PRESSURE AND PRESSURE DERIVATIVE ANALYSIS FOR HYDRAULICALLY FRACTURED VERTICAL WELLS WITH FACE SKIN

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1 VOL., NO. 3, JULY 06 IN ARPN Jounal o Engineeing and Applied ciences Asian Reseach Publishing Neok (ARPN). All ighs eseved..apnjounals.com PREURE AND PREURE DERIVATIVE ANALYI FOR HYDRAULIALLY FRATURED VERTIAL WELL WITH FAE KIN Feddy Humbeo Escoba, Ruben Albeo Gonzalez, Laua Maia Henandez and laudia Macela Henandez Univesidad ucolombiana/enigaa, Avenida Pasana, Neiva, Huila, olombia The Univesiy o Oklahoma, Mebone chool o Peoleum and Geological Engineeing, E. Boyd. E Rm, Noman, Ok, Unied aes o Ameica escoba@usco.edu.co ABTRAT Pseudolinea lo akes place in inie-conduciviy acues hen he acue ace possesses a modeae o high ace skin. Then, an appaen linea lo ill be developed once ellboe soage has decayed. I given he case, he inepee should avoid inepeing he es using he model o an ininie-conduciviy acue. Alhough, he saighline convenional analysis has been included, his pape pesens a mehodology ha uses chaaceisic poins and lines ound on he pessue deivaive plos. The omulaed echnique as applied o a synheic example and successully compaed o convenional analysis. Keyods: acued ells, acue skin, TD Technique, linea lo, bilinea lo.. INTRODUTION Undoubedly, he ok conduced by Gingaen, Ramey and Raghavan (974) as a beakhough in he ield o ell es inepeaion o veical acued ells. They used Geen uncions o develop he pessue soluion o ininie-conduciviy and uniom-lux hydaulic acues. Then, inco, amaniego and Dominguez (978) povided a semi analyic soluion o descibing he pessue behavio o an oil ell dained by a inie-conduciviy acue. They povided he onse o deining hehe o no a acue has ininie o inie conduciviy. The limiing value as se o 00 hich o pacical sense became 300. Then, a acue has ininie conduciviy hen is dimensionless conduciviy value, D, is geae han 300. I has inie conduciviy, oheise.\ eveal eseaches have been conduced on he omulaion o mehodologies o ell es inepeaion o hydaulically acued veical ells. To name a e o hem: Escoba e al. (999) applied he TD Technique, Tiab (993a), o inie-conduciviy acued veical oil ells. They successully applied he developed expessions o ield and synheic daa and poined ou he pacicabiliy o his mehodology. Lae, Nunez, Tiab and Escoba (003) exended he ome ok o gas ells using he pseudopessue and pseudopessue deivaive uncion. Tiab (993b) inoduced he ellipical lo egime and povided is chaaceizaion via TD Technique applied o oil ells. The convenional analysis mehodology as lae pesened by Escoba, Monealege-M and anillo (006). Escoba e al. (0) ound o gas ells ha he pseudoime uncion helps o bee chaaceized boh acued ells and naually acued omaions. They povided TD echnique as he inepeaion ool. The same yea, Escoba, Bonilla and icey (0) chaaceized heavy oil omaions dained by ininie-conduciviy hydaulically acued veical ells. And, ecenly, Escoba, Zhao and Zhang (04) pesened an inepeaion mehodology o bi-zonal (composie) gas esevois also using he TD Technique. Escoba, Ghisays and Bonilla (04) pesened a ne model o he ellipical lo egime and homogeneous and acued occuing omaions hen hey ae poduced hough acued ells. Also, Escoba e al. (05) pesened ell es inepeaion in acued ells in hich he heshold pessue gadien plays an impoan ole. All he ome oks eeed o ansien pessue analysis. As a as ansien ae analysis is concened, Escoba, aso and Mosquea (04) pesened he TD mehodology o ininie and inie-conduciviy hydaulically acued gas and oil ells. The model o pseudolinea lo egime as inoduced by Wong, Haingon and inco-ley (984) and hey applied convenional analysis o ell es inepeaion in such sysems. A ypical o case o a damaged acued ell is given in Figue-. Ealy daa is aeced by pseudolinea lo egime and he bilinea behavio shos up once ellboe soage eec ae vanished. The pupose o his ok is o exend TD Technique o pseudolinea lo. The developed expessions ee successully esed ih synheic examples.. MATHEMATIAL FORMULATION.. Mahemaical model 868

2 VOL., NO. 3, JULY 06 IN ARPN Jounal o Engineeing and Applied ciences Asian Reseach Publishing Neok (ARPN). All ighs eseved..apnjounals.com P & *P ', psi Pseudolinea lo egime Wellboe soage zone Bilinea lo egime , h Figue-. Peasall ield hoizonal ell daa. Ae ea (0). Wong e al. (984) pesened he mahemaical soluion o pessue behavio unde pseudolinea condiions: In-siu poppan pemeabiliy, k Facue idh, Aleed zone hickness, a Facue ace aleed zone pemeabiliy, k a Figue-. Finie-conducive acue ih acue ace skin. Ae ea (0). x s a k ka ubsiuing he dimensionless quaniies given by Equaions (3), (4), (6) and (7) ino Equaion (), i yields: (7) ( P ) ( ) 4 3/ / D Dx s D hich pessue deivaive is: ( * P ') ( ) 8 3/ / D D Dx s D hee he dimensionless quaniies ae deined by k D x = cx s D () () (3) k a k khp k a 4.qB k 4 3/ xk k cx a ka Equaion (8) implies ha a aesian plo o pessue o pessue dop vesus he squae-oo o ime ill yield a saigh line hich geneal equaion is given by: k (8) khp PD (4) 4.qB D kh(* P ') * PD' (5) 4.qB D kx The sysem unde sudy is given in Figue-. The acue is suounded by a zone o educed pemeabiliy acue ace skin- hen lo oads he acue ace is impaied, hen only lo along he acue is seen. This is eleced as a linea lo egime... onvenional analysis Wong e al. (984), ho inoduced convenional analysis o he case unde sudy, deined he poduc o acue ace skin by he hal-acue lengh by: (6) P b m (9) Noice ha o pessue buildup es, he andem squae oo is ploed insead. Ae subsiuing Equaion (7) in Equaion (9), he inecep and slope can be used o esimae acue conduciviy and he a(k/k a-) poduc, especively: 798.7q B B mbh kc k b h k k ka 33.7qB a (0) ().3. TD Technique The pessue and pessue deivaive behavio geneaed by he model poposed by Wong e al. (984) is given in Figue-3. A linea lo along he acue chaaceized by a slope o 0.5 on he pessue deivaive is given insead o a bilinea lo egime ih slope o ¼ 869

3 VOL., NO. 3, JULY 06 IN ARPN Jounal o Engineeing and Applied ciences Asian Reseach Publishing Neok (ARPN). All ighs eseved..apnjounals.com on pessue deivaive. I he skin and damage zone idh ae small; hen, he bilinea lo egime can be seen as shon by he ield example povided by ea (0). The pseudolinea behavio is mahemaically eaed belo. P & *P ' D Dx D.E+0.E+00.E-0.E-0.E-03 Pseudolinea lo egime ` Radial lo egime.e-04.e-03.e-0.e-0.e+00.e+0.e+0.e+03.e+04.e+05 Dx Figue-3. Dimensionless pessue and pessue deivaive vesus dimensionless ime behavio o a inieconduciviy acued veical ell ih ace skin, s = 0.08 and D = 00. Ae subsiuing Equaions (3), (5) and (6) ino he pessue deivaive govening equaion, Equaion (), e obain, kh P 4.qB 8 k c x 3/ ( * ') k s () ubsiuing Equaion (7) and solving o he acue conduciviy, i yields:.66 q B (3) k c (* ') h P a k a Equaion () is used by eading an abiay poin duing he pseudolinea lo egime. Hoeve, i is bee o ead an aveaged poin hich is obained by exapolaing he one-hal saigh line o he ime o one hou, hen, Equaion (3) becomes: ubsiuing he dimensionless paamees in he above expession: s k ( ) P x c ( * P') / (6) Use o Equaion (7) in Equaion (6) leads o ind he a(k/k a-) poduc: k k ( ) P (7) a ka c ( * P') Duing adial lo he dimensionless pessue deivaive is a saigh line ih zeo slope hich akes he value o one hal. In such case, a hoizonal line is dan along he adial lo egime and he pessue deivaive coesponding o such line is ead and hoizonal pemeabiliy can be obained om he olloing expession developed by Tiab (993a): 70.6qB k h (* P') / (8) Tiab (993a) also povided an expession o ind he skin aco by eading he pessue dop a any abiay ime duing adial lo: m P k 0.5 ln 7.43 (* ') P c (9) inco e al. (978) pesened he pessue model o inie-conduciviy acues:.45 P 4 D / Dx D (0) Assuming he acue skin is negligible and subsiuing he dimensionless quaniies in he above expession and solving o he acue conduciviy, Tiab e al. (999) ound a ay o ind acue conduciviy: (4).66 qb k (* ') h P c a k a qb ckh( P) BL The pessue deivaive o Equaion (0) is: () Dividing he pessue equaion, Equaion (), by he pessue deivaive expession, Equaion (), i yields: ( P ) 8 ( * P ') ( ) D s / D D Dx (5) 0.67 * P ' 4 Dx D / Dx D () Also, Tiab e al. (999) subsiued he dimensionless paamees anohe om o obaining acue conduciviy: 870

4 VOL., NO. 3, JULY 06 IN ARPN Jounal o Engineeing and Applied ciences Asian Reseach Publishing Neok (ARPN). All ighs eseved..apnjounals.com.74 qb ckh (* P') BL (3) Equaions () and (3) use, especively, values o pessue and pessue deivaive a a ime o h. This is pacical since allos aveaging he pessue and pessue deivaive daa values: Tiab e al. (999) also pesened an impoan equaion ha enables us o ind acue conduciviy om he inesecion poin beeen he pessue deivaives o adial lo and bilinea lo egimes, RBLi, c k k RBLi 677 ( ) 3 (4).3.. Relaion beeen Pseudolinea and Bilinea lo egimes In his ok, e ound ha he poin o inesecion beeen he pessue deivaives o pseudolinea lo egime, Equaion (), and bilinea lo egime, Equaion (), o also esimae a(k/k a-) poduc by he applicaion o Equaion(7); k k a ka c PFLBLi (5).3.. Relaion beeen Pseudolinea and Radial Flo Regimes As menioned beoe, he dimensionless pessue deivaive akes he value o one hal duing adial lo egime. Then, i he igh-hand side o Equaion () is se o 0.5, he olloing expession can be obained: k a k kpflri c ( k ) a (6).3.3. Ohe Useul Relaionships Tiab (997) pesened an excellen expession o ind hal-acue lengh hen acue conduciviy is knon o vice vesa: x m e k (7) Economides, Waes and Dunn-Noman (988) pesened he belo o ind he skin aco hen adial lo is unseen bu he hal-acue lengh is knon: m ln x (8) inco-ley e al. (978) also pesened he olloing equaion o he same pupose: m ln x u0.u u ln x 0.8u 0.064u u 3 (9) (30) 3. YNTHETI EXAMPLE A synheic es as simulaed ih Equaion () ih he inomaion given belo: B =. bbl/tb q = 500 TB/D h = 30 =.5 cp x = 00 c = x0-5 psi - P i = 950 psi = 0 % k = 50 md = 0.3 D = 80 s = 0. Esimae acue conduciviy and he a(k/k a-) poduc using boh TD Technique and convenional analysis. oluion by TD Tecnique The olloing daa ee ead om Figue-4. = h P = 3.6 psi (*P ) = 64.5 psi =.07 h P = 35. psi (*P ) = 7.9 psi Ri = 3 h P & *P ', psi P 3.6 psi ( * P') 7.9 psi P 35. psi ( * P') 64.5 psi.07 h 3 h Ri h , h Figue-4. Pessue and pessue deivaive vesus ime log-log plo o he synheic example. Use o Equaions (8) and (9) allo inding pemeabiliy and mechanical skin aco: 70.6qB 70.6(500)(.5)(.) k 50. md h ( * P') (30)(64.5) 87

5 VOL., NO. 3, JULY 06 IN ARPN Jounal o Engineeing and Applied ciences Asian Reseach Publishing Neok (ARPN). All ighs eseved..apnjounals.com m 3.6 (50.)(436.4) 0.5 ln (0.)(.5)(0 )(0.3 ) m = -6.5 Equaion (7) is use o ind he a(k/k a-) poduc: k 50.(.07) 35. a 5 ka (0.)(.5)(0 ) 3.6 a(k/k a-) = 0.9 Deemine acue conduciviy ih Equaion (3) (.5)(.) (0.)(.5)(0 ) 30(3.6) k = md- Deemine he dimensionless acue conduciviy ih Equaion (6); D kx (50.)(00) Equaion (6) allos veiying he a(k/k a-) poduc: k (50. )(3) a 5 ka (0.)(.5)(0 )(7057.) a(k/k a-) = 0.46 oluion by onvenional analysis The slope and inecep om he aesian plo o Figue-5 ae: m = 6. psi/h 0.5 b = 8. psi Find acue conduciviy and he a(k/k a-) poduc using Equaions (0) and (); P, psi m 6. psi / h 8. psi b , h Figue-5. Pessue dop vesus squae oo o ime aesian plo o he synheic example. / 798.7(500 )(.5)(.) (.5)(.) 5 (6.)(8.)(30 ) (50.)(0.)(0 ) k = md- k (8.)(30) a (50.)( ) k a 33.7(500)(.5)(.) a(k/k a-) = 0.67 Find dimensionless acue conduciviy ih Equaion (6); D kx (50.)(00) The esimaion o acue conduciviy in he example is 80. The TD echnique povided a close value han convenional analysis. ONLUION Ne expessions o he esimaion o acue conduciviy and he a(k/k a-) poduc (a oal o hee) ee developed o chaaceize he pseudolinea lo egime ound on hydaulic acue ih ace skin. The expessions ee succesully esed ih a simulaed example poviding accepable esuls compaed o he inpu-simulaed values and convenional analysis povided by inco e al. (978). AKNOWLEDGEMENT The auhos hank Univesidad ucolombiana, and he Univesiy o Oklahoma o poviding inancial suppo o he compleion o his sudy. Nomenclaue B Volume aco, b/tb b Inecep o he P vs. 0.5 aesian plo D Facue conduciviy c Toal sysem compessibiliy, psi - h Resevoi hickness, k Resevoi hoizonal pemeabiliy, md k a Facue ace aleed zone pemeabiliy, md m lope o he P vs. 0.5 aesian plo P Pessue, psi P i Iniial esevoi pessue, psi P Wellboe loing pessue, psi q Flo ae, BPD Wellboe adius, kin aco m Mechanical kin aco s kin acue in acue ace Facue skin Time, days D Dimensionless ime coodinae D*P D Dimensionless pessue deivaive (*P ) Pessue deivaive x Hal-acue lengh, Aleed zone hickness, a 87

6 VOL., NO. 3, JULY 06 IN ARPN Jounal o Engineeing and Applied ciences Asian Reseach Publishing Neok (ARPN). All ighs eseved..apnjounals.com D i BLi Ri REFERENE Geeks Poosiy, acion Viscosiy, cp uices Dimensionless Iniial Pseudolinea lo egime Pseudolinea an bilinea inesecion Pseudolinea an adial inesecion Radial inco L., H., amaniego V., F. and Dominguez A. N Tansien Pessue Behavio o a Well ih a Finie-onduciviy Veical Facue. ociey o Peoleum Enginees. doi:0.8/604-pa. Economides M.J., Waes Dunn-Noman Peoleum Well onsucion. John Wiley and ons, Ne Yok, UA. p. 6. Escoba F.H., Monealege-M. M. and anillo J.H onvenional Analysis o haaceizaion o Bi-Radial (Ellipical) Flo in Ininie-onduciviy Veical Facued Wells. T and F - iencia, Tecnología and Fuuo. 3(): IN Escoba F.H., Mainez L.Y., Méndez L.J. and Bonilla L.F. 0. Pseudoime Applicaion o Hydaulically Facued Veical Gas Wells and Heeogeneous Gas Resevois Using he TD Technique. Jounal o Engineeing and Applied ciences. 7(3): Escoba F.H., Bonilla D.F. and icey Y.Y. 0. Pessue and Pessue Deivaive Analysis o Pseudoplasic Fluids in Veical Facued Wells. Jounal o Engineeing and Applied ciences. 7(8). Escoba F.H., Zhao Y.L. and Zhang L.H. 04. Inepeaion o Pessue Tess in Hydaulically-Facued Wells in Bi-Zonal Gas Resevois. Ingenieía e Invesigación. 34(4): Escoba F.H., aso J.R. and Mosquea J Rae- Tansien Analysis o Hydaulically Facued Veical Oil and Gas Wells. Jounal o Engineeing and Applied ciences. 9(5): Escoba F.H., Zhao Y.L. and Fahes M. 05. haaceizaion o he naually acued esevoi paamees in ininie-conduciviy hydaulically-acued veical ells by ansien pessue analysis. Jounal o Engineeing and Applied ciences. 0(): Escoba F.H., Zhao Y.L., Pounik M., Liu Q.G. and Olaya-Main G. 05. Inepeaion o Pessue Tess in Uniom-Flux Facued Veical Wells ih Theshold Pessue Gadien in Lo Pemeabiliy Resevois. Jounal o Engineeing and Applied ciences. 0(0): Gingaen A.., Ramey H.J., J. and Raghavan R Applied Pessue Analysis o Facued Wells. J. Pe. Tech., July: Nunez W., Tiab D. and Escoba F. H. 003, Januay. Tansien Pessue Analysis o a Veical Gas Well Ineseced by a Finie-onduciviy Facue. ociey o Peoleum Enginees. doi:0.8/8095-m. ea G. 0. Well Tes Design and Analysis. Penn Well opoaion. Tulsa, OK. UA. p Tiab D. 993a. Analysis o Pessue and Pessue Deivaive ihou Type-uve Maching: - kin and Wellboe oage. Jounal o Peoleum cience and Engineeing, Vol., pp Also Pape PE 543 (993) Poducion Opeaions ymposium held in Oklahoma iy, OK. Tiab D. 993b. Analysis o Pessue and Pessue Deivaive ihou Type-uve Maching - III. Veically Facued Wells in losed ysems. ociey o Peoleum Enginees. doi:0.8/638-m. Tiab D Advances in Pessue Tansien Analysis. UPTE Taining Manual, Noman, OK. p Tiab D., Azzougen A., Escoba F. H. and Beumen. 999, Januay. Analysis o Pessue Deivaive Daa o Finie-onduciviy Facues by he Diec ynhesis. Technique. ociey o Peoleum Enginees. doi:0.8/50-m. Wong D.W., Haingon A.G. and inco-ley H Applicaion o he Pessue Deivaive Funcion in he Pessue Tansien Tesing o Facued Wells. PE 3056, PEFE. (5): Escoba F.H., Ghisays-Ruiz A. and Bonilla L.F. 04. Ne Model o Ellipical Flo Regime in Hydaulically- Facued Veical Wells in Homogeneous and Naually- Facued ysems. Jounal o Engineeing and Applied ciences. IN (9):

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