sudy he pessue behvo nd omule nepeon mehodologes o moe ccue chcezon o gh hydocbon-beng omons. Ths ppe pesens n nlycl mehodology o nepeon usng he pseudopessue nd pseudopessue devve log-log plo o he chcezon o hydulclly cued (plly o ully peneng) vecl wells compleed n b-zonl gs esevos. The mehodology povded uses chcesc pons nd lnes ound on he pseudopessue nd pseudopessue devve plo so h new nlycl epessons nd coelons e developed o esme hl-cue lengh, cue peneon o, mobly nd sovy os, dl pemebly, sze o he nne zone, well dnge e, vecl pemebly nd sn co. The new epessons wee successully veed nd esed wh synhec emples. Keywods: essue devve, cued wells, mobly o, sovy o. RESUMEN d l cecene demnd en el cumeno de omcón de bj pemebldd, l nepecón memác de dos de pozos se h ondo mpone p l genecón de modelos memácos que esuden el compomeno de l pesón y se omulen meodologís p un ccezcón más ec de omcones peds que conenen hdocbuos. o ende, en ese ículo se pesen un meodologí p nepe puebs de pesón en ycmenos gsíeos b-zonles dendos po un pozo vecl hdáulcmene cudo (penecón pcl o complee) que ulz el gáco log-log de pseudopesón y devd de pseudopesón. meodologí dd us línes y punos cceíscos hlldos en dcho gáco de modo que se desollon nuevs epesones y coelcones p longud med de cu, elcón de penecón de l cu, elcones de movldd y lmcenje, pemebldd dl, mño de l zon neo, áe de denje, pemebldd vecl y coes de dño. s epesones nuevs se vecon eosmene medne ejemplos snécos. lbs clve: evd de pesón, pozos cudos, elcón de movldd, elcón de lmcenje. Receved: Augus h 03 Acceped: Febuy h 04 Inoducon 3 Hydulc cung hs been used eensvely ove he ps 50 yes o smule low pemebly hydocbon wells. Cuenly, cung s vey cve o hozonl wells n gs shle omons. The sudy o he pessue behvo o cued wells s o ge mponce o he peoleum ndusy. Tnsen ess n wells n low-pemebly gs omons wee conduced by ee nd Holdch (98) usng boh convenonl nlyss nd ype-cuve mchng. They lso pesened sevel eld emples. The mponce o cung low pemebly gs esevos on he ecovey co o gs sysems ws ddessed by emon, el nd empsey (974). Rghvn, Ue nd Thoms (976) pesened n nlycl soluon o sudy he pessue behvo o vecl cued wells nsde o n nne-sze esevo when he well does no penee he ene py zone. They povded ype cuves o he nepeon o pessue ess n such sysems o cses on unom-lu nd nne-conducvy cues. Holdch, ee nd Gs (983) used convenonl nlyss o esmng esevo nd cue pmees om pessue buldup ess usng convenonl echnques by noducng n eve echnque. Rodguez, Hone, nd Cnco-ey (984) noduced sem-nlycl soluon o sudy he nsen low behvo o plly peneng neconducvy vecl cue. Fo such sysems dung he elyme peod, he low behvo s equvlen o ully peneng Feddy Humbeo Eob. h.., Olhom Unvesy, USA. Alon: oesso, Unvesdd Sucolombn/CENIGAA, Colomb. E-ml: Yu ong Zho. h.., Se Key booy o Ol nd Gs Resevo Geology nd Eploon, Souhwes eoleum Unvesy, Chn. Alon: ecue, Souhwes eoleum Unvesy, Chn. E-ml: 3 e Hu Zhng. h.., Alon: Se Key booy o Ol nd Gs Resevo Geology nd Eploon, Souhwes eoleum Unvesy, oesso, Souhwes eoleum Unvesy, Chn. How o ce: Eob, F. H., Zho,. Y., & Zhng,. H. (04). Inepeon o essue Tess n Hydulclly Fcued Wells n B-Zonl Gs Resevos. Ingeneí e Invesgcón, 34(),

2 ESCOBAR, ZHAO AN ZHANG cue. In hs peod, ehe blne o lne low peods e obseved, dependng upon cue conducvy. Tb (994) nd Tb e l. (999) pesened nepeon echnques o nne- nd ne-conducvy cues n ol wells bsed on he pessue nd pessue devve plo whou usng ype-cuve mchng. An eenson o he ome wo o gs wells ws peomed by Nunez-Gc, Tb nd Eob (003). Fo cued wells n compose esevo, Chu nd Shn (993) pesened mhemcl model o boh ne-conducvy o unom-lu veclly cued wells whn compose esevo. Chen nd Rghvn (995) developed model o cued well poducng n compose esevo. They lso ddessed some compuonl ssues dung he compuon o he poducs In() Kn(). Feng e l (009) poposed seepge low model o cued heeogeneous compose esevo usng he equvlen lowng essnce mehod. Howeve, ew o hese uhos consdeed he eec o wellboe soge nd sn cos n he models. They lso ecluded he pl peneon eec o he hydulc cue. Zho e l. (03) pesened novel nlycl soluons o boh plly o ully peneng vecl cued wells n compose (b-zonl) gs esevo, s depced n Fgue. The omulon used he connuous pon souce uncons n n nsoopc esevo on he bss o souce uncon heoy long wh he plce nsomon mehod nd he uhmel s pncple. The nlycl soluons wee obned usng he consucng uncon nd he connuous pon souce uncons o wells n b-zonl gs esevos wh uppe nd lowe boundes closed. K0( s) K0( m s) K0 ( m s ) - m( ) K0( m s) K0( m s ) d s () - I0( s) I0 ( m s) K0( m s ) - I0( m s) K0 ( m s ) Fo he cse o plly peneng cue well, Zho e l. (03) lso pesened he ollowng soluon: m( ) K0 ( - ) y s s - 0 I0 ( - ) y s n I0 ( - ) y sn 4h h n K0 ( - ) y sn z h h cos( n ) cos( n ) sn( n ) d h h h zm n () n ( n 0,,,...) (3) n K ( s ) K ( s ) n 0 m n 0 m K ( s ) K ( s ) 0 m n 0 m I ( s ) K ( s ) n 0 m n 0 m I ( s ) K ( s ) 0 m n 0 m n n n n (4) (5),,( c ) w,,( c ) m Zone Zone Fgue. Sech o b-zonl gs esevo The pesen wo uses he soluons pesened by Zho e l. (03) o undesnd pseudopessue nd pseudopessue devve behvos so sevel unque eues on hs plo e used o genee sevel epessons o chcezng he esevo. The epessons h we developed wee esed successully wh synhec emples. Mhemcl model Zho e l. (03) pesened n nlycl soluon o he pessue esponse o ully peneng hydulclly cued gs well n b-zonl esevo, e n π sn s (6) h s s (7) seudopessue nd seudopessue evve behvos Alhough Zho e l. (03) epln he pessue behvo o he sysems unde consdeon, s mpon o esblsh he mn low egmes h cn be pesened n cued wells whn bzonl esevos. The dmensonless pseudopessue nd pseudopessue devve plo s shown n Fgue. The dshed lnes coespond o cue h penees hl o he esevo hcness, whle he sold lne s o ully peneng cue. These cuves llow he cle obsevon o wellboe soge eecs vey ely me. Then, he omon lne low egme o he hydulc cue s obseved, chcezed by slope o one-hl on he pseudopessue devve cuve. The b-dl low egme my lso be pesened n cued wells hvng nne conducvy, s shown n Fgue, obseved once lne low vnshes. Fo he plo povded, b-dl low occus dung dmensonless pseudome beween 0.0 nd nd s ecognzed by slope o 0.36 on he devve cuve. Fo he cse o ully peneng cue, dl low ollows once he cue eecs e no longe el n he es. Howeve, s obseved n he dshed cuves o Fgue coespondng o plly peneng cue, sphecl low es plce. The sphecl low possesses negve one-hl slope on he devve cuve. In boh cses, dl low n zone develops dmensonless pseudome o ppomely 0, ollowed by hozonl pen o he devve nown s dl low egme n zone (see Fgue 3). A he end INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84) 77

3 INTERRETATION OF RESSURE TESTS IN HYRAUICAY FRACTURE WES IN BI-ZONA GAS RESERVOIRS o he es (_ > ~0000), second pleu s obseved coespondng o he dl low egme n zone. As Fgue 3 shows, dependng on he vlue o M, he second pleu wll be bove o below he level o he dl low egme dung zone. Fo vlues o M >, he second pleu s hghe hn he s pleu nd vce ves. m () nd *m (' ) _.E+0 00.E+0 0.E+00.E-0 0..E Bdl low Sphecl low Rdl low n zone Rdl low n zone Wellboe ne.e soge low m 50, C 0.000, s 0., M, W 0-6.E-06.E E E E-0.E-0 0-.E+00.E+0 0.E+0 00.E E E E _ Fgue. mensonless pseudopessue nd pseudopessue devve vs. dmensonless pseudome log-log plo dsplyng he ypcl behvo o boh ully nd plly peneng cued well n b-zonl esevo m () nd *m (' ) _.E+0.E+0.E+00.E-0.E-0 M m 00, C , s 0.0, W.E-04-4.E-03.E E-0 0-.E+00.E+0 0.E+0 00.E E E E E _ Fgue 3. Eec o mobly o on he dmensonless pseudopessue nd pseudopessue devve behvo o ully peneng cued well n b-zonl esevo m () nd *m (' ) _ E+0 00.E+0 0.E+00.E-0 0. W , C , s 0.0, M m.e E E E-0 0-.E-0 0-.E+00.E+0 0.E+0 00.E E E E E E _ Fgue 4. Eec o sovy o on he dmensonless pseudopessue nd pseudopessue devve behvo o ully peneng cued well n b-zonl esevo Anohe emble eue s povded n Fgue 4. I he sovy o s smlle hn uny, he devve dsplys mnmum, hen connues o he second pleu. Fo sovy os gee hn one, he devve dsplys mmum pon beoe omng he second pleu. As he vlues e he hn one, boh he mmum nd mnmum vlues e moe ponounced. Noce h he deences o he dmensonless wellboe soge coecen vlues nd dmensonless d n Fgues o 4 do no epesen ny eec on he govenng equons o he geneon o he TS Technque. mensonless mees The dmensonless qunes e dened s: m m (8) cg cg (9) M (0) y () y () z z z (3) wh s he eecve pemebly o he b-zonl esevo, h h (4) hzm h e z hzm (5) h (6) h (7) h Fo compessble luds, he pseudopessue, m(), noduced by Agwl (949) s gven by: m( ) d ( ) Z( ) (8) 0 Agwl (949) lso noduced he pseudome uncon o ccoun o he me dependence o gs vosy nd ol sysem compessbly: d ( ) c ( ) (9) o seudome s bee dened s uncon o pessue s new uncon gven n h ps/cp: p ( d / d) ( ) d ( ) c ( ) (0) po seudome nsed o goous/noml me s used n hs wo becuse Eob e l. (0) demonsed h he hydulc cue pmees e bee esmed usng pseudome hn goous me. The sysem unde consdeon s ssumed o possess plly peneng hydulc cue n vecl well h hs hllengh,, wdh, w, nd pemebly,. The dmensonless pseudome, pseudopessue nd pseudopessue devve e dened s: _ A A () 78 INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84)

4 ESCOBAR, ZHAO AN ZHANG _ _ w m h m m () (3) (4) 4.5qT * m ' _ * ' h m (5) 4.5qT Inne-conducvy cues ne low egme The nepeon mehodology ollows he phlosophy o he Tb s ec Synhess echnque noduced by Tb (993). Rodguez e l. (984) pesened n nlycl soluon o he well pessue behvo o plly peneng cue h s dped hee s: m ( ) _ And he pseudopessue devve s (6) * m( )' 0.5 (7) Ae eplcng he dmensonless qunes gven by Equons 3 o 5 no Equons 6 nd 7, he esuls shown below e obned. whee m( ) m ( ) (8) [ ( )* m( )'] 0.5 m ( ) (9) m qt h (30) Solvng o om Equons 8 nd 9 yelds; qt ( ) h[ m( ) ] 0.47 qt ( ) h[ * m( ')] Equons 3 nd 3 llow he esmon o he hl-cue lengh usng ehe he vlue o deenl pseudopessue o pseudopessue devve ed ny by pseudome pon dung he lne low egme. Epolon o he pseudopessue devve he pseudome vlue o h ps/cp s ecommended. Red he pseudopessue devve pseudome vlue o h ps/cp. Unde hese condons, Equon 3 becomes: 0.47qT h[ * m( ')] I s peeed o leve he em n Equons 3 o 3 so h es he vlue o one o ully peneng hydulc cues. (3) (3) (33) The soluon o he dusvy equon dung he dl low egme o he nenl zone, zone, s: * m( )' 0.5 _ (34) Fom such n equon, Eob, opez nd Cnllo (007) developed he equons o pemebly nd sn co, ncludng pseudome: 7.6qT h m * ' * ' m s ' 0.5 ln 7.43 m w Fgue 3 shows h he soluon o he dusvy equon dung he dl low egme o zone eled o he dl low egme o zone s: * m( )' 0.5M (35) (36) _ (37) Fom whch he pemebly o zone s ound s: 7.6qTM h m * ' Alhough he pseudosedy se egme s no shown n he plos gven, he govenng equon dung he le pseudosedy-se egme s used by Eob e l. (007) s gven by: _ A _ A (38) * m ' (39) An epesson om Equon 39 obns he esevo dnge e om n by pessue devve pon dung he pseudosedy se: qt pss * '.357 A h m Agn, s bee o epole he pseudopessue devve pseudome o h ps/cp. The esevo dnge e cn lso be deemned om he necep o Equon 39 wh Equons 34 nd 37: pss (40) p A (4) p A (4) 30.77M The hl-cue lengh cn lso be veed o e-esmed om he necep pon beween he lne nd he wo dl lnes, v.g. Equons 9, 34 nd 37: ( ) (43) ( ) 34.74M (44) INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84) 79

5 INTERRETATION OF RESSURE TESTS IN HYRAUICAY FRACTURE WES IN BI-ZONA GAS RESERVOIRS I s necessy o chnge by when M<. Addonlly, he vecon o he hl-cue lengh om he necep pon beween he lne nd pseudosedy-se lnes, v.g. Equons 9 nd 39: A (45) ( ) p Inne-conducvy cues B-dl low egme The govenng dmensonless pessue nd pessue devve equons pesened by Tb (994) dung b-dl low ppled o gs sysems wee enged by Eob e l. (0) o gs low: e m _ A 0.7 e * m' _ A _ A Once he dmensonless qunes gven by Equons, 4 nd 5 e eplced no Equons 46 nd 47 nd hen solved o he hl-cue lengh, he ollowng equons e obned: Beng: G BR 0.694e * m' BR G BR.8695e mbr G BR 73.4qT h A Equon 47 s se equl o 0.5 nd 0.5 M dung he dl low egmes o zones nd, especvely, coespondng o he nesecon pon o he dls nd b-dl lnes. Then, he ollowng equons e deved: (46) (47) (48) (49) (50) BR (5) BR ( M) (5) The nesecon pon o he b-dl nd le pseudosedy-se lnes, Equons 47 nd 39, leds o nohe epesson o nd he esevo pemebly: A e BRp I s lso necessy o chnge by when M<. The nesecon beween he lne nd b-dl lnes, Equons 7 nd 47, poduces he ollowng epesson:.5 (53) ( ) BR (54) 39 Sphecl low egme The nlycl soluon pesened by Joseph (984) nd eended o gs low wh he pseudome uncon condon s gven below: h h m( ) (55) 3/ ssp 3/ sp sw sp w _ Whch pseudopessue devve s: * m( )' 3/ h (56) 4 3/ sp w _ Moncd e l. (004), usng he soluon pesened by Joseph (984), povded n nepeon echnque usng he pessue devve cuve o sphecl nd hemsphecl low o ol nd gs esevos. The soluon o gs usng goous me s eended hee o pseudome, so he sphecl pemebly nd he sn co cused by sphecl low cn be esmed usng ny by pseudome pon dung sphecl low on whch vlues o pseudopessue nd pseudopessue devve cuves e ed o be used n he ollowng epessons: In wch; sp qt ( * m ( )') ( ) sp sp m ( ) sw sp ssp sp ( ) sp * m( )' sp 3 3 sp y z h v /3 (57) (58) (59) sw h (60) ln h / The nesecon o he dl lows nd he sphecl low lnes,.e., Equons 34, 37 nd 56, obn: sp w h ( ) (6) sp 3 sp h ( ) (6) 3 sp I s lso ound n hs wo h he nesecon pon omed by he lne nd he sphecl low lnes, Equons 7 nd 56, povde new epesson o esme he hl-cue lengh: 3/ sp ( ) sp (63) h The pon o necepon beween Equons 47 nd 56, b-dl nd sphecl low lnes, llows obnng nohe epesson o he esmon o he hl-cue lengh:.0834 sp ( ) BRsp ( h).9445 (64) 80 INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84)

6 ESCOBAR, ZHAO AN ZHANG Tb (003) developed he ollowng equon o ele he hlcue lengh, omon pemebly, cue conducvy nd pos-cue sn co: w s e.93 w The deemnon o he sn cue eques unnng wo low ess deen gs low es. A sngle es povdes he pseudosn co s gven by Equon 36. Wellboe Soge Eecs I wellboe soge eecs e pesened, he soluon o he elyme dusvy equon (Tb (993)) educes o: C (65) m ( ) (66) C m'( ) C Fo whch he dmensonless wellboe soge s gven by: C C hcw Epessons o esme he wellboe soge coecen e ound by eplcng he especve dmensonless qunes no he bove equons so h: ( ) C 0.49qTc m ( ) ( ) C 0.49qTc [ ( )* m( )] The nesecon o he ely un-slope lne, Equon 68, wh he dl hozonl sgh lne, Equon 34, gves: C 0.5 Fom hs epesson, n equon o esme ehe pemebly o wellboe soge s obned once he dmensonless pmees e eplced: N N (67) (68) (69) (70) (7) 695cC ( ) us (7) h Fuhemoe, nohe epesson s obned om he nesecon pon o he un-slope lne, Equon 68, wh he dl low o zone, Equon 37, o gve: 695cCM ( ) us (73) h Mobly o, Inne dus nd Sovy Fgue 3 shows h hee ess elonshp beween he mobly o nd he pessue devve n he second pleu. Ths empcl elonshp wh coelon coecen (R ) o s gven by: Fomng he dl low o zone eques lmos wo log cycles h nvolve mny esng mes. Thee my be cses n whch he second pleu omed by he dl low n zone my no be obseved o s oo nosy. In such cses, s useul o use he nlecon pon omed beween he wo dl low lnes. Ths nlecon pon s nomlly dcul o see. A bee opon s o e he second devve nd ele hs second devve o he vlue o he pessue devve o he dl low n zone. Ths vlue wll be used n coelon wh coelon coecen (R ) o o deemne he vlue o M so M 3.85R , M (75) Anohe coelon wh coelon coecen (R ) o s ound below. M.4R.3347 R.074, M (76) Fgue 5 llows he obsevon o he duon o he dl low o zone s uncon o he dus o zone. The dus cn be esmed wh he ollowng coelon h possesses coelon coecen (R ) o nd uses he endng me o he dl low egme n zone ( ) e ( ) e m.79686* Fnlly, wh obsevons om Fgue 4, s possble o esblsh elonshp (usng moe cuves hn he cuves gven n he plo) beween ehe he mmum o mnmum, he mobly o, M, nd he sovy o, s ollows. (77) W *( R ) b (78) Whee consns nd b e epessed s: M M * M *, 0.5 (79) b M M M * *, 0.5 (80) 4.040* M 0.005, M 0.5 (8) b.584* M 0.30, M 0.5 (8) An vege coelon coecen o s ound o he se o empcl Equons 78 hough 8. m () nd *m (' ) _.E+0 00.E+0 0.E+00.E-0 0. m C , s 0.0, M, W.E E E E-0 0-.E-0.E+00.E+0 0.E+0 00.E E E E E E _ Fgue 5. Eec o he sze o he dus o zone on he dmensonless pseudopessue nd pseudopessue devve behvo o ully peneng cued well n b-zonl esevo ( * m( )' ( ( )* m( )' M ( * m ( )' ( ( )* m ( )',, (74) Fgue 6 llows he obsevon o nohe mpon eue. As he peneon o, h/h, eeed o hee s, educes s vlue, he devve dsplys hunch beoe he dl low egme INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84) 8

7 INTERRETATION OF RESSURE TESTS IN HYRAUICAY FRACTURE WES IN BI-ZONA GAS RESERVOIRS ss. The mmum vlue o he pseudopessue devve ws coeled gns, gvng he epesson below h possesses coelon coecen (R ) o : m () nd *m (' ) _.E+0 0.E R (83) Mmum pons 3 R3 C , s 0., M, W.E-0 0..E E-0 0-.E-0.E+00.E+0 0.E+0 00.E E E E _ Fgue 6. Eec o he cue peneon on he dmensonless pseudopessue devve behvo o plly peneng cued well n b-zonl esevo m(), () *m(' ) nd [ ()*m(' )]', ps /cp.e m ( ) E Mmum ( ) BR 30pons ( )* m( )' sp.e E ( ) 90 ( ) 38 m ( ) ( ) sp 40 ( ) 760 sp ( ) BRsp 760 ( ) 3400 sp ( ) BR 8 ( ) sp 905 ( ) 75 ( ) 380 m ( ) sp ( )* m( )' ( )* m( )' ( ) e ( )* m( )' ' n 6790 ( )* m( )' 0400 ( ) E E-03.E-0.E-0.E+00.E+0 0.E+0 00.E E+04.E+05.E+06.E+07.E+08.E+09 (), h h ps/cp Fgue 7. seudopessue, pseudopessue devve nd second pseudopessue devve vs. pseudome o emple Synhec Emples Tble. mees o emples BR Vlue mee Emple Emple, md 5, % 4 7 h, q, M/ Emple T, R , w, m, M 0.5 W 0.5 The d povded n he second column o Tble wee used o smule pessue es whee pseudopessue, pseudopessue devve nd second pseudopessue devve gns pseudome e gven n Fgue 7. eemne he esevo pemebly, mobly o, pseudosn cos, cue lengh nd he dus o zone. Soluon. Sevel low egmes e obseved n hs es. In chonologcl ode, hese low egmes e lne, b-dl, mmum cused by he pl peneon eec o he cue, dl low n zone, nson peod nd nl dl low egme n zone. The ollowng nomon ws ed om Fgue 7, nd uns e no epoed o spce-svng puposes: ()= 380 h*ps/cp m() = ps /cp ()* m() = ps /cp ()* m() = ps /cp [()* m() ] n= 6790 ps /cp ()* m() m = ps /cp ()= h*ps/cp m()= ps/cp ()* m() = 0400 ps /cp () = 38 h*ps/cp () = 90 h*ps/cp ()sp = 3400 h*ps/cp ()sp = 600 h*ps/cp ()sp = 905. h*ps/cp m()sp= ps /cp ()* m() sp = 0400 ps/cp ()sp = 40 h*ps/cp ()spbr = 40 h*ps/cp ()e = h*ps/cp ()BR = 75 h*ps/cp ()BR = 30 h*ps/cp Tble. Resuls o Emple mee Vlue Equon No., md 5 35, md M R M sp, md sp, md sp, md s ssp sw,.4 60, , 5.8 3, , , , , , , m, Usng he pseudopessue devve vlue dung dl low o zone n Equons 35 nd 38, vlues o = 5 nd = 9.98 md wee ound. Then, he mmum pon ound e he b-dl low egme dvded by he pseudopessue devve vlue ed dung he dl low egme o zone llows o he esmon o pseudopessue devve o, R3, o Ths o leds o clculon o vlue o by mens o Equon 77. The o beween he wo vlues o he dl low egmes povdes vlue o M o.9947 wh Equon 74. The by vlues o pseudome, pseudopessue nd pseudopessue devve dung he lne low egme e used o clcule hl-cue lengh,, o 4.97 nd 5.8 usng Equons 3 nd 3. The esmon o ohe pmees long wh he equons used s gven n Tble. 8 INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84)

8 ESCOBAR, ZHAO AN ZHANG Emple Fgue 8 pesens pseudopessue nd pseudopessue devve vesus pseudome smuled wh he d povded n he hd column o Tble. The esmon o esevo pemebly, pseudosn co, hl-cue lengh, sovy nd he dus o zone s equed. Tble 3. Resuls o Emple mee Vlue Equon No., md.05 35, md M s , , , , , , , m, R W Soluon. Fgue 8 shows h he ollowng low egmes e plce: lne, dl, dl low n zone, hen mnmum low cused by he sovy o ollowed by he dl low n he second zone. The ollowng pmees, whch e useul o he clculons, e obned om Fgue 8. ()= 9840 h*ps/cp m() = ps /cp ()* m() = ps /cp ()* m() = ps /cp ()* m() m_mn = ps /cp ()= h*ps/cp m()= 405 ps /cp ()* m() = 6780 ps /cp () = 7800 h*ps/cp () = 400 h*ps/cp ()e = h*ps/cp ()BR = h*ps/cp ()BR = 9300 h*ps/cp m() nd, () *m(' ) ps /cp.e E E E m ( ) 409 ( ) ( ) 7800 ( ) 400 BR ( )* m( )' 6780 ( ) 9840 ( ) 9300 ( ) m ( ) ( )* m( )' 5700 ( )* m( )' ( )* m( )' m_ mn.e E+00.E+0 0.E+0 00.E E E+05.E+06.E+07.E+08.E+09.E+0.E+.E+ (), h ps/cp Fgue 8. seudopessue, pseudopessue devve nd second pseudopessue devve vs. pseudome o emple As shown n emple, he s pmees o be clculed e he pemebles n boh zones by usng Equons 35 nd 37. Then, n esmon o he mobly o wh Equon 74 s equed. The esuls o hese clculons e epoed n Tble 3 long wh some ohe vlues. BR Anlyss o Resuls Vey close geemen ws ound n he esmon o he hl-cue lengh, pemebly, nd mobly o o he gven emples comped wh he cul vlues. By smple nspecon, smll deences e obned. Alhough he esmon o he sovy o s no ec becuse o s sensvy, he uhos gee h he vlues povded by he empcl epessons e ccepble. Concluson Sevel new epessons wee obned o chcezng pessue ess un n b-zonl gs esevo dned by hydulclly cued well consdeng ehe ull o pl peneon. The esmed pmees wee soy comped o cul vlues deemned by wong synhec pessue ess. The nepeon echnque povded llows o he he vecon o some pmees. A mnmum, en deen epessons o esmng hl-cued lengh wee povded. emebly nd mobly cn lso be veed. Acnowledgmens The uhos hn he sponsoshp o The Nonl Scence Fund o sngushed Young Schols o Chn (Gn No. 5509), he Cene o Resech n Scences nd Geo-Ago-Envonmenl Resouces (Ceno de Invesgcón en Cencs y Recusos Geogombenles), CENIGAA, nd Unvesdd Sucolombn. Reeences Agwl, G. (979, Sep.). Rel Gs seudo-me New Funcon o essue Buldup Anlyss o MHF Gs Wells (SE 879). pe pesened he 54h echncl coneence nd ehbon o he Socey o eoleum Engnees o AIME held. s Vegs, NV. Chen, C. C., & Rghvn, R. (995). Modelng cued well n compose esevo. SE Fomon Evluon, (0), Chu, W. C., & Shn, G.. (993).A new model o cued well n dl, compose esevo (ncludes ssoced ppes 799, 8665 nd 9). SE Fomon Evluon, (8), 5-3. Eob, F. H., ópez, A. M., & Cnllo, J. H. (007). Eec o he seudome Funcon on Gs Resevo nge Ae eemnon., CT&F Cenc, Tecnologí y Fuuo, 3(3), 3-4. Eob, F. H., Mnez,. Y., Méndez,. J., & Bonll,. F. (0). seudome Applcon o Hydulclly Fcued Vecl Gs Wells nd Heeogeneous Gs Resevos Usng he TS Technque. Jounl o Engneeng nd Appled Scences, 7(3), Feng, J.., uo, R.., Chen,. S., Chng, Y. W., & Yu,. J. (009, Mch). A compose seepge model o cued esevo (SE 955). pe pesened SE Annul Techncl Coneence nd Ehbon. Bhn. Holdch, S.A., ee, J., & Gs, S. R. (983, My). An Impoved echnque o Esmng emebly, Fcue engh, nd Fcue Conducvy om essue-buld Tess n ow-emebly Gs Wells (pp ). Socey o eoleum Engnees o AIME. Joseph, J. A. (984). Unsedy-Se Cylndcl, Sphecl nd ne low n oous Med. h.. dsseon, U. o Mssou-Roll. ee, J., & Holdch, S. A. (98). Fcue Evluon wh essue Tnsen Tesng n ow emebly Gs esevos. Jounl o eoleum Technology, emon, R. F., el, H. J., & empsey, J. R. (974). Eecs o Fcue nd Resevo mees on ecovey om ow emebly Gs Resevos (SE 5). Moncd, K., Tb,., Eob, F. H., Monelege-M, M., Chcon, A., Zmo, R. A., & Nese, S.. (005). eemnon o Vecl nd Hozonl emebles o Vecl Ol nd Gs Wells wh l Compleon nd l eneon usng essue nd INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84) 83

9 INTERRETATION OF RESSURE TESTS IN HYRAUICAY FRACTURE WES IN BI-ZONA GAS RESERVOIRS essue evve los whou Type-Cuve Mchng. CT&F Cenc, Tecnologí y Fuuo, (6), Nunez-Gcí, W., Tb,., & Eob, F. H. (003, Mch). Tnsen essue Anlyss o Vecl Gs Well Ineseced by Fne- Conducvy Fcue (SE 8095). pe pesened SE oducon nd Opeons Symposum. Olhom Cy, Olhom, USA. Rghvn, R., Ue, A., & Thoms, G. W. (976, Oc.). Vecl Fcue Hegh: Eec on Tnsen Flow Behvo (SE 606). pe s pesened he SE-AIME Annul Fll Techncl Cone- Rodguez, F., Hone, R. N. & Cnco-ey, H. (984, Apl). lly eneng Fcues: essue Tnsen Anlyss o n Inne Conducvy Fcue (SE 743). pe pesened he 984 Clon Regonl Meeng. ongbech, CA. Tb,. (994). Anlyss o essue evve whou Type-Cuve Mchng: Veclly Fcued Wells n Closed Sysems. Jounl o eoleum Scence nd Engneeng,, Tb,. (993). Anlyss o essue nd essue evve whou Type-Cuve Mchng: - Sn nd Wellboe Soge. Jounl o eoleum Scence nd Engneeng,, 7-8. Tb,. (003). Advnces n pessue nsen nlyss TS Technque. ecue Noes Mnul. Nomn, Olhom, USA: The Unvesy o Olhom. Tb,., Azzougen, A., Eob, F. H., & Beumen, S. (999, Mch). Anlyss o essue evve o Fne-Conducvy Fcues by he ec Synhess Technque (SE 50). pe pesened he 999 SE Md-Connen Opeons Symposum. Olhom Cy, OK. Zho, Y.., Zhng,. H., Hu, S. Y., Zho, J. Z., & Zhng, B. N. (04). Tnsen pessue nlyss o cued well n b-zonl gs esevos. pe sen o CT&F o eques publcon. Nomenclue A Resevo dnge e, C Wellboe soge coecen, M/ps R [()* m() ] n/[ ()* m() ] R [()* m() ]m_mn/[ ()* m() ] R3 [()* m() ]m/[ ()* m() ] c Compessbly, /ps h Fomon hcness, h Fcue hegh, hzm snce om he cue mddle pon o he esevo s boom boundy,, Rdl/hozonl pemebly o zone, md, Rdl/hozonl pemebly o zone, md w Fcue conducvy, md- h Hozonl pemebly o zone, (y)0.5 md sp Sphecl pemebly o zone, md -decon pemebly, md y y-decon pemebly, md v, z Vecl pemebly o zone, md e Resevo lengh, M Mobly o, / m() seudopessue uncon, ps /cp m() seudopessue n plce domn essue, ps q Gs low e, M/ m snce om well o he end o zone, w Well dus, sw Sphecl low dus, s Sn co ssp eudosn co n he sphecl low s seudosn co T Tempeue, R Tme, h () seudome uncon, ps h/cp [()* m() ] seudopessue devve uncon, ps /cp [()* m() ] Second pseudopessue devve uncon, pps /cp [()* m() ]m_mn Mmum o mnmum pseudopessue w devve uncon e dl low o zone, ps /cp [()* m() ]m Mmum pseudopessue devve uncon beoe dl low o zone, ps /cp _ mensonless pseudome wh espec o w _A mensonless pseudome wh espec o A _ mensonless pseudome wh espec o e Resevo hl lengh, Hl-cue lengh, R R R3 e [()* m() ] n/[()* m() ] [()* m() ]m_mn[()* m() ] [()* m() ]m/[()* m() ] Resevo lengh n -decon Hl-esevo lengh y z Z Resevo lengh n y-decon Resevo lengh n z-decon Gs devon co Gee Fcue peneon o, h/h Chnge, dop oosy, con Vosy, cp usvy consn o zone Sues Zone Zone BR B-dl BR B-dl pseudome o ps*h/cp BRp Inesec o b-dl nd pseudosedy-se lnes BR Inesec o b-dl nd lne lnes BRsp Inesec o b-dl nd sphecl lnes mensonless e Eenl g Gs Inesecon o nl condons n Inlecon pon ne ne low pseudome o ps*h/cp p Inesec o lne nd pseudosedy-se lnes mn Mnmum N A pon on he ely un-slope lne o Reeence vlue p, pss seudosedy se dl low Rdl low n zone Rdl low n zone us Inesecon o dl low n zone wh ely unslope lne us Inesecon o dl low n zone wh ely unslope lne BR Inesecon o dl nd b-dl low egmes e End o dl low egme n zone Inesecon o dl nd lne low egmes p Inesecon o dl nd pseudosedy-se lnes Sndd condons sp Sphecl low sp Inesecon o sphecl nd dl n zone lnes sp Inesecon o sphecl nd dl n zone lnes Tol us Ely un-slope dung wellboe soge eecs w Well 84 INGENIERÍA E INVESTIGACIÓN VO. 34 No., AUGUST - 04 (76-84)

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