PRESSURE AND PRESSURE DERIVATIVE ANALYSIS FOR A VERTICAL WELL IN WEDGED AND T-SHAPED RESERVOIRS

Transcription

1 VOL. 9, NO. 6, JUNE 014 ISSN Asian Reseach Publishing Newok (ARPN). All ighs seved. PRESSURE AN PRESSURE ERIVATIVE ANALYSIS FOR A VERTICAL WELL IN WEGE AN T-SHAPE RESERVOIRS Fddy Humbeo Escoba, Maía Paula Polanco and Wilson enavides Univesidad Sucolombiana/CENIGAA, Avenida Pasana - Ca 1, Neiva, Huila, Colombia fescoba@usco.edu.co ASTRACT I is difficul o idenify he mechanics of an undegound sevoi and is even mo difficul o idenify he qualiies of sies ha have unusual foms. Pssu ansien analysis is a ool ha helps o undesand wha is happening inside he sevoi, undesand he flow mechanisms a a maco level, and someimes, howeve, hey a no enough o idenify flow channels geneaed by complex faul sysems. Cunly, he is no a mehodology o chaaceize eihe wedge o T-shaped deposis by pssu ansien esing. Then, his pape conains a caful analysis of diffen pssu behavios on such sysems, so numeical simulaions we un fo sevois sysems consideing T and wedge geomeies having a well inside hem in a vaiey of locaions. The final poduc consiss of developing an inepaion mehodology using he pssu and pssu deivaive log-log plo o chaaceize hese ypes of sevois. The developed equaions we succesfully esed wih synheic examples. Keywods: sevois, faul, wedge, pssu, channels, TS echnique. 1. INTROUCTION Pssu ansien ess a pefomed wih he pupose of evaluaing and deemining popeies of hydocabon-beaing fomaions. Howeve, some sevois may have unusual shapes caused by complex fauling o delaic channels. They possess he condiions fo he fomaion of wedge o T-shaped sysems. The complexiy of he geology of such sysems can be chaaceized by ansien-pssu analysis. Howeve, he idenificaion and quanificaion of he developed sysems qui complex analyses, mainly, since he is no an analyical mehodology fo achieving he inepaion in such sysems in which fauls can ac as leaky o sealing baies and, depending upon hei geomey and locaion, fauling can fom elongaed deposis o channels developing T o wedge shapes. Non-linea gssion analysis which is by iself laed o nonuniqueness is he mos common way fo inepaion pssu ess in he above menioned sysems. Some sudies have been conduced on he above geomeical siuaions. Hone, Kawaku and Temeng (1981) povided a novel analysis based upon an analyical soluion o obseve he pssu behavio of a well poducing fom a non-unifom hick saum nea o a pinch-ou. Sewa and Whaballa (1980) descibed a new soluion fo compamenalized sysems based upon maeial balance consideaions. Anisu-Rahman and Ambasha (1997) we focused on he effecs of ock-fluid popeies on he pssu ansien in compamenalized sevois. Fo such pupose hey developed an analyical soluion. They also found ha ceain paamees associaed wih he geological sucu of compamens can be esimaed via pssu esing. Mijinyawa and Gingaen (008) invesigaed by numeical simulaion fou mo common geomeies found in complex sevois. They showed ha all hese configuaions poduce chaaceisics behavios on well-es daa which we analyzed by analogy of simple sysems. Chales, Rieke, and Puushohaman (1999) analyzed and ineped pssu ess of wo wedge off-sho sevois locaed in he souh of Louisana. Anisu- Rahman, and ensen (003) uilized an inegaeansfomaion (ITT) useful o develop new analyical soluions mo poweful han he exising soluions o evaluae T and wedge shaped sevois. In his pape, T and wedge geomey condiions, wedge angles and well locaions we simulaed o obseve pssu behavio and idenify unique chaaceisics fo each sysem so an analyical mehodology using he log-log plo of he pssu and pssu he deivaive vesus ime is fomulaed and successfully esed wih simulaed cases.. MATHEMATICAL FORMULATION.1. asic equaions The dimensionless ime and pssu quaniies used in his sudy a given below: k = (1) φµc w kh P = P 141.qµ Tiab (1993) demonsaed ha he dimensionless cipocal ae deivaive duing adial flow gime akes he value of 0.5, () [ * P '] = 0.5 (3) Fom which he pemeabiliy is solved once he pssu deivaive of Equaion () is placed ino Equaion (), such as: 845

2 *P ' VOL. 9, NO. 6, JUNE 014 ISSN Asian Reseach Publishing Newok (ARPN). All ighs seved. 70.6qµ k = h ( P) ' Whe (* P ) is he value of he pssu deivaive duing adial flow gime. Tiab (1993) also found an expssion fo he skin faco by dividing he dimensionless pssu duing adial flow by Equaion (4) o give; (4) Table-1. Well locaion and sevoi geomey. 90 ANGLE INTERCEPTIN FAULTS WELL LOCATION VORTEX P k s = 0.5 ln ( * P' ) φµ c w (5) 75 CENTERE eing P he pssu dop ad a any abiay ime,, duing adial flow. The govening pssu deivaive equaion duing pseudoseady-sae gime is given by: [ ] * P ' = π ( ) (6) A p A p NEAR LEFT FAULT NEAR RIGHT FAULT As indicaed by Tiab (1994), an equaion fo he deeminaion of dainage aa, A, is found fom he inesecion poin of he adial flow gime govening equaion, Equaion (3), and he pssu deivaive equaion duing pseudoseady-sae, Equaion (6), as follows: kpi A = (7) φµc Whe pi is he poin of inesecion beween he adial flow pssu deivaive and he pseudoseady-sae pssu deivaive (exapolaed) lines... Pssu deivaive behavio fo wedge-shaped sevois enavidez and Polanco (014) pefomed seveal simulaions un we pefomed fo wedge sevois unde vaiaion of well locaion and he angle fomed by he ineceping sealing fauls so diffen empiical expssions we developed o find he angle beween he ineceping fauls o he disance fom he well o he cene of he T as will be indicaed below. The diffen geomey combinaions a given in Table Well locaed in he voex As descibed in Figu-1, he well pssu deivaive ace displays iniial adial flow gime and hen psens a consecuive behavio as he angle changes incases fom 30, 45, 60, 75 y 90 so does he ime o ach he boundaies; hefo, ending of adial flow ime can be colaed wih he angle as given in Table-. Then, an equaion fo he esimaion of he angle is given by: = (8) RAIAL FLOW = = ( * P' ) + + ( * P' ) ( * ') P 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 Figu-1. Pssu deivaive behavio fo a well locaed in he voex of wo ineceping fauls. eing he ime a which he adial flow gime ends and is he dimensionless ime fo ha given ime. Table-. Values of angle and ending dimensionless ime of adial flow gime fo a well locaed in he voex of a wedge-shaped sevoi., 4x 3 30 x x x x 5 90 Anohe esimaion of he ineceping angle is found fom he imum value of he pssu deivaive befo pseudoseady-sae develops. These imum poins agains angle a given in Table-3 fom which he following expssion was obained: 846

3 *P ' *P ' VOL. 9, NO. 6, JUNE 014 ISSN Asian Reseach Publishing Newok (ARPN). All ighs seved. ( P ) kh * ' qµ qµ ( qµ ) + kh( * P ') kh( * P ') = Table-3. Values of angle and imum dimensionless pssu deivaive values fo a well locaed in he voex of a wedge-shaped sevoi. ( *P ) ( *P ) min, The minimum value of he pssu deivaive ha shows up jus befo he developmen of he pseudoseady-sae gime was also colaed o find he inesecing angle and hei values a poed in Table-3. The suling colaion is given by: ( P ) min ( P ) ( P ) = * ' 4 6 min min * ' * ' (9) ()... Well locaed nea he igh faul I is obseved in Figu- ha once he nea faul is fel a ansiion peiod develops and a hemi-adial flow gime shows up and vanishes once he ohe faul is ached by he disubance. efo he pseudoseady-sae, a imum shows up as he ineceping angle is smalle han 60. Fo his case, he chaaceisic poin used was he dimensionless ime fo he ansien wave o ach he fa faul which values fo each sudy angle a given in Table (11) = le 7.0 le le..3. Well locaed a he lef faul This case which dimensionless pssu deivaive poed in Figu-3 is vey simila o he fome one which means ha Equaion (11) can be applied. Howeve, in his case he ending ime of he adial flow gime agains he angle, see Table-5, was used o develop he following expssion: = Table-4. Values of angle and dimensionless inflecion ime afe aching he second faul fo a well locaed in he voex of a wedge-shaped sevoi. le, 1.11x x 6 30.x x x x 6 90 le (1) Table-5. Values of angle and ending dimensionless ime of adial flow gime fo a well flow a well locaed nea he lef faul of a wedge-shaped sevoi.,.8x x x x 5 90 RAIAL FLOW = le 7.0 le le + 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 Figu-. Pssu deivaive behavio fo a well locaed a he igh faul of he voex. Equaion (11) was obained fo esimaing he inesecing angle fo his case. le RAIAL FLOW = E+0 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 Figu-3. Pssu deivaive behavio fo a well locaed a he lef faul of he voex. 847

4 *P '' *P ' *P ' VOL. 9, NO. 6, JUNE 014 ISSN Asian Reseach Publishing Newok (ARPN). All ighs seved...4. Well locaed a he cene The pssu deivaive behavio of a well locaed in he middle poin of wo inesecing fauls is given in Figu-4 fo angles of 0, 30, 45, 60, 75 and 90. As he inesecing angle incases he ime quid o finish he adial flow gime also incases. See also Table-6. Pseudoseady sae develops once all he sevoi boundaies a ached. Table-6. Values of angle and ending of adial flow fo a well locaed in he cene of wo sealing fauls in a wedge-shaped sevoi.,.79x x x x x x 6 90 Since a chaaceisic paen was no obseved, he nex sep was o sudy he second dimensionless pssu deivaive behavio. Case E was aken ou of he analysis since is locaed oo close o he wes bounday, hen, is behavio does no follow an esablished paen as given in Figu-6, fom which daa poed in Table-7 was ad. L E C A L pf Figu-4. Well locaion inside he T-shaped sevoi. 1.E+0 FLUJO RAIAL A C E E C A = E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 Figu-5. Pssu deivaive behavio fo a well locaed along a T-shaped sevoi. Figu-4. Pssu deivaive behavio fo a well locaed a he cene of wo sealing fauls. E C A Equaion (13) was obained using he daa fom Table = + (13) A C.3. Pssu deivaive behavio fo T-shaped sevois Fo his ype of sysems, simulaions we pefomed fo diffen well locaions inside he sysem accoding o labels A hough E as depiced in Figu-4. The pssu deivaive behavio fo each case is poed in Figu-5. Noice ha he close o he cene of he T, he longe he duaion of he adial flow. Afe he adial flow vanishes, Case A psens a combinaion of wo pependicula linea flow gimes while case E has an iniial pdominan linea flow ha lae combines wih anohe adial flow coming fom he lowe pa of he T. As a final behavio, pseudoseady-sae is developed. L k = * ( * P'') min φµ c w E+06 1.E+07 1.E+08 Figu-6. Second pssu deivaive behavio fo a well locaed along a T-shaped sevoi. A dimensionless disance was defined as he aio beween he disance fom he wes o eas bounday o he cene, hus; L L pf = (14) L 848

5 *P ' VOL. 9, NO. 6, JUNE 014 ISSN Asian Reseach Publishing Newok (ARPN). All ighs seved. Infomaion fom Table-7 led o develop an expssion fo he esimaion of he disance fom he well o one of he boundaies along he uppe gion of he T, see Figu-4, as follows: qµ k L = kh( * P'') c 1 min φµ w (15) Table-7. Coodinaes of he minimum value of he second pssu deivaive fo each dimensionless lengh. ( *P ) min ( ) min L EXAMPLES The synheic examples we un wih he infomaion given below: φ = 0% w = 0.5 f q =500 ST/ h = 0 f o = 1.15 bls/st µ = 5 cp c = 4x -6 psi -1 A = f 3.1. Example-1 Pssu and pssu deivaive agains ime daa a poed in Figu-7 fo he case of a well locaed nea he voex of wo sealing fauls. I is quid o esimae pemeabiliy, skin faco, dainage aa and he angle fomed beween he wo fauls. 1.E+04 ( 70.6)( 500)( 5)( 1.15) ( 0)( 60.9) k = = 33.3 md ( 33.3)( 1) s = ln = ( 0.)( 5)( 4 )( 0.5 ) ainage aa is found using Equaion (7), ( 33.3)( 0) A = = f ( )( )( ) Equaion (1) allows anslaing he acual ime o dimensionless ime, ( )( 33.3)(.6) = = 3000 ( 0.)( 1.15)( 4 6 )( 0.5 ) The ineceping angle is found afe placing he above value ino Equaion (8), = (3000) 5.0 (3000) (3000) = The angle of inesecion can be veified using he imum poin of he pssu deivaive in Equaion (9), hus, (33.3)(0)(500) = (500)(5)(1.15) (500)(5)(1.15) (500)(5)(1.15) + = 43.4 (33.3)(0)(500) (33.3)(0)(500) P & * P', psi 1.E+03 P = psi 1.E+0 = 1h =.6 h (* P') = 500psi Flujo adial (* P') = 60.9psi = 3600 h pi 3.. Example- The same inpu daa fo he fome example was applied fo a wedge-shaped sevoi wih he well cened. The inpu ineceping angle was 70. Pssu deivaive daa is poed in Figu-8. I quid finding he inesecing angle. 1.E+0 1.E+03 1.E+04 1.E+05, h Figu-7. Log-log plo of pssu and pssu deivaive vs. ime fo example-1. Soluion The following infomaion was ad fom Figu- 7: = 1 h P = psi (* P ) = 60.9 psi =.6 h (* P ) = 500 psi Pemeabiliy and skin faco a esimaed wih Equaions (4) and (5), specively, hus: = 133 h 1.E+0 1.E+03 1.E+04 1.E+05, h Figu-8. Log-log plo of pssu deivaive vs. ime fo example-. 849

6 VOL. 9, NO. 6, JUNE 014 ISSN Asian Reseach Publishing Newok (ARPN). All ighs seved. Soluion A value of of 133 h was ad fom Figu-8 which anslaes ino a dimensionless ime of 1.16x 6. This allows finding an inesecion angle of 67 by means of Equaion (13) Example-3 A pssu es of T-shape sevoi was simulaed using he daa menioned a he beginning of he examples. The well is locaed in he posiion given in Figu-4 and he oal lengh fom he well o he middle posiion (poin A in Figu-4) was 600 f. aa of he pssu deivaive and second pssu deivaive agains ime a poed in Figu-9. I is quid o find he lengh of he posiion. * P' & * P", psi 1.E+03 1.E+0 1.E-0 = 0 h ( * P") min = psi 1.E+0 1.E+03 1.E+04 1.E+05, h Figu-9. Log-log plo of pssu deivaive and second pssu deivaive vs. ime fo example-3. Soluion A value of of 0 h and a minimum second deivaive of 63 psi we ad fom Figu-9. The cosponding dimensionless values found wih Equaion 1 and he second deivaive of Equaion 1 sul ino: = ( *P ) min = Use of Equaion (15) leads o find a dimensionless lengh of 0.44 which muliplied by 600 anslaes ino f. 4. COMMENT ON THE RESULTS The values of inesecing angles and disance fom bounday o well we in close agemen wih he values used fo he simulaion which veify he validiy of he given colaions. 5. CONCLUSIONES AN RECOMMENATIONS Pssu behavio was sudied fo a well locaed inside boh wedge-shape (simulaed wih inesecing sealing fauls) and T-Shaped sevois which led o he developmen of new equaions o find he inesecing angle and he disance fom he bounday o he well. The developed expssions we successfully esed wih synheic examples. ACKNOWLEGEMENTS The auhos gaefully hank he Mos Holy Tiniy and he Vigin May mohe of God fo all he blessing ceived duing hei lives. The auho also hanks Univesidad Sucolombiana fo poviding financial suppo fo he complemen of his sudy. Nomenclau A aining aa, f c h k P P wf q e w s (* P ) Oil volume faco, b/st Compssibiliy, 1/psi Fomaion hickness, f Fomaion compssibiliy, md Pssu, psi Well-flowing pssu, psi Flow ae, ST/ ainage adius, f Wellbo adius, f Skin faco Tes ime, h Pssu deivaive, psi ( * P ) Second pssu deivaive, psi ( *P ) imensionless pssu deivaive ( *P ) imensionless second pssu deivaive L pf φ L isance fom bounday o cene, f Lengh fom bounday o middle of Change Gek Poosiy, faccion µ Viscosiy, cp le pi w p Suffixes imensionless inflecion ime afe aching he second faul Maximum Radial End of adial flow Inesec of adial-pseudoseady sae lines well iempo Pseudoseady-sae 850

7 VOL. 9, NO. 6, JUNE 014 ISSN Asian Reseach Publishing Newok (ARPN). All ighs seved. REFERENCES Anisu-Rahman A. and ensen New Analyical Soluions fo Pdicing Pssu isibuion and Tansien ehavio in Wedges and Tuncaed Wedges. Pape SPE 71585, SPE Jounal, Sepembe. enavides W. and Polanco M. P Inepación de Puebas de Psión en Yacimienos Acuñados y en Foma de T..Sc. Thesis. Univesidad Sucolombiana, Febuay. Chales.., Rieke H. H. and Puushohaman R Well Tes Chaaceizaion of Wedge-Shaped, Fauled Resevois. Pape SPE Ocobe. Mijinyawa A. and Gingaen A. C Influence of geological feaus and well es behavio. Pape SPE , Roma, Ialia junio de. Roland N., Hone and Kawaku O. and Temeng Recogniion and Locaion of Pinchou oundaies by Pssu Tansien Analysis. Pape SPE 9905, Mach. Sewa G. and Whaballa A. E Pssu ehavio of Compamenalized Resevois. Pape SPE 19779, Ocobe. Tiab Analysis of Pssu and Pssu eivaive wihou Type-Cuve Maching: 1- Skin and Wellbo Soage. Jounal of Peoleum Science and Engineeing. 1: Tiab Analysis of Pssu and Pssu eivaive wihou Type Cuve Maching: Veically Facud Wells in Closed Sysems. Jounal of Peoleum Science and Engineeing. 11: