RATE-TRANSIENT ANALYSIS FOR HYDRAULICALLY FRACTURED VERTICAL OIL AND GAS WELLS

Transcription

1 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. RATE-TRANSIENT ANAYSIS FOR HYRAUICAY FRACTURE VERTICA OI AN GAS WES Feddy Humbeo Escoba, Jhon Richad Caso and Juan Sebasian Mosquea Univesidad Sucolombiana/CENIGAA, Avenida Pasana - Ca 1, Neiva, Huila, Colombia escoba@usco.edu.co ABSTRACT Seveal common esevoi poducion condiions esul in low a a consan pessue; hen, a consanly changing well-lowing pessue is ecoded. Nowadays, mos well-es analysis mehods assume consan-ae poducion especially since gas shale wells ae nomally esed by ecoding he low ae values unde consan pessue condiions. In such cases, well esing could be eliminaed in many cases as being o lile value o economically unjusiiable because o he esuling poducion loss when compaed wih wha can be obained om consan wellboe pessue poducion daa. Then, his pape pesens a ansien-ae analysis o aiicially acued veical wells lowing unde consan pessue in homogenous deposis wih cicula/squae shape. Epessions o esevoi chaaceizaion using boh TS and convenional echniques ae inoduced and successully esed wih ield and synheic eamples. Keywods: acued wells, RTA, TS, consan-pessue condiions, acue conduciviy. INTROUCTION Muli-ae esing can be se as he main esing ool o esevoi chaaceizaion since i compises he emaining ess. A pessue dawdown es can be deined as he simples muli-ae es wih a single non-zeo low ae. Besides, a pessue buildup es can be deined as a muli-ae es having wo low aes: one dieen han zeo and anohe one wih a value o zeo. Muli-ae ess can have seveal low ae vaiaions eihe wih egula o iegula changes in low ae. I he low ae changes coninuously, hen, he case o ansien-ae analysis is obained. The behavio o a well opeaing a consan sandace pessue is analogous o ha o a well opeaing a consan low ae. In a consan pessue low esing, he well poduces a a consan boom-hole pessue and low ae is ecoded wih ime. Since ae soluions ae ound on basic low pinciples, low ae daa can be used o esevoi chaaceizaion and dieen popey esimaions. Hence, his echnique can be consideed as an alenaive o convenional well esing echniques: consan low ae cases. Howeve, hey ae cusomay used in decline-cuve analysis. Aps (1945) developed he sandad eponenial, hypebolic, and hamonic decline equaions. Feovich (1980) geneaed he dimensionless ae-ime ype cuves o decline cuve analysis o wells poducing a a consan boom-hole pessue. He demonsaed ha decline cuve analysis no only has a solid undamenal basis bu also povides a ool wih moe diagnosic powe han had been peviously nown. These ype cuves combined analyical soluions o he low equaions in he ansien egion and empiical ae elaionships, poposed by Aps (1945), in he pseudoseady sae egion. A mehod o deemining he sin eec om ae-ime daa was given by Ealoughe (1977). Wih egad o heeogeneous esevois, numeous analyical and numeical soluions o consan pessue poducion condiions have been published using he Van Evedingen and Hus (1949) amous soluion. Escoba, Rojas and Canillo (01) eended he convenional echnique o ae-ansien analysis in long and naow homogeneous and naually acued esevois. Also, Escoba, Rojas and Bonilla (01) and Escoba, Sanchez and Canillo (008) povided mehodologies o ansien-ae inepeaion o elongaed homogeneous and heeogeneous esevoi sysems and gas esevois, especively, ollowing he philosophy o he TS echnique, Tiab (1993). Fo hydaulically acued wells, Cinco-ey and Samaniego (1978) and Cinco-ey, Samaniego and ominguez (1978) pesened one o he mos impoan indings so-called inie conduciviy. This model epesens he geneal case compaing wih he pevious published models. In addiion, Tiab (1994) applied he TS echnique, Tiab (1993), o acued wells. Unde a consan pessue poducion, he well ineceped by a veical acue has been also discussed in he lieaue. Eension o he TS echnique o hydaulically acued oil wells was peomed by Aab (003). In his wo, boh convenional and TS echniques ae eended o inepeaion o ae-ansien ess un in hydaulically veical acued gas and oil wells. The poposed mehodologies wee successully esed wih acual and synheic daa. MATHEMATICA FORMUATION The ollowing dimensionless paamees ae used o he mahemaical developmen: imensionless ime based on aea, A: A (1) φµ c A imensionless ime based on hal-acue lengh,, 739

2 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved () φµc *(1/ q )' (8) 0.5 C imensionless pseudoime, Agawal (1979), based on hal-acue lengh,, a a( P) ( P ) (3) φ imensionless oil low ecipocal ae, 1/q, 1 q ( i Pw ) h P (4) 141.qµ B imensionless gas low ecipocal ae, 1/q, 1 q ( i) ( w ) h mp mp (5) 144qT imensionless acue conduciviy, C, C w (6) 1.E+0 1.E+01 1.E+00 1.E-01 1.E-0 1.E-03 Bilinea low m0.5 Radial low m 0 Pseudoseady sae 1.E-05 1.E-0 1.E+00 1.E+0 1.E+04 1.E+06 1.E+08, h Figue-1. imensionless ecipocal ae and ecipocal ae deivaive o a veical well wih a inieconduciviy acue. Tansien-ae analysis o oil wells having a inieconduciviy acue by he TS echnique In such cases, he mos emaable low egime seen a ealy imes, i wellboe soage allows, is he bilinea. This is ecognized by a slope o one ouh on he ecipocal ae deivaive as indicaed in Figue-1. The dimensionless ecipocal ae and ecipocal ae deivaive govening equaions o such low ae: 1.7 q C 0.5 (7) An epession in oil-ield unis will esuls ae plugging he dimensionless paamees given by Equaions (), (4) and (6) ino Equaion 6, hus, µ B q h P w c 0.5 (9) 0.5 ( φµ ) Which ecipocal ae deivaive is given by 1.4µ B *(1/ q)' 0.5 (10) 0.5 h P w ( φµ c ) Fom which an epession o esimae he acue conduciviy is obained using he ecipocal ae deivaive ead a a ime o 1 hou. w µ B φµ c h P[ *(1/ q)' ] B1 (11) Tansien-ae analysis o oil wells having inieconduciviy acues by he convenional mehod Equaion (7) suggess ha he slope m B om a Caesian plo o he one-ouh oo o ime vesus he ecipocal ae can be used o esimae acue conduciviy: w µ B 0.5 mbh P( φµ c ) (1) Tansien-ae analysis o gas wells having a inieconduciviy acue by he TS echnique Once he dimensionless quaniies given by Equaions (), (5) and (6) ae eplaced ino Equaion (7), he esuling epession o bilinea gas low and is ecipocal deivaive ae: T q h m P w c [ ] ( ) 4 φµ ( ) i T *(1/ q)' (13) 0.5 (14) h P 4 w φµ ( c) i Solving o he acue conduciviy om he above epession, ( w) app T 0.5 ( φµc) h[ m( P) ][ *(1/ q)' ] B1 (15) 740

3 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. I using pseudoime, as deined by Equaion (3), he above epession becomes: ( w) app T 0.5 ( φ) h[ m( P) ][ *(1/ q)' ] B1 (16) Equaions (15) o (16) ae used o esimae he appaen acue conduciviy om he ecipocal ae deivaive ead a eihe he ime o 1 hou o he pseudoime o 1 h-psi/cp. Tansien-ae analysis o gas wells having a inieconduciviy acue by he convenional echnique Equaion (13) indicaes ha he slope m B om a Caesian plo o he one-ouh oo o eihe ime o pseudoime vesus he ecipocal ae can be used o esimae acue conduciviy om he ollowing epessions using ime and pseudoime, especively, *(1/ q )'.391 (0) 0.5 Ae plugging he dimensionless quaniies deined by Equaions () and (4) ino he above epession will yield: B µ q h P φc Which ecipocal ae deivaive esuls: B *(1/ q)' h P µ φc (1) () Theeoe, solving o he hal-acue lengh esuls in an epession ha uses he ecipocal ae deivaive, eapolaed i necessay, a he ime o 1 hou: ( w) app ( w) app T m 4 Bh[ m( P) ] φµ ( c ) i T m 4 Bh[ m( P) ] φ (17) (18) B µ (3) h P q φc [ *(1/ ) '] 1 Tiab (1994) inoduced he concep and deiniion o biadial (o ellipical) low egime. The deiniion o pessue deivaive behavio is adaped hee o ansienae analysis, such as: 1.E+04 Pseudoseady sae m e *(1/ q )' A A (4) 1.E+03 1.E+0 1.E+01 inea low m 0.5 Biadial low m Radial low m 0 1.E-0 1.E-01 1.E+00 1.E+01 1.E+0 1.E+03 1.E+04, h Figue-. imensionless ecipocal ae and ecipocal ae deivaive o a veical well wih an ininieconduciviy acue. Tansien-ae analysis o oil wells having an ininieconduciviy acue - TS echnique Alhough, linea low egime can also appea in low o medium inie-conduciviy acues, i is moe common ha eihe linea and/o biadial, Tiab (1994), develop as depiced in Figue-. The dimensionless ecipocal ae govening equaion o linea low, Aab (003), is given as ollows: q (19) Which deivaive esuls; Fom inegaion o Equaion (4), i yields, 1 q e A (5) Replacing in Equaion (5) -i he sysem is squae- hen A 4 e, he dimensionless paamees given by Equaions (1) and (4) will esul: 0.64 B µ q h P ( φc ) Which ecipocal ae deivaive is given by [ *(1/ )'] µ µ q h P ( φc ) (6) (7) Solving o, when he ecipocal deivaive is ead a a ime o 1 h, 16/ B µ [ *(1/ q)' ] h P( φc ) BR1 (8) 741

4 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. Equaion (8) is useul o esimae he halacue lengh by eading he ecipocal ae deivaive duing biadial low egime a a ime o 1 hou, eapolaed i needed. Tansien-ae analysis o oil wells having an ininieconduciviy acue by convenional analysis Equaion (19) suggess ha he slope m om a Caesian plo o he squae oo o eihe ime o pseudoime vesus he ecipocal ae can be used o esimae hal-acue lengh by means o he ollowing epession; 95.05T 1/ q h m( P) [ φµ ( c ) ] [ ] i (35) The ecipocal ae deivaive wih espec o ime o Equaion (35) is given as ollows: 34.19T [*(1/ q)'] BR h m P c BR [ ( )][ φµ ( ) i] (36) Solving o he hal-acue lengh, we obain o ime and pseudoime, especively: B mh P µ φc (9) 34.19T 0.64 h [ m( P) ][ φµ ( c) i] [ *(1/ q)'] BR1 (37) Fo ininie-conduciviy cases, when biadial low is pesen, he hal-acue lengh can be esimaed om he slope m BR o a Caesian plo o he ime o he powe 9/5 o agains he ecipocal ae, using he ollowing epession: 16/5 9.46B µ mbrh P( φc ) (30) Tansien-ae analysis o gas wells having an ininieconduciviy acue by he TS echnique The below epession is obained once he dimensionless paamees given by Equaions () and (5) ae eplaced ino Equaion (19): T q h mp ( ) φµ ( c) [ ] i Which deivaive is: T *(1/ q)' h m( P) φµ ( c ) [ ] 0.5 i Solving o he hal-acue lengh; T h m( P) *(1/ q)' φµ ( c ) [ ][ ] i (31) (3) (33) As deal beoe, i he value o he ecipocal ae deivaive is ead a he ime o one hou, eapolaed i necessay, Equaion (33) becomes: T h m P q c [ ( )][ *(1/ )'] ( φµ ) i i (34) I biadial low is pesened and consideing a squae esevoi, hen A 4 e, hen, he govening equaion esuling om subsiuing Equaions (1) and (5) ino Equaion (5), yields; 34.19T 0.64 h [ m( P) ] φ [ *(1/ q)'] BR1 (38) Tansien-ae analysis o gas wells having an ininieconduciviy acue by convenional analysis The slope om a Caesian plo o he squae-oo o ime (o pseudoime) agains he ecipocal ae allows obaining he hal-acue lengh om obsevaion o Equaion (31), hus: T mh mp φµ c [ ( )] T mh mp c [ ( )] (39) (40) Noice ha Equaion (40) is given o pseudoime and Equaion (36) is o egula ime. Also, i ollows o he case o biadial low egime ha Equaion (35) suggess ha a Caesian plo o eihe ime o pseudoime o he powe 9/5 (o ) will povide a slope, m BR, which leads o ind he hal-acue lengh: 95.05T 0.64 mbrh [ m( P) ][ φµ ( c ) i] 95.05T 0.64 mbrh [ m( P) ] φ (41) (4) Again, Equaion (4) is given o pseudoime and Equaion 41 is o egula ime. Pseudoadial low egime Aab (003) demonsaed ha he dimensionless ecipocal ae deivaive duing adial low egime aes he value o 0.5, 74

5 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. [ *(1 / q )'] R 0.5 (43) Fom which, he pemeabiliy is esimaed by eading he value o he ecipocal ae deivaive duing adial low egime, [*(1/q) ], and using he ollowing epession: 70.6µ B (44) h P[*(1/ q)'] R And he sin aco is esimaed by eading he ecipocal ae, (1/q), ead a any abiay ime,, duing he adial low egime, hus: (1 / q) R R s 0.5 ln [*(1/ q)'] R φµ ( c) iw (45) The pemeabiliy Equaion o gas well was pesened by Escoba e al. (008), T h[ m( P) ] ( 1/ q) ' R (46) Noice ha Equaion (45) can be applied o gas wells, bu in his case, i is eeed as appaen sin since ineial low condiions have o be aen ino accoun. Howeve, o gas low using pseudoime he esuling appaen sin equaion is: (1 / q) R a( P) R s ' 0.5 ln [*(1/ q)'] R φw (47) Inesecion poins I was ound in his sudy ha he govening equaion o he ecipocal ae duing pseudoseady-sae egime is given by: [ ] *(1/ q )' 3 π ( ) (48) PSS A PSS Taing he deivaive o Equaion (4) (oil wells) and plugging his esul ino Equaion (48) along wih he dimensionless ime quaniy given by Equaion (1) and solving o A allows o obain an epession o ind he dainage aea by eading he ecipocal ae deivaive [*(1/q) ] pss a any abiay poin duing he uni-slope lae pseudoseady: BPSS A.85 h Pφc[ *(1/ q)'] PSS (49) Fo he case o gas wells, he esuling equaion is given o acual ime by: 3.54TPSS A (50) h m ( P ) φµ ( c ) [ *(1/ q )'] And o pseudoime; i PSS 3.54 Ta( P) PSS A (51) h m ( P ) φµ ( c ) [ *(1/ q )'] i PSS Equaions (49) hough (51) ae also used by eading he ecipocal ae deivaive a a value o 1 h o 1 h-psi/cp. The poin o inesecion omed by he pseudoseady-sae ecipocal deivaive saigh line (Equaion 48) wih he bilinea low egime deivaive, Equaion (8), allows inding epessions o esimaing he dainage aea: w BPSSi A φµc (5) Equaion (5) applies also o gas wells using acual ime consideing ha he gas viscosiy and oal compessibiliy ae evaluaed a iniial condiions. Fo pseudoime, he esuling equaion is: w a( P) BPSSi A φ (53) The poin o inecep beween he biadial (Equaion 4) and pseudoseady sae (Equaion 48) deivaive lines allows inding anohe useul equaion o esimae he dainage aea in oil and gas esevois: 16/ BRPSSi A φµc (54) Fo gas wells wih pseudoime, 16/ a( P) BRPSSi A φ (55) The poin o inesecion omed by he pseudoseady-sae ecipocal deivaive saigh line (Equaion 48) wih he linea low egime ecipocal ae deivaive, Equaion (0), leads o he esimaion o he dainage aea: PSSi A (56) 5.79 φµc Which is also good o gas wells i he viscosiy and oal compessibiliy ae given a iniial condiions. I pseudoime is used, he esuling equaion would be: 743

6 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. a( P) PSSi A (57) 5.79 φ Inesecion o boh bilinea (Equaion 8) and linea ecipocal ae deivaive (Equaion 0) lines will povide: w ' Bi φ µ c (58) Inesecion o boh bilinea (Equaion 7) and linea ecipocal ae (Equaion 19) lines will povide: w Bi φ µ c (59) Inesecion o boh bilinea (Equaion 8) and linea ecipocal deivaive (Equaion 0) lines wih biadial (Equaion 4) will povide: ( w) app ' Bi φ (65) ( w) app Bi φ ( w ) app BBRi φ (66) (67) BRi (67) 39φ φ 1677 ( w ) (68) RBi 3 app Ri (69) 107φ φµ c w BBRi (60) RBRi (70) 4587φ The above equaion assumes ha A 4 e, hen, boh aea and esevoi lengh cancelled ou. Inesecion o linea (Equaion 0) and biadial ecipocal deivaive (Equaion 4) lines leads o: The inecepion o he ecipocal ae deivaives omed by he adial low egime, Equaion (43), and he uni-slope line oced o daw by he lae pseudoseadysae line, Equaion (48), leads o: BRi 39φµ c (61) RPPSi A (71) 01.φµc The inesecion poin beween he adial low dimensionless ecipocal deivaive line (Equaion 43), wih he bilinea low (Equaion 8), linea low (Equaion 0) and biadial low (Equaion 4) ecipocal deivaive lines allow o obain: φµ c RBi 1677 ( ) 3 w (6) Ri (63) 107φµ c Which wos o oil and gas wih eal ime. Fo gas wih he pseudoime uncion, Equaion (71) becomes, a( P) RPPSi A (7) 01.φ Finally, i bilinea low eiss, hen, acue conduciviy can be esimaed. I linea low egime eiss hen acue hal-lengh can be esimaed. Fo cases such cases whee eiss only one o hese wo low egimes, he acue paamees can be ound om anohe by: RBRi 4587φµ c (64) Equaions (58) hough (64) apply o boh oil and gas using eal ime. Fo pseudoime, hese equaions become, especively: w e s e s w w w (73) (74) 744

7 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. EXAMPES Eample-1 Aab (003) epos a ansien-ae es un in a acued well which ecipocal ae and ecipocal ae deivaive ae povided in Figue-3. Relevan inomaion is given in Table-1. I is equied o chaaceize such es. 1/q & *(1/q )', day/stb 1.E-03 1.E-05 1.E-06 (1/ q ) [ *(1/ q ) '] h Bi B1 [ *(1/ q )'] h 1.E-05 1.E-03 1.E-0 1.E-01 1.E+00 1.E+01 1.E+0 1.E+03, h 390 h RBi [ *(1/ q ) '] Figue-3. Recipocal ae and ecipocal ae deivaive vesus ime o eample-1. Table-1. Relevan inomaion o eample-1. Paamee Value Paamee Value C (bbl/psi) 0 µ o (cp) 0.85 w () 0.3 () 110 h () 30 C 10 φ (%) 0 P i (psi) 500 T ( F) 1 P w (psi) 3500 B, b/stb 1.05 c (psi -1 ) (md) 15 Soluion by TS echnique The lowing inomaion was ead om Figue-3. R 40.7 h Bi h (1/q) R day/stb [*(1/q) ] R day/stb [*(1/q) ] B day/stb [*(1/q) ] day/stb 1/q, day/stb 9.E-05 8.E-05 7.E-05 6.E-05 5.E-05 4.E-05 3.E-05.E-05 1.E-05 m day/stb/ h 0.E `, h Figue-4. Recipocal ae and ecipocal ae deivaive vesus he squae oo o ime o eample 1. Pemeabiliy and sin aco ae ound om Equaions (44) and (45), especively; 70.6(0.85)(1.05) 16 md (30)(1700)( ) (16)(40.7) s 0.5 ln (0.)(0.85)( )(0.9 s 4.64 (3), The hal-acue lengh is ound wih Equaion 3.19(1.05) (0.85) 11 5 (30)(1700)( ) (0.)( )(16) 1/q, day/stb m B day/stb/ h Equaion (11) is used o ind he acue conduciviy; (0.85)(1.05) w 1 5 (0.)(0.85)( )(16) (30)(1700) w md- Finally, acue conduciviy is esimaed wih Equaion (6), 0.E , h Figue-5. Recipocal ae and ecipocal ae deivaive vesus he ouh oo o ime o eample C 9.1 ( 111)( 16.5) Soluion by convenional analysis The slope, m, o he Caesian plo given in Figue-4 is day/stb/h 0.5. Equaion (9) is used o ind he hal-acue lengh: 745

8 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved (1.05) 0.85 ( )(30)(1700) (15)(0.0)( ) Also, he slope m B o day/stb/h 0.5 ead om he Caesian plo o Figue-5 allows deemining he acue conduciviy using Equaion (1), (0.85)(1.05) w (30)(1700)(0.*0.85* *15) w md- Finally, Equaion (6) is used o esimae de dimensionless acue conduciviy; C 9.3 ( 119.9)( 15) Eample Figue-6 pesens synheically geneaed ecipocal ae and ecipocal ae deivaive o a veical gas well having a inie-conduciviy acue. Ohe impoan daa concening his es is pesened in Table-. Table-. Fluid, esevoi and well inomaion o eample. Paamee Value Paamee Value e, 4000 µ g (cp) w () 0.3 () 00 h () 0 w, (md-) 100 φ (%) 5 P i (psi) 000 T ( F) 1 P w (psi) 1800 B, b/stb 1.05 c g (psi -1 ) (md) 1.5 γ g 0.9 m(p), (psi /cp) Soluion by TS echnique The lowing inomaion was ead om Figue h RBi 00 h RPSSi 400 h BPSSi 6100 h (1/q) day/stb [*(1/q) ] day/stb [*(1/q) ] B day/stb Equaions (46) and (45) lead o deemine a pemeabiliy value o 1.6 md and a pseudo sin aco o -5.5, especively. Then, an appaen acue conduciviy value o 1180 md- is ound wih Equaion (17). Equaion 74 leads o esimae a hal-acue lengh o The ime o inesecion beween he ecipocal ae deivaives o adial and bilinea low egimes esuls o be h which is vey close o he value o 00 h ead om Figue-6. A dainage aea o 99.5 aces is esimaed wih Equaion (53) which coesponds o an eenal adius o Again, Equaion (71) is used o obain a dainage aea o 98.4 Aces wih anslaes ino /q, day/stb.0e E-03 1.E E E-04 m B day/stb/ h 0.0E , h Figue-7. Recipocal ae and ecipocal ae deivaive vesus he ouh oo o ime o eample- 1.E-0 1.E-03 (1 / q ) /MMsc [ *(1 / q )] /MMsc 6100 h BPSSi Soluion by convenional analysis The slope, m B, o he Caesian plo given in Figue-4 is day/stb/h 0.5. Equaion (17) is used o ind he appaen acue conduciviy which esuls o be md-. 1.E-05 [ *(1 / q )'] /MMsc B1 00 h 1000 hs 1.E-0 1.E-01 1.E+00 1.E+01 1.E+0 1.E+03 1.E+04, h RBi 400 h RPSSi Synheic eample-3 A Tansien ae es was simulaed wih he inomaion given in Table-4. Figue-8 pesens ecipocal ae and ecipocal ae deivaive agains ime o his simulaion. I is equied o chaaceize such es. Figue-6. Recipocal ae and ecipocal ae deivaive vesus ime o eample-. 746

9 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. Table-3. Well, luid and esevoi paamees o synheic eample-3. Paamee Value Paamee Value C (bbl/psi) 0 (md) 0.8 w () 0.6 µ o (cp) h () 16.4 () 00 φ (%) 1 P i (psi) 5000 e () 5000 P w (psi) 500 B, b/stb 1. c (psi -1 ) Soluion by TS echnique The lowing inomaion was ead om Figue-8. R h (1/q) R day/stb [*(1/q) ] R day/stb [*(1/q) ] BR day/stb [*(1/q) ] day/stb 1/q, day/stb m day/stb/ h , h Figue-9. Recipocal ae and ecipocal ae deivaive vesus he squae oo o ime o eample-3. Soluion by convenional analysis The slope, m, o he Caesian plo given in Figue-9 is day/stb/h 0.5 which allows inding a hal-acue lengh o 06.9 by mans o Equaion (9). 1.E-0 Pemeabiliy and sin aco ae esimaed wih Equaions (44) and (45). These ae 0.93 md and -.1, especively. A acue-hal lengh o md was ound wih Equaion (3). Also, Equaion (8) allowed o ind anohe esimaion o he hal-acue lengh o Facue conduciviy is ound wih Equaion (73) o be 5. Md-. 1/q, day/stb 1.E-0 8.E-03 6.E-03 4.E-03.E-03 m BR day/stb/ h 1.E-01 1.E-0 [ *(1 / q )] /STB 1.E-03 (1/ q ) / STB [ *(1/ q )'] /STB [ *(1/ q )'] /STB BR1 795h 1.E , h Figue-8. Recipocal ae and ecipocal ae deivaive vesus ime o eample-3. 0.E , Figue-10. Recipocal ae and ecipocal ae deivaive vesus he ime o he powe 9/5 o eample-3. Moeove, he slope, m BR, o he Caesian plo given in Figue-10 is day/stb/h. This is used o ind a hal-acue lengh o by mans o Equaion (30). COMMENTS OF RESUTS Table-4 pesen a summay o he main esuls obained om he woed eamples and compaed o he eeence values. I is obseved a good mach beween he esuls and he eeence values. A highe deviaion is seen in eample- (gas well) since he obained values coesponds o appaen acue conduciviy which has o be coeced due o ineial eecs. h 747

10 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. Tabla-4. Compaison o main esuls om eamples. Paamee Reeence value This sudy TS This sudy Convenional Eample-1, w, md Eample-, ( w ) app, md e, Eample-3, CONCUSIONS Boh TS and convenional echniques wee complemened o chaaceize ansien ae ess in hydaulically-acued veical hydocabon Wells. The new epession was successully esed wih ield and simulaed daa. ACKNOWEGEMENTS The auhos gaeully han he Mos Holy Tiniy and he Vigin May mohe o God o all he blessing eceived duing hei lives. REFERENCES Agawal G Real Gas Pseudo-ime a New Funcion o Pessue Buildup Analysis o MHF Gas Wells. Pape SPE Jounal o Engineeing and Applied Sciences. 7(3): Escoba F.H., Rojas M.M. and Canillo J.H. 01. Saigh-ine Convenional Tansien Rae Analysis o ong Homogeneous and Heeogeneous Resevois. yna. Yea 79, No. 17: , Apil. Escoba F.H., Sanchez J.A. and Canillo J.H Rae Tansien Analysis o Homogeneous and Heeogeneous Gas Resevois using The TS Technique. CT and F - Ciencia, Tecnología y Fuuo. 4(4): Feovich M.J ecline Cuve Analysis Using Type Cuves. Jounal o Peoleum echnology. pp Tiab Analysis o Pessue and Pessue eivaive wihou Type Cuve Maching: Sin and Wellboe Soage. Pape 5476 pepaed o he pesenaion a he poducion Opeaion Symposium held in Olahoma Ciy, OK, USA. pp Tiab Analysis o Pessue and Pessue eivaive wihou Type Cuve Maching: Veically Facued Wells in Closed Sysems. Jounal o Peoleum Science and Engineeing. 11: Van Evedingen A. F. and Hus. W The Applicaion o he aplace Tansomaion o Flow Poblems in Resevois. Tans., AIME. 186: Tiab Advances in pessue ansien analysis - TS echnique. ecue Noes Manual. The Univesiy o Olahoma, Noman, Olahoma, USA. p Aps J. J Analysis o ecline Cuves. Peoleum Tansacion, AIME. 160: Aab N Applicaion o Tiab s iec Synhesis Technique o Consan Boom Hole Pessue Tes. M.Sc. Thesis. The Univesiy o Olahoma, Noman, OK. USA. Cinco-ey H., Samaniego F. and ominguez N Tansien Pessue Behavio o a Well wih a Finie Conduciviy Veical Facue. Sociey o Peoleum Engineeing Jounal. pp , Augus. Cinco-ey H., Samaniego F. and ominguez N Tansien Pessue Behavio o a Well wih a Finie Conduciviy Veical Facue. Soc, Pe, Eng. J , Augus. Ealoughe R.C., J Advances in Well Tes Analysis. Monogaph seies, SPE o AIME, allas. 5. Escoba F.H., Rojas M.M. and Bonilla.F. 01. Tansien-Rae Analysis o ong Homogeneous and Naually Facued Resevoi by he TS Technique. 748

11 VO. 9, NO. 5, MAY 014 ISSN ARPN Jounal o Engineeing and Applied Sciences Asian Reseach Publishing Newo (ARPN). All ighs eseved. Nomenclaue A ainage aea, B Oil volumen aco, b/stb C Wellboe soage coeicien, bbl/psi C imensionless acue conduciviy c Sysem compessibiliy 1/psi h Fomaion hicness, Pemeabiliy, md w Facue conduciviy, md- ( w ) app Appaen acue conduciviy, md- 1/q Recipocal ae, /STB 1/q imensionless ecipocal ae P i Iniial esevoi pessue, psi P w Well-lowing pessue, psi P Pessue, psi e ainage adius, w Wellboe adius, s Sin aco s Appaen sin aco Time, h Time ead on deivaive cuves o bilinea Tiempo adimensional *1/q imensionless ecipocal ae deivaive *(1/q) Recipocal ae deivaive, ay/stb Gee Change γ g Ga speciic gaviy φ Poosiy ρ ensiy, lbm/ 3 µ Viscosiy, cp Suijos g i BPSSi B B1 BBRi Bi BPSSi BR BR1 1 BRi Ri PSSi o PSS R, RBi RBRi RPSSi w Gas Inecep, iniial imensionless Bilnea-pseudoseady-sae inecep Bilinea Bilinea a 1 h o 1 psi-h/cp Bilinea-biadial inecep Bilinea-linea inecep Bilinea-pseudoseady sae inecep Biadial Biadial a 1 h o 1 psi-h/cp inea inea a 1 h o 1 psi-h/cp inea-biadial inecep inea-adial inecep inea-pseudoseady-sae inecep Oil Pseudoseady-sae Pseudoadial Radial-Bilinea inecep Radial-Biadial inecep Radial-pseudoseady-sae inecep Well 749