REFORMULATION OF THE ELLIPTICAL FLOW GOVERNING EQUATION FOR A MORE COMPLETE WELL TEST DATA INTERPRETATION IN HORIZONTAL WELLS

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1 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com REFORMUATION OF THE EIPTICA FOW GOVERNING EQUATION FOR A MORE COMPETE WE TEST DATA INTERPRETATION IN HORIZONTA WES Javier Andre Marine, Freddy Humbero Ecobar and ui Fernando Bonilla Univeridad Surcolombiana, Av. Parana - Cra., Neiva, Huila, Colombia fecobar@uco.edu.co ABSTRACT Before year 000, ellipical flo regime a conidered a a raniion period found in a horional ell ranien e. I i recognied by a -lope (or 0.35-lope endency on he preure derivaive plo. Fe reearche have been conduced on he ranien analyi for uch flo regime. Since he fir model doe no provide a pracical ay of obaining he reervoir horional permeabiliy, a ne model hich i funcion of he reervoir lengh along he -direcion, reervoir hicne, horional ellbore lengh, ell radiu and horional reervoir permeabiliy, a preened in 004. Our eperience, hoever, ha indicaed o u ha hi model omeime fail o provide accurae value of horional permeabiliy. Therefore, he fir model preened in he lieraure, hich depend upon he permeabiliie in he horional direcion, ha been reaen in hi udy ih a ligh modificaion and uccefully eed in many cenario. The convenional raigh-line and TDS mehod are ued a inerpreaion echnique for crude and ga flo. Boh real ime and peudoime ere implemened for ga flo. Keyord: horional ell, ellipical flo, radial, anioropy, peudorradial, permeabiliy, inerecion poin, characeriic line.. INTRODUCTION When permeabiliy varie from one direcion o anoher, he reervoir i conidered o be anioropic. Permeabiliy meaured parallel o a bedding plane may be everal ime larger han ha meaured perpendicular o he bedding plane. I i normally acceped ha mo reervoir roc have a loer verical permeabiliy han horional permeabiliy. Thi anioropy ha been recognied and i ofen facored ino he analyi of parially compleed verical ell e, for eample. While i i agreed ha horional permeabiliy anioropy can occur, i i rouinely negleced in verical ell e analyi bu i i more common in horional ell e analyi. Conideraion on he ellipical flo in horional ell are fairly ne. The fir or of hi ind hoed up in 000, Iaaa e al. (000, in hich hey inroduced he ellipical flo regime in horional ell. Their preure derivaive governing equaion a a funcion of he areal permeabiliy and he / y raio. They found ou ha he horional permeabiliy anioropy ha no influence on he duraion of hi flo period bu i affec i aring ime. They did no ue non-linear regreion analyi a he inerpreaion echnique, bu ue a poin on he preure derivaive during he ellipical flo o eimae he permeabiliy anioropy raio ih an epreion hey provided in heir or. They aed ha he lope of he preure derivaive during uch flo regime had a value of 0.35 hich i lighly differen o he value of a aen by Tiab (994 for verical ell and and Ecobar e al. (004 for horional ell. Chacon e al. (004 inroduced a ne preure derivaive equaion hich depend upon he reervoir lengh along he -direcion, reervoir hicne, horional ellbore lengh, ell radiu and horional reervoir permeabiliy. Thi flo regime correpond o he raniion period beeen early linear and peudorradial flo regime. I i characeried for a -lope line on he preure derivaive log-log plo (Figure-. Ecobar e al. (004 obained he governing preure equaion by inegraing he model preened by Chacon e al. (004. They applied he TDS, Tiab (993, mehodology for characeriing he ellipical flo regime o ne equaion uing characeriic line and poin found on he preure and preure derivaive plo ere developed o obain areal permeabiliy, ( y 0.5, he reervoir lengh along he - direcion, h, permeabiliy in he y-direcion, y, and he geomerical in facor,, caued by he convergence from early-linear flo o ellipical flo regime. The mo recen reearch on hi iue a preened by Ecobar and Monealegre (008. Uing he model developed by Chacon e al. (008, hey implemened he convenional-raigh line mehod for characeriing he ellipical flo regime, o he reervoir areal permeabiliy and ellipical in facor can be eimaed. Hoever, in any of he above o cae, i.e., TDS or convenional analy i ha been noed ha hee epreion fail omeime o provide accurae value of he parameer, hen, a ne formulaion i preened here. Therefore, he model propoed by Iaaa e al. (000 a ued for a beer characeriaion of he ellipical-flo by mean of boh he TDS echnique and convenional analyi. Hoever, a ligh variaion of he model a inroduced. The 0.35 eponen a changed for. Pracical eample are given o demonrae he accuracy of he preened mehodology.. FUNDAMENTA EQUATIONS The dimenionle quaniie ued in hi or for he oil phae are defined a, 304

2 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com P D h y P ( 4.qµB D y ( φµcr.e+0 P D and D*P' D.E+0.E+00.E-0 Early Radial Early inear ipical Flo Peudo-eady ae ae radial (peudorradial Flo ae inear.e-0.e+00.e+0.e+0.e+03.e+04.e+05.e+06.e+07.e+08.e+09 Figure-. Idenificaion of he early-radial, early-linear, ellipical and peudorradial flo regime on he preure and preure derivaive plo. D Preure i implici in Equaion ( ince for gae, boh vicoiy and compreibiliy are funcion of preure. In fac, ha i hy peudoime a inroduced. For ga flo, he definiion of he peudopreure funcion, Al-Huainy e al. (966, i given by: h y mp ( md ( P (3 4.5qT c To accoun for variaion of he oal compreibiliy-vicoiy produc in ga flo behavior, Agaral (979 inroduced he peudoime concep, a follo: y Da ( a P φr 3. MATHEMATICA DEVEOPMENT There are o ype of ellipical flo regime ha may occur in horional ell. The fir one i in he verical plane a early age hich rani ino early radial a can be oberved in Figure-. I i no pracical ince i canno be oberved in any field e becaue of ime and ellbore orage. The econd ellipical flo occur in a region before he peudorradial flo regime. Thi depend upon he horional ell lengh compare ih reervoir idh in he direcion of he horional lengh, and he horional permeabiliy. Thi ype of ellipical flo hich e delled on ill occur if he horional lengh i mall compared ih he reervoir idh in he horional ell direcion. The inde 0.35 or ill vary depending on hi feaure. I i no a fied inde. Thi ellipical flo i purely in horional plane and depend upon horional (4 lengh compare ih he reervoir idh and horional permeabiliy anioropy. For uch cae, Chacon e al. (004 propoed he folloing preure governing equaion: P D 0.7 D hh + S 5 r π Thi model, hoever, fail omeime o provide accurae value of he eimaed parameer, hen, he model propoed by Iaaa e al. (000 a ued uing an inde of o be conervaive ih he original or of Tiab (994 and Ecobar e al. (004, D* PD' 0.74 D y 0.35 A een in Equaion 6, he preure derivaive i a funcion of he horional anioropy, y /, and he areal permeabiliy, ( y 0.5, maing difficul o olve for permeabiliy in any direcion, or y, hen, a ne dimenionle preure equaion i preened here: 0.7 r P.5 + y D D Afer plugging Equaion y ino Equaion 7 yield, q B q B P + (8 f µ 4. µ 0.7 h y ( φc h y P m + b f (5 (6 (7 305

3 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com For preure buildup eing, afer applying he uperpoiion concep, he folloing epreion i obained: 5.47 qµ B P ( + h c y ( φ p (9 Equaion 8 and 9 ugge ha a Careian plo of P v. (for dradon or P v. [( p + - ] (for buildup give a linear rend hich lope and inercep allo for he eimaion of hen horional anioropy and ellipical in facor, uch a: y qµ B (0 h c m ( φ 0.7 hb y 4.qµ B ( Characeriic poin and line a Taing derivaive o Equaion 7 ih repec o dimenionle ime yield: 0.7 r D* P' D D y ( Afer plugging Equaion y ino Equaion and olving for he horional permeabiliy yield, 5.554qµ B 0.64 y 0.7 h( φc ( * P' We can alo olve for he horional ell lengh, : qµ B 0.5 h( φc ( y ( * P' (3 (4 b The in facor caued by he ellipical flo regime reul from dividing he dimenionle preure equaion, Equaion 7, by he dimenionle preure derivaive, Equaion. Afer plugging he dimenionle quaniie, Equaion and, ino he above epreion and olving for he in facor e obain: P ( * P' 5.44 φµ c (5 Being P and (* P are he preure and preure derivaive value read on he ellipical flo line a any convenien ime,. The oal in facor i he ummaion of he mechanical in ( m, and he geomerical in facor: he -direcion peudoin, he - direcion peudoin and he in caued by he ellipical flo regime. c An equaion for he -direcion permeabiliy reul from he inerecion poin of he early radial line ih he ellipical line i given a: 0.4 h φµ c er ell (6 d Permeabiliy in he -direcion can be eimaed from he inerecion poin beeen he early- linear and ellipical flo regime: φµ c 9.79 iel, ell (7 e The inerecion poin of he ellipical flo regime and he peudorradial flo regime, or lae-radial flo regime, alo provide an epreion o eimae horional ell lengh: 0.7 ell pr φµc (8 f The la inerecion poin correpond o ha of he lae linear and ellipical flo regime. Thi alo provide an epreion o eimae he -direcion permeabiliy: 0.7 φµ c 9.79 h ell ll (9 The epreion for ga flo ih boh rigorou ime and peudo ime are provided in appendi A. 4. APPICATIONS 4.. Eample Goode and Thambynayagam (987 preened an eample of a preure buildup e of a horional oil ell compleed in he cener of a emi-infinie anioropic reervoir. Relevan informaion i given in Table-. Deermine -, y- and -direcion permeabiliie and in facor. Soluion Three ell-defined flo regime are oberved in Figure-: early radial, early linear and ellipical. The peudorradial flo regime i een only a he final period of ime and i i no ell defined. The folloing informaion a read from Figure-: (* P ell pi ell 0.79 hr er-ell 0. hr Tel-ell hr er-el 0.8 hr ell-pr 0.8 hr 306

4 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com Tel-pr 8 hr Deermine and from he inerecion ime of he linear line ih eiher early-radial or peudorradial raigh line. h φµ c ( ( 0. (.5 ( md 0.8 iel, pr ier, el ( 30.77( 0. (.5 ( ( φµ c md 8 Eimae ( y 0.5 uing Equaion 3, 5.554qµ B 0.64 y 0.7 h( φc ( * P' y (3000(.5 (.5( (60 (0.(3 0 (000 ( md Verify and from he inerecion ime of he ellipical line ih eiher early radial line or peudorradial line, er-ell (Equaion 6 and ell-pr (Equaion h φµ c er ell (60(48.49 (0.(.5( md ( ell pr φµc 000 el - pr 8.0 hr P and * P', pi 00 0 (* P' ell pi el - ell hr ell - pr 05 hr er - ell 0. hr er - el 0.8 hr ell 0.79 hr , hr Figure-. Preure and preure derivaive for eample. 0.7 (48.49( f 5 0.(3 0 (.5 Verify from Equaion 7, φµ c 9.79 iel, ell 5 (0.(30 (.5( md The mechanical in facor i found from he early radial flo period: P 55.7 (00 5(0. m - ln ln (* ' 9.4 (0.(.5( er y er 5 P er hr φµ cr ( Eimae he in facor from he linear-flo regime, ( 5 ( el P el m+.3 5 h φµ c ( * P' el 60 (0.(.5( The geomerical in facor caued by he ellipical flo regime i found from Equaion 5: P ( * P' 5.44 φµ c S 57.5 (47.5( (0.(.5 ( 3 0 (

5 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com 4.. Eample Deermine ( y 0.5 and in facor for he previou eample uing he convenional mehod. Soluion A hon in Figure-, he ellipical flo characeried by a -lope line on he preure derivaive rend indicae ha hi flo ei approimaely beeen and 50 hr. A he ame period of ime, he lope and inercep of he linear plo of Figure- 3 ere m pi/hr and b 7. pi P, pi b 7. pi m pi/hr ipical flo dominaed period Figure-3. Careian plo of preure drop v. ime o he poer 9/5 for eample., hr (Ky 0.5 i obained From Equaion 0, qµ B 5.47(3000(.5 (.5(50 y 70.8md h( φc m (60 (0.(3 0 (000 (58.33 The in facor caued by he ellipical flo regime i obained from Equaion, hb y (70.8(60( qµ B 4.(3000(.5( Eample 3 Oan e al. (989 preened a ell preure e for a horional oil ell compleed a he cener of a emiinfinie anioropic reervoir. Table- ho informaion of reervoir and ell properie. Deermine -, y- and - direcion permeabiliie. Soluion The preure and preure derivaive curve of Figure-4 ehibi five ell-defined flo regime a follo: early radial, early linear, ellipical, peudorradial and peudoeady ae. The folloing informaion i read from Figure-4, (* P ell 8.4 pi ell 30. hr er-ell.6 hr Tell-pr 860 hr (* P pr 7.7 pi (* P er.9 pi Tel-pr 40 hr er-el.5 hr el-ell 8.0 hr Deermine and from he inerecion ime of he linear line ih eiher early-radial or peudorradial raigh line. h φµ c ( 30.77( 0.4(. ( md.5 iel, pr ier, el ( ( ( ( 30.77φµ c ( md 40 Eimae ( y 0.5 uing Equaion 3, 5.554qµ B 0.64 y 0.7 h( φc ( * P' y 5.554(5000(. (.(4.85 ( (66 (8.4 ( md Verify ( y 0.5 from he peudorradial flo regime, Engler and Tiab (996, 70.6qB µ 70.6(5000(.(. y 03.9 md h ( * P' (84(7.7 pr Eimae y from Early radial flo regime 308

6 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com 70.6 qµ B 70.6(5000(.(. y md ( * P' erhr.6(66(.9 Verify and from he inerecion ime of he ellipical line ih eiher early radial line or peudorradial line, er-ell (Equaion 6 and ell-pr (Equaion h φµ c er ell (84(4.85 (0.4(.( md ( P and * P', pi 00 ell - pr 900 hr (* P' pr 7.7 pi el - pr 40 hr (* P' ell 8.4 pi 0 el - ell 8.0 hr (* P' er.9 pi ell 30. hr er - ell.6 hr er - el.5 hr , hr Figure-4. Preure and preure derivaive for eample ell pr φµc 0.7 (4.85( f 5 0.4(5 0 (. Verify from Equaion 7, φµ c 9.79 iel, ell 5 (0.4(50 (.( md Eample 4 A ell preure e a imulaed in a ga reervoir uing he informaion from Table-. Preure and preure derivaive daa are repored in Figure-5. Deermine -, y- and -direcion permeabiliie. Soluion The log-log plo of peudopreure and preure derivaive again peudoime i given in Figure-5. From ha plo he folloing informaion a read: Deermine and from he inerecion ime of he linear line ih eiher early radial or peudorradial line, h φ ( 30.77( md ( P 9000 a a el pr er el ( 30.77( 0.5( φ.73md 6 ( P 5 0 Eimae ( y 0.5 uing he Equaion A.7, qT P g y ( 0.7 a hφ ( a( P* m( P' (000( (.73 y 5 ( md 0.7 (50(0.5 (00 (6933 Eimae y from early radial flo regime, 7.6qT g 7.6(000( y 4.4 md ( a( P* mp ( ' 9.93(00(500 er 309

7 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com.e+07 m(p and *m(p', pi /cp a.e+06.e+05 ( a( P * m( P' ell 6933 pi /cp ( a( P * m( P' er 500 pi /cp.e+04 a( P er el 9000pi hr/cp a( P er ell 9000pi hr/cp 5 a( P ell pi hr/cp.e+03.e+0.e+0.e+03.e+04.e+05.e+06.e+07.e+08.e+09.e+0 a, Pi hr/cp P 6 a( el pr 5 0 pihr/cp P P 7 a( ell pr.9 0 pi hr/cp 5 a ( el ell.7 0 pi hr/cp Figure-5. Peudopreure and Peudopreure derivaive for eample 4. Verify from Equaion A.0, φ 9.79 a ( P el ell (0.5( md 5.70 Verify and from he inerecion ime of he ellipical raigh line ih eiher early-radial or peudorradial line, er-ell (Equaion A.9 and ell-pr (Equaion A h φ a( P er ell ( ( md ( a( P ell pr φ ( f ( ANAYSIS OF RESUTS The reul obained from he ored problem agree very ell ih eiher hoe from he lieraure ource or he inpu value ued for imulaion implying he accuracy of he ne propoed governing equaion, herefore, here i no need of imulaing eample and 3 for verificaion. 30

8 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com Table-. Reervoir and ell parameer for eample. Parameer Value, oil flo Value, ga flo Eample and Eample 3 Eample 4, md 50 y, md , md 5 0 q, BPD MSCF/D µ, cp B, bbl/stb.5. - h, f , f φ, % c, (pi h, f h, f r, f m T, F - 50 Nomenclaure B Oil formaion facor, bbl/stb c Compreibiliy, /pi h Formaion hicne, f Permeabiliy, md h Areal permeabiliy ( y 0.5, md Horional ell lengh, f m(p Peudopreure funcion, pi /cp P Preure, pi P D Dimenionle preure derivaive P D Dimenionle preure q Flo rae, bbl/d. For ga reervoir he uni are Mcf/D r Well radiu, f Sin facor m Mechanical in facor Geomerical in facor or -direcion peudoin facor Geomerical in facor or -direcion peudoin facor T Reervoir emperaure, ºR Time, hr * P Preure derivaive funcion, pi * m(p Peudopreure derivaive funcion, pi /cp a (P Peudoime funcion, pi h/cp * m(p Peudopreure derivaive funcion ih repec o rigorou/real ime a * m(p Peudopreure derivaive funcion ih repec o peudoime Dimenionle ime D 3

9 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com Gree φ Suffice Change, drop Flo ime, hr Poroiy, fracion µ Vicoiy, cp hr Time of hr D el ell er g h i ll pr o c y Dimenionle Early linear flo period ipical flo period Early radial flo period Ga Horional Inerecion ae linear flo period Peudoradial flo period Oil Sandard condiion Toal Well -direcion inde y-direcion inde -direcion inde 6. CONCUSIONS An eiing governing equaion for ellipical flo preened in horional ell a uilied and lighly modified o provide a more complee TDS and convenional mehodologie for he eimaion of he horional permeabiliy, -, and direcion permeabiliie, ellipical in facor and horional ellbore lengh are accuraely eimaed for oil and ga reervoir. Thee equaion ere uccefully eed ih eample provided in he lieraure and one ynheic eample. ACKNOWEDGEMENTS The auhor ould lie o han and acnoledge he financial uppor of Univeridad Surcolombiana for he compleion of hi udy. REFERENCES Agaral G Real ga peudo-ime a ne funcion for preure buildup analyi of MHF ga ell. Paper preened a he SPE 54 h Technical Conference and Ehibiion of he Sociey of Peroleum Engineer of AIME, a Vega, NV. Sepember 3-6. SPE 879. Al-Huainy R., Ramey H.J. and Craford P.B The flo of real gae hrough porou media. Journal of Peroleum Technology. May. pp , Tran. AIME, 37. Chacon A., Djebrouni A. and Tiab D Deermining he Average Reervoir Preure from Verical and Horional Well Te Analyi Uing Tiab Direc Synhei Technique. SPE Proceeding, SPE Aia Pacific Oil and Ga Conf. and Ehibiion, Perh, Auralia. 8-0 Ocober. Engler T.W. and Tiab D Analyi of Preure and Preure Derivaive ihou Type-Curve Maching. 6- Horional Well Te in Anioropic Reervoir. Journal of Peroleum Science and Engineering. 5: Ecobar F.H., Muño O.F. and Sepulveda J.A Horional Permeabiliy Deerminaion from he ipical Flo Regime for Horional Well. CT and F - Ciencia, Tecnología y Fuuro. (5: 83-95, December. Ecobar F.H. and Monealegre-M M Deerminaion of Horional Permeabiliy from he ipical Flo of Horional Well Uing Convenional Analyi. Journal of Peroleum Science and Engineering. 6: 5-0. Goode P. A. And Thambynayagam R. K Preure Dradon and Buildup Analyi of Horional Well in Anioropic Media. SPE Formaion Evaluaion. pp Iaa M. B., Zaoral K., Ambaha A. K. and Maar Deerminaion of Horional Permeabiliy Anioropy from Horional Well Te. Paper preened a he 000 SPE Saudi Arabia Secion Technical Sympoium held in Dhahran, Saudi Arabia. -3 Ocober. Oan E., Raghavan R. and Johi S. D Horional Well Preure Analyi. SPEFE Tiab D Analyi of Preure and Preure Derivaive ihou Type-Curve Maching: -Sin and Wellbore Sorage. Journal of Peroleum Science and Engineering. : 7-8. Tiab D Analyi of Preure Derivaive ihou Type-Curve Maching: Verically Fracured Well in Cloed Syem. Journal of Peroleum Science and Engineering. :

10 VO. 7, NO. 3, MARCH 0 ISSN Aian Reearch Publihing Neor (ARPN. All righ reerved..arpnjournal.com APPENDIX A. GAS RESERVOIRS EQUATIONS Dimenionle preure and preure derivaive 0.7 r mp ( y D D 0.7 r D* m( P' D D y Convenional mehod q g T y 0.7 h φ ( µ c i m (A. (A. (A.3 hb y (A.4 4.5qT g TDS rigorou ime qT g y 0.7 ( ( * ( ' h φ µ c i m P mp ( ( * m( P' 5.44 φµ c TDS peudoime q T P g y ( 0.7 a hφ ( a( P* m( P' (A.5 (A.6 (A.7 mp ( a ( P ( a( P* m( P' 5.44 φ (A h φ a( P er ell φ 9.79 a( P el ell 0.7 a( P ell pr φ 0.7 φ 9.79 h a( P ell ll (A.9 (A.0 (A. (A. 33