Freddy Humberto Escobar 1, Yu Long Zhao 2 and Mashhad Fahes 3 1.

Transcription

1 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. CHARACTERIZATION OF THE NATURALLY FRACTURED RESERVOIR PARAMETERS IN INFINITE-CONDUCTIVITY HYDRAULICALLY- FRACTURED VERTICAL WELLS BY TRANSIENT PRESSURE ANALYSIS Freddy Humbero Eobar 1, Yu Long Zhao and Mashhad Fahes 3 1 Universidad Surcolombiana/CENIGAA, Avenida Pasrana - Cra 1, Neiva, Huila, Colombia Sae Key Laboraory o Oil and Gas Reservoir Geology and Eploiaion, Souhwes Peroleum Universiy, Xindu Sree, Xindu disric, Chendu, Sichuan, P. R. China 3 The Universiy o Oklahoma, Mewbourne School o Peroleum and Geological Engineering, E. Boyd S. SEC Rm, Norman, Ok, USA eobar@uo.edu.co ABSTRACT I has become common o hydraulically racure a naurally racured ormaion o increase he well s producion poenial. Since he mass ranser beween racured nework and hydraulic racure is much higher han ha rom mari o racures, he hydraulic racure-racure nework inerporosiy low parameer is much higher ha ha o mari-racure nework. As a resul, he ransiion period behavior rom naurally racured o homogeneo akes place beore radial low regime during he early bilinear, linear or ellipical (birradial) low regimes. The purpose o his paper is o provide epressions by boh convenional analysis and TDS echnique or characerizing he naurally racured parameers when he ransiion period inerrups he response o an ininie-conduciviy racure. The developed epressions or boh mehodologies were saisacorily esed wih simulaed eamples. Keywords: pressure ransien analysis, inerporosiy low parameer, TDS echnique, ellipical low, naurally racured reservoirs. 1. INTRODUCTION Even wells in naurally racured ormaions are being subjeced o hydraulic racuring since he bulk permeabiliy is low and well produciviy is no as high as epeced. Typical such cases have been seen in some wells o he Orinoco basin oohill in Colombia where he wells were racured j aer being drilled. Tiab (1994) was he irs o ideniy he ellipical low regime in pressure es daa o ininie-conduciviy hydraulically racured verical wells. He called i birradial low, provided he governing pressure and pressure derivaive equaions and implemened he TDS echnique or is characerizaion. This low regime can also be seen in horizonal wells beore he lae radial low regime shows up and is due mainly o horizonal anisoropy. The las model or he ellipical low in horizonal wells was presened by Marinez, Eobar and Bonilla (01). The original pressure governing model or ellipical low inroduced by Tiab (1994) involves he drainage area, which may aec he calculaions i he pressure es is oo shor or he developmen o he lae pseudoseady-sae period. An eperienced er o he TDS echnique may deal wih ha problem ing he poin o inersecion beween he radial low and he birradial low regime lines, Tiab (1994). However, o overcome his problem, Eobar, Bonilla and Ghisays-Ruiz (014) ormulaed a new ellipical/birradial low model or verical wells in eiher racured or unracured ormaions. Alhough Tiab s model, Tiab (1994), works perecly, i requires he knowledge o well-drainage area or esimaing he hal-racure lengh. However, someimes his condiion canno be me becae he pressure es needs o be run long enough or he developmen o he lae ime pseudoseady-sae period or he deeraion o he drainage area. The new model presened by Eobar e al. (014) ecludes he reservoir drainage area and slighly modiies Tiab s model, Tiab (1994), o accoun or naurally-racured double-porosiy sysems. Tiab and Beam (007) presened a pracical inerpreaion o he pressure behavior o a inieconduciviy hydraulically racured verical well locaed in a naurally racured reservoir. The inerpreaion is based on analyical equaions derived o deere permeabiliy, racure sorage capaciy raio, inerporosiy low coeicien, skin and wellbore sorage rom he pressure derivaive plo wihou ing ype-curve maching echnique. In oher words, hey implemened he TDS echnique, Tiab (1993), or such sysems. Par o he work presened by Eobar, Marinez and Monealegre (009) was oced on he implemenaion o convenional analysis or he work presened by Tiab and Beam (007). In heir work, Tiab and Beam (007) developed epressions or bilinear and linear low regimes bu ecluded he presence o he ellipical or birradial low regime. Thereaer, his work consiues an eension o 535

2 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. he research o Tiab and Beam (007) or ininieconduciviy racures where early birradial low is seen. This implemenaion was saisacorily esed wih synheic eamples.. MATHEMATICAL FORMULATION.1. Governing equaion The dimensionless pressure behavior in he Laplace domain or a hydraulically racured well in double-porosiy reservoirs was inroduced by Ozkan and Raghavan (1988): 1 s s ( s)[1 0.73) PD () s K 0 0( ) d (1) s s( s) khp PD (6) 141.qB D kh(* P ') * PD' (7) 141.qB Finally, he dimensionless racure conduciviy inroduced by Cinco-Ley, Samaniego and Doguez (1976) is deined as: k w C D (8) k For a pseudoseady-sae mari low model, he uncion o sorage capaciy and inerporosiy low parameer is given by: (1 ) () s () (1 ) s 9/5 D *P D ' * Eobar e al. (014) provided he ollowing pressure and pressure derivaive governing equaions or ellipical low regime: 5 D PD 9 6 D D* PD' 6 When = 1, Equaions (3) and (4) accoun or homogeneo reservoirs. For he case o naurallyracured ormaions, =, which is he dimensionless soraiviy coeicien. Consequenly, or he purpose o his work, will be replaced by he soraiviy raio... Dimensionless quaniies The dimensionless ime, based on hal-racure lengh, is given below: D (3) (4) k (5) c The dimensionless pressure and pressure derivaive parameers or oil reservoirs are given by: 1.E-0 [( 9/5 * PD ')* ] 0.03 D 1.E-06 1.E-05 1.E-04 1.E-03 1.E-0 1.E+0 1.E+03 1.E+04 1.E+05 9/5 * D Figure-1. Eec o he inerporosiy low parameer on he pressure and pressure derivaive during he ellipical low regime, = TDS echnique By plugging he dimensionless parameers in Equaions (3) and (4) and solving or he hal-racure lengh, we obain: qb hpbr1 c k qb h (* P') BR1 c k (9) (10) Figures-1 hrough 4 were generaed ing he analyical soluion provided by Ozkan and Raghavan (1988), Equaion (1), which considers low beween hydraulic racure and racure nework. As seen in Figure-1, once he ransiion period vanishes, he reservoir behaves as i i were homogeneo. In his case, Equaions (9) and (10) become: 5353

3 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. qb hpbr1 c k (11) qb.03 (* P') h ( c ) k (14) qb h (* P') BR1 c k (1) Where BR1 corresponds o he sraigh line drawn on he second ellipical low regime which occurs aer he ransiion period. However, beore he ransiion period eiher ellipical low or linear low may ake place. I linear low occurs a early ime, hen Equaions (9) and (10) m be replaced by hese wo epressions developed by Tiab and Beam (007); Fracure conduciviy can be ound ing an epression presened by Tiab (003); kw k e r s w (15) I is observed in Figure-1 ha he imum poin o he pressure derivaive imes he inerporosiy low parameer o he power is a consan: qb Ph ( c ) k (13) 9/5 ( D * D ') 0.03 P (16) The ollowing epression o deere he inerporosiy low parameer is obained once Equaion (7) is subsiued in Equaion (16): 9/5 D *P D ' * 9/5 [P & *P ' ]* D D D 1.E-0 [( 9/5 * PD ')* ] ma 0. D ( * / ) 0.475hr D ma E-03 1.E-0 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 * D Figure-. Correlaion o he maimum and imum pressure derivaive coordinaes during ellipical low period, = E+0 1.E-0 Heerogeneo behavior E-04 1.E-03 1.E-0 1.E+0 1.E+03 1.E+04 * D Homogeneo behavior Uni-slope line during ransiion period Figure-3. Eec o he soraiviy raio on he pressure and pressure derivaive during he ellipical low regime, = 100. qb kh(* P ') 5/9 (17) Figure- conains a log-log plo o pressure and pressure derivaive boh muliplied by he inerporosiy low parameer raised o he power 9/5 (or ) vers dimensionless ime muliplied by inerporosiy low parameer and divided by he soraiviy raio. During he early heerogeneo behavior, all curves uniy no maer he values o soraiviy raio. A maimum poin is displayed beore he ransiion period which coordinaes are given by: [( * P ') ] 0. (18) 9/5 D D ma D ma (19) Once Equaion (7) is subsiued ino Equaion (18), we obain: qb kh(* P ') ma 5/9 (0) Subsiuing Equaion (5) ino Equaion (19) yields wo epressions o ind eiher inerporosiy low parameer or soraiviy raio one as a uncion o he oher so; 5354

4 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. k ma (1) c is inerruped by he ransiion period, ollowed by he ellipical or birradial low. In such cases, dividing Equaion (14) by Equaion (1) leads o: c () k ma h * P' BR1 k qb * P ' (8) Figure-3 was prepared o uniy he uni-slope line showing up during he ransiion period. Such sraigh line was ied o: ( * P ') (3) 16/5 D D D Aer subsiuing he dimensionless parameer and solving or he inerporosiy low parameer, i yields: c h (* P') qb 5/16 (4) Equaion (4) works ing he coordinaes o any poin along he uni-slope line. During radial low regime, he value o he dimensionless pressure derivaive is one hal. Equaing his value o Equaion (3) and solving or he inerporosiy low parameer, we ge: The poin o inersecion beween he early birradial line, Equaion (4), and he uni-slope line during he ransiion period, Equaion (3), leads o he ollowing epression: 1 k BRi c 9/16 (9) Tiab and Beam (007) inroduced he governing pressure derivaive equaion or early linear low: D o: 1 D * PD' (30) Which, in combinaion wih Equaion (3), leads c k i 5/16 (5) c (31) k 3/5 Li Where i is he inersecion poin o he uni-slope line and he radial low regime. From observaion o he imum poin in Figure-, an epression wih a correlaion coeicien o was obained: 9/5 P D * P 9/5 Dpss ma ma (6) Dividing Equaion (10) by Equaion (1) and solving or he soraiviy raio, we ge:.7778 * P' BR1 * P' BR1 (7) Equaion (7) is valid whenever birradial/ellipical low eiss a boh side o he ransiion period. However, in ininie-conduciviy racure, i is possible ha he linear low occurs a early ime, and hen 1.E-0 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 * D Figure-4. Dimensionless pressure imes he inerporosiy low parameer o he power vers dimensionless ime muliplied by /, = Convenional analysis As seen in Figure-4, here is a poin in which he dimensionless pressure converges o a value during he ransiion period. From here we can read ha: 9/5 P (3) Dpss 5355

5 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. Subsiuing Equaion (3) in and solving or he inerporosiy low parameer yields: qb khppss 5/9 (33) As seen in Figure-4, once he ransiion period vanishes a line wih slope o 0.4 is developed. Draw such line and draw a parallel line hroughou he poins j beore he developmen o he ransiion period. Read wo poins in boh lines a he same ime value and ake is raio, which is ed in he ollowing iing equaion ha has a correlaion coeicien o The soraiviy raio is ound rom such separaion: Phigh Plow 5e (34) 3. SYNTHETIC EXAMPLES Table-1. Inpu daa or eamples. Eamples Parameer 1 k, md 5 1 h, 100 0, % 5 10 B, rb/stb , cp c, 1/psi q, BPD 30 50, /9 (50)(1.5)(1.) (5)(100)(3.) 3.1. Synheic eample 1 Figure-5 repors simulaed pressure drop vers ime daa obained wih he inormaion given on he second column o Table-1. I is required o ind he naurally-racured reservoir parameers. Soluion by Convenional Analysis In Figure-5 a horizonal line was drawn going hrough he ransiion poins. The ollowing value o pressure drop was read: P pss = 3. psi Using he above value in Equaion (33), i yields: Then, a 0.4-slope line is drawn aer he ransiion period. Anoher parallel line goes j aer he iniiaion o he ransiion period so he ampliude can be esimaed. Two pressure drop values are read a he same ime rom boh sraigh lines, so: P high = psi P low = 6 psi Equaion (34) will lead o he esimaion o he soraiviy raio: e E+0 P, psi P pss 3. psi ( P) psi high ( P) 6psi low The hal-racure lengh is ound rom he slope o a Caresian plo (m L= psi/hr 0.5 ) o pressure vers he square roo o ime, Figure-7, ing Equaion (A.7): 30(1.) (100)(360.8) (0.05)(0.0001)(5)(0.01) Soluion by TDS Technique 1.E-06 1.E-05 1.E-04 1.E-03 1.E-0, hr Figure-5. Pressure drop vers ime log-log plo or eample

6 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. P & *P ', psi 1.E+03 1.E+0 ( * P') 0.75 psi ma ( * P') psi ma hr 1.E-0 1.E-06 1.E-05 1.E-04 1.E-03 1.E-0, hr ( * P') 190 psi i hr Li 0.06 hr 1. hr BRi hr (* P') 8.5psi ( * P') psi hr Figure-6. Pressure and pressure derivaive vers ime log-log plo or eample 1. 5 BR1 The inerporosiy low parameer is esimaed wih Equaions (17), (0), (), (4) and (5): 5/9 30(1.5)(1.) (100)(0.085) 5/9 30(1.5)(1.) (100)(0.75) (0.1)(0.05)(1.5)(0.0001)(10 ) ( ) 5/16 (0.05)(0.0001)(10 )(100)(0.305) (30)(1.)(0.003) P, psi m psi/ hr L 5/16 (0.05)(0.0001)(10 ) (0.06) The soraiviy raio is esimaed wih Equaions (6), (8), (9) and (31): , hr Figure-7. Caresian plo o pressure vers he square roo o ime or eample 1. The ollowing inormaion was read rom he pressure and pressure derivaive log-log plo provided in Figure-6. ma = hr (*P ) ma= 0.75 psi (*P ) = psi (*P ) = psi BRi = hr (*P ) BR1= 8.5 psi Using Equaion (14); (*P ) BR1= 8.5 psi (*P ) = 190 psi = hr i = 0.06 hr Li = 1. hr = hr 30(1.) (190)(100) (0.0001)(5) The hal-racure lengh is also ound wih Equaion (1); 30(1.) (8.5) 0.05(0.0001) (8.5) (1.) (190) (50)(0.015) (0.05)(1.5)(0.0001)(10 ) (0.05)(1.5)(0.0001)(10 ) 3/5 9/ (5)(50 )(1.) Synheic eample Figure-8 presens synheic pressure drop vers ime daa obained wih he inormaion given on he hird column o Table-1. I is required o ind he naurallyracured reservoir parameers. Soluion by Convenional Analysis In Figure-8 a horizonal line was drawn going hrough he ransiion poins. The ollowing value o pressure drop was read: P pss = 184 psi P high = 1600 psi P low = 373 psi 5357

7 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. m L = psi/hr 0.5 read rom Figure-10. Equaion (33) leads o an esimaion o a value or he inerporosiy low parameer and Equaion (34) provides a soraiviy raio o A hal-racure lengh o 35.1 is ound wih Equaion (A.7) 1.E+04 The ollowing inormaion was read rom he pressure and pressure derivaive log-log plo provided in Figure-9. ma = hr (*P ) ma= 44 psi (*P ) =.4 psi BRi = hr (*P ) = 6.53 psi (*P ) BR1=105 psi (*P ) = 1400 psi = 0.09 hr Li = 340 hr = 0.04 hr P, psi 1.E+03 P pss 184 psi ( P) high 1600 psi ( P) 373psi low The calculaions were perormed in he same way as in eample 1 and a summary o he resuls is given in Table-3. 1.E+0 Table-. Resuls rom convenional analysis. 1.E-04 1.E-03 1.E-0 1.E+0 1.E+03 Figure-8. Pressure drop vers ime log-log plo or eample 1., hr Parameer Eample 1 Inpu Eample Inpu Equaion Number, A E+05 Li 340 hr Table-3. Resuls rom TDS echnique. P & *P ', psi 1.E+04 1.E+03 (* P') 44psi 1.E+0 ma ( * P') 1400 psi ( * P') BR1 105 psi i 7hr (* P').4psi ma hr 0.09 hr ( * P') 6.53 psi 0.04 hr 1.E-04 1.E-03 1.E-0 1.E+0 1.E+03, hr BRi hr Figure-9. Pressure and pressure derivaive vers ime log-log plo or eample. P, psi m psi/ hr L Parameer Eample 1 Eample, Equaion Number Inpu Inpu Inpu , hr Figure-10. Caresian plo o pressure vers he square roo o ime or eample. Soluion by TDS Technique 4. COMMENTS ON THE RESULTS The provided eamples in his work demonsrae he applicabiliy o he developed mahemaical epressions. I is worh o red ha he esimaion o he naurally racured parameers is very sensiive, and as a resul, deviaion errors o one order o magniude are allowed. The mehod provides saisacory resuls especially or he inerporosiy low parameer. I urher 5358

8 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. accuracy is required, hen, e o compuer program is highly recommended. These eamples were perormed by hand and he values orm he plos were read by eye, consequenly, deviaion rom inpu values is epeced. Equaions or gas low are given in Appendi B and he equaions or he esimaion o hal-racure lengh are given in Appendi A. CONCLUSIONS a) New mahemaical epressions or boh TDS echnique and convenional analysis were inroduced and successully esed wih synheic cases or he ull characerizaion o pressure ess conduced in naurally-racured reservoirs whose wells have been subjeced o hydraulic racuring. b) For he case o he TDs echnique, ive epressions or he esimaion o he inerporosiy low parameer were developed. Four equaions or he esimaion o he soraiviy raio were also inroduced in his work. ACKNOWLEDGEMENTS The auhors hank Universidad Surcolombiana, he Universiy o Oklahoma and he Naional Science Fund or Disinguished Young Scholars o China (Gran No ) and he Naional Program on Key Basic Research Projec (973 Program, Gran No. 011CB01005), and he Naural Science Foundaion o China (Gran No ). Nomenclaure A Draining area, B b C D c h k k w m P P w q q g r w s Oil volume acor, rb/stb Shores disance rom a laeral bounday o a well, Dimensionless racure conduciviy Compressibiliy, 1/psi Formaion hickness, Formaion compressibiliy, md Fracure conduciviy, md- Slope Pressure, psi Well-lowing pressure, psi Oil low rae, STB/D Gas low rae, MSCF/D Wellbore radi, Hal-racure lengh, Skin acor. Laplace parameer Tes ime, hr T (* P ) Reservoir emperaure, R Pressure derivaive, psi ( D*P D ) Dimensionless pressure derivaive Greek Change Porosiy, raccion Mari-racure nework inerporosiy low parameer Fracure nework-hydraulic racure inerporosiy low parameer μ Viosiy, cp Suies BR BR BR BR1 BRi D D ell Li ma w i Variable o ideniy homogeneo (=1) or heerogeneo (=) reservoirs Dimensionless soraiviy coeicien Birradial low in he homogenoe región (second birradial) Second birradial low a he ime o 1 hr Birradial Birradial a 1 hr Inersec o birradial and uni-slope lines Dimensionless Dimensionless based on hal-racured lengh Ellipical (same as birradial) Linear a 1 hr Inersec o linear and uni-slope lines Maimum poin Minimum poin Well Time REFERENCES Uni-slope during heerogeneo o homogeneo ransiion period Inersec o radial al uni-slope lines Agarwal G Real Gas Pseudo-ime a New Funcion or Pressure Buildup Analysis o MHF Gas Wells. Paper SPE 879 presened a he 54 h echnical conerence and 5359

9 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. ehibiion o he Sociey o Peroleum Engineers o AIME held in Las Vegas, NV, Sepember 3-6. Cinco-Ley H., Samaniego F. and Doguez N Transien Pressure Behavior or a Well wih a Finiy- Conduciviy Verical Fracure. Paper SPE 6014 presened a he SPE-AIME 51 s Annual Fall Technical Conerence and Ehibiion, held in New Orleans, LA, Ocober 3-6. Eobar F.H., Monealegre-M. M. and Marínez J.A Convenional Pressure Analysis or Naurally Fracured Reservoirs wih Transiion Period beore and Aer he Radial Flow Regime. CT and F - Ciencia, Tecnología y Fuuro. 3(5): Eobar F.H., Ghisays-Ruiz A. and Bonilla L.F New Model or Ellipical Flow Regime in Hydraulically- Fracured Verical Wells in Homogeneo and Naurally- Fracured Sysems. Journal o Engineering and Applied Sciences. ISSN Vol. 9. Nro. 9: Marinez J.A., Eobar F.H. and Bonilla L.F. 01. Reormulaion o he Ellipical Flow Governing Equaion or Horizonal Wells. Journal o Engineering and Applied Sciences. 7(3): Ozkan E. and Raghavan R. 1988, Aug 17. Some New Soluions o Solve Problems in Well Tes Analysis: Par - Compuaional Consideraions and Applicaions. Sociey o Peroleum Engineers. doi:null Tiab D Analysis o Pressure and Pressure Derivaive wihou Type-Curve Maching: 1- Skin and Wellbore Sorage. Journal o Peroleum Science and Engineering. 1: Tiab D Analysis o Pressure and Pressure Derivaive wihou Type Curve Maching: Verically Fracured Wells in Closed Sysems. Journal o Peroleum Science and Engineering. 11: Tiab D Advances in pressure ransien analysis - TDS echnique. Lecure Noes ManualThe Universiy o Oklahoma, Norman, Oklahoma, USA. p Tiab D. and Beam Y Pracical Inerpreaion o Pressure Tess o Hydraulically Fracured Wells in a Naurally Fracured Reservoir. Sociey o Peroleum Engineers. doi:10.118/ ms. January 1. Appendi-A. Esimaing hal-racure lengh by Convenional Analysis Eobar e al. (014) provided he ollowing equaion; qb k P9.486 kh c Or, ell (A.1) P m (A.) Which implies ha a Caresian plo o P vs. (or drawdown) or ΔP vs. [( p+δ) - Δ ] (or buildup) provides a sraigh line which slope, m ell, provides he halracure lengh, qb k khmell c (A.3) During he homogeneo par, Equaion (A.3) becomes: qb k khmell c I he early low regime is linear, hen: qb P h ck P m (A.6) L (A.4) (A.5) So rom he Caresian plo o eiher pressure or pressure drop vers he square roo o ime: qb hml c k (A.7) Appendi-B. Gas low Equaions (1), () and (3) are applied o gas wells i he viosiy and oal sysem compressibiliies are given a iniial condiions, i means (c ) i. However, i hose epressions are epressed ing he pseudoime concep, Agarwal (1979), i yields: 5360

10 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved k ( ) Da a P (B.1) For gas wells, Agarwal (1979) also included he pseudopressure deiniion, mp ( ) D hk mp ( i ) mp ( ) (B.) 14.5qT which dimensionless pseudopressure derivaive is given by: hk * m( P)' D* m( P) D' (B.3) 14.5qT Aer replacing Equaions (B.1), (B.) and (B.3) ino Equaions (3) and (4) leads o obain: a( ) BR BR qt P h( m( P) k (B.4) qt k khmell For gas wells, Equaions (13) and (14) become: qt 1 h( m( P)) k qt 1 h* m( P)' k 0.5 (B.11) (B.1) (B.13) The analogo orm o Equaion (A.7) or gas low is hen: qt 1 hml k The gas version o Equaions (0) o (3) is: 0.5 (B.14) qt ( P) h (* mp ( )' k a BR BR qt h (* mp ( )' BR1 k (B.5) (B.6) qt kh[* m( P ') ma ] (B.15) ka( P) ma (B.16) 5/9 For he homogeneo zone: a( ) BR BR qt P h( m( P) k a( ) BR BR qt P h (* mp ( )' k qt h (* mp ( )' BR1 k (B.7) (B.8) (B.9) In convenional analysis or gas low case, Equaion (3) becomes, (B.17) k ( P) a ma Equaion (4) or gas low is: h [* mp ( ')] qba ( P) 5/16 Epressions (6) and (7) or gas wells: a( P) a( P) a( P) ma a( P) ma (B.18) (B.19) Dividing Equaion (B.6) by Equaion (B.9) and solving or he soraiviy raio: qt k P ( ) a P kh 0 Then, he slope o he Caresian plo will give, (B.10).7778 * m( P)' BR1 * m( P)' BR1 (B.0) 5361

11 VOL. 10, NO. 1, JULY 015 ISSN ARPN Journal o Engineering and Applied Sciences Asian Research Publishing Nework (ARPN). All righs reserved. Dividing Equaion (B.13) by Equaion (B.9) and solving or he soraiviy raio: [* m( P)'] BR1 h k qb [* m( P)'] (B.1) Equaions (9) and (31) rewrien or gas low are: 1 k a ( P) BRi /16 (B.) (B.3) k P 3/5 a( ) Li are: Finally, he gas versions or Equaion (33) and (34) qt khm( P) pss 5/9 (B.4) m( P) high m( P) low 5e (B.5) 536