A PRACTICAL METHOD TO DETERMINE AQUIFER LEAKAGE FACTOR FROM WELL TEST DATA IN CBM RESERVOIRS

Transcription

1 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. A PRACTICAL METHO TO ETERMINE AQUIFER LEAKAGE FACTOR FROM WELL TEST ATA IN CBM RESERVOIRS Feddy Humbeo Escoba 1, Piyank Sivasav and Xingu Wu 1 Univesidad Sucolombiana/CENIGAA, Avenida Pasana, Neiva, Huila, Colombia Mewbone School of Peoleum and Geological Engineeing, The Univesiy of Oklahoma, Noman, OK, USA fescoba@usco.edu.co ABSTRACT Wae influx is an impoan faco which needs o be quanified duing ealy sages of esevoi developmen o jusify pojec economics. Fo a CBM esevoi i is much moe impoan o quanify degee of connecion beween Coal seams and aquife (if any) as is poducion mechanism is based on efficien dewaeing pocess. Howeve, i is difficul o quanify connecion faco values ealy in field life. Many adiional models anging fom simples seady sae model by Schilhuis and fekovich uilizing maeial balance, o unseady sae soluion of diffusiviy equaion by Van-Evedingen and Hus exiss fo finding wae influx bu nealy all of hem have inhei assumpions elaed o aquife/esevoi bounday pessue o influx ae and equies accuae hisoical poducion daa fo esimaion of coec influx which is ofen no available. Howeve, duing Appaisal/exploaoy sage we do have accuae measuemen of pessue a wellboe and poducion/injecion ae when we conduc pessue ansien esing. Well es analysis plays an impoan ole in esevoi chaaceizaion and can aid in coec wae influx calculaions. Cuenly, pessue falloff es esponses o quanify wae influx in he esevoi wih wells ha exis nea consan pessue bounday can be analyzed by eihe ypecuve maching o non-linea egession analysis. The fome is basically a ial-and-eo pocedue and he lae can lead o incoec/impacical esuls. So, we need a moe obus and accuae mehods fo wae influx calculaion. In his wok, a pacical mehod is developed o inepe he injecion-falloff es esponse fo a CBM esevoi in connecion wih aquife o quanify connecion faco beween aquife and esevoi using pessue ansien ess which ae usually conduced duing field appaisal/exploaion phase. Besides complemening he convenional saigh-line mehod fo he deeminaion he leakage faco, we also povide a soluion using chaaceisic poins found on he pessue, pessue deivaive and second pessue deivaive log-log plo of a leaky aquife esevoi model (Cox and Onsage, 00) which allowed us o develop elaionships fo he accuae esimaion he leakage faco. An exemely useful applicaion of he second pessue deivaive was also included fo esimaion he unknown esevoi pemeabiliy fo cases in which adial flow egime is compleely masked by ohe flow egimes. The povided inepeaion mehodologies wee successfully esed wih synheic examples. Keywods: aquife pemeabiliy, wae influx, esevoi chaaceizaion, leakage faco, CBM. INTROUCTION Unconvenional gas esevois have become an inegal pa of enegy supply baske due o eve inceasing demand of oil and gas and decease in new convenional discoveies. In fuue, hese unconvenional esevois ae expeced o become moe significan as ea of convenional/easy oil and gas comes o an end. So i is vial o devise new mehods fo chaaceizaion of hese complex esevois because accuae knowledge is no pesen abou hei poducion mechanism and pefomance wih ime. Coalbed mehane (CBM) has developed ino an impoan pa of unconvenional esouces. A he ime esevoi is discoveed, nealy all hydocabon esevois ae suounded by poous ock conaining wae - Schafe, Howe and Ownes (1993), CBM esevois mosly conain naual facues called cleas which ae in mos cases filled wih wae, since CBM esevois opeae on pinciple of desopion of gas fom coal seam suface due o depessuizaion, mos CBM esevois equie efficien dewaeing befoe hey can poduce commecial volume of gas -Onsage and Cox (000). In some insances, wae influx fom ohe aquife unis can inhibi he dewaeing of he coal and heeby limi coal bed mehane ecovey - Onsage and Cox (000). So i becomes exemely impoan o come up wih new mehods o chaaceize he degee of connecion beween hese coals and ohe unis ealy in field life. I has been long ecognized in goundwae indusy ha many aquife have impefec seals and ae in connecion wih ohe aquife unis hough a low pemeabiliy confining laye -Cox and Onsage, (00), paiculaly fo shallow esevois like CBM small aquifes conneced wih esevoi can be in communicaion o ohe aquife unis hough low pemeabiliy leaky confining layes, his implies aquife connecion wih esevoi is no pefec (ΔP es. ΔP aquife). Tadiional Mehods used in Peoleum Indusy fo wae Influx quanificaion like fekovich and Schilhuis uilizes maeial balance wih assumpion of ΔP es.= ΔP aquife, wheeas Van Evedingen-Hus (VEH) is based on diffusiviy equaion and gives soluion fo condiions of consan eminal wae influx and consan eminal pessue a aquife esevoi bounday fo adial models. Cae-Tacy mehod is jus modificaion ove 4763

2 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. VEH o save compuaion ime. Accoding o Fekovich, M.J. (1971) The las hee mehods have poved useful fo pedicing wae dive pefomance afe suffien hisoical daa have been obained o fix necessay influx consans wih wha some conside o be disappoining. Coas (196) and Allad and Chen (1988) consideed - model o povide wae influx soluions fo boom wae dive esevoi wih consan eminal ae and pessue case. Some ohe ecenly developed mehods like GRACE ansfom by Al-ghanim, Nashawi & Malallah, (01) can also be used fo wae influx calculaions. Howeve, all mehods ae based on eihe consan eminal pessue a aquife/esevoi bounday o consan eminal influx ae which may o may no be ue. In 00, Cox and Onsage ook a slighly diffeen appoach han VEH o obain soluions fo diffusiviy equaion, fo a CBM esevoi conneced wih aquife. Which could be easily used in pessue ansien esing o come ou wih amoun of pessue suppo povided by aquife. This infomaion can be vey ciical befoe incopoaing any wae influx models fo calculaion of amoun of wae influx. The consan eminal ae bounday condiion (as duing falloff/buildup ae =0) is much valid a well locaion ahe han a aquife/esevoi bounday which VEH assumed. In his Pape we build-up on soluions given by Cox and Onsage, 00 fo pessue falloff es and pesen a new mehod ha can povide a moe pacical way o quanify aquife esevoi conneciviy fo a esevoi wih boom wae dive ealy in field life using pessue ansien daa. Hanush and Jacob (1955) published a mehodology known in goundwae indusy as leaky aquife model. The idealized Hanush-Jacob leaky aquife model assumes a consan pessue bounday a op/boom of confining laye. Leaky aquife models moe accuaely descibes boom/op wae dive CBM esevois in which coal is only poducing laye his case can also be modelled by mehods given by Neuman, S.P. and P.A. Wihespoon (197) and Guo, Sewa and Too (00) which is essenially same as ha of Hanush and Jacob (1955). Hanush and Jacob (1955) developed soluion fo non-seady disibuion of dawdown caused by pumping a well a a consan ae fom an effecively infinie and pefecly elasic aquife of unifom hickness in which leakage akes place in popoion o he dawdown. They came up wih paamee b called as leakage faco which is a funcion of hydaulic conduciviy and hickness of confining bed hough which leakage occus. Leakage coefficien is defined as he quaniy of flow ha cosses a uni aea of he ineface beween he main aquife and is semi- confining bed, if he diffeence beween he head in he main aquife and in ha supplying leakage is uniy -Hanush (1956). Using Laplace ansfom mehod Cox and Onsage (00) came up wih dimensionless pessue soluions fo diffusiviy equaions of Leaky Aquife model, and demonsaed he applicaion of leaky aquife model o injecion falloff ansien es in CBM wells wih use of ype cuves o quanify leakage faco of aquife in esevoi. In his pape, based on soluions given by Cox and Onsage, we popose a moe pacical mehod which uses chaaceisics poins found on pessue and second pessue deivaives log-log cuve and y o demonsae he obusness of poposed mehod using synheic pessue falloff daa fo a CBM esevoi. The objecive of his pape is o povide newe ways fo analyically chaaceizing he leakage faco fom a pessue es which is defined as degee of connecion beween song aquife uni and main esevoi. Fis, he convenional analysis mehod was implemened o find such paamee based upon he seady-sae pessue dop which can be easily found fom eihe he Caesian o semilog o log-log plo of pessue vesus ime. Nex, we fomulaed a pacical echnique o find he leakage faco using he maximum poin found on he second pessue deivaive once adial flow is vanished. A pacical equaion fo he deeminaion of esevoi pemeabiliy was also inoduced using he menioned maximum poin. Such mehodology uses he TS echnique inoduced by Tiab (1995). We use wo chaaceisic feaues (maximum pessue deivaive and inesecion beween boh deivaives) o develop fou expessions fo esimaing he leakage faco. We also unified he lae seady-sae cuve by muliplying he ime by he leakage faco, so an expession using a negaive wo slope was developed fo he leakage faco deeminaion. The poenial use of he second deivaive was used o find a new expession o esimae pemeabiliy fom he maximum second pessue deivaive which is so vial fo cases when he adial flow is unseen. The equaions wee successfully veified by using synheic examples. Model descipion leaky aquife concep An ideal leaky aquife model is depiced in Figue-1. Thee layes ae consideed Poducing Laye, Confining Laye and Aquife laye which povides pessue suppo o main poducing laye hough a confining laye. I is inees o quanify amoun of pessue suppo povided by aquife o main esevoi. Figue-1. Schemaic of a leaky aquife model. 4764

3 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. Assumpions fo leaky aquife model No fee gas pesen, so single phase flow of only wae is occuing in he esevoi. Vaious layes ae homogenous and have unifom esevoi popeies and Rock Compessibiliy is negligible. The pessue a boom of confining laye is consan because of aquife suppo (Cox and Onsage, 00). Veical pemeabiliy of suppoing aquife is simila o hoizonal pemeabiliy also capillay and gaviy effecs ae ignoed. Radial flow analysis The soluion fo Laplace ansfom of wellboe pessue fo a leaky aquife model Equaion (1) was povided by Cox and Onsage (00).This fomula was inveed using Sehfes algoihm o obain dimensionless pessue and ime cuves fo well wih no wellboe soage and skin having diffeen values of leakage faco anging fom 1x10 - o 1x10-6. A ealy ime, cuves show chaaceisic adial flow behavio wih zeo slope bu depending on he quaniy of suppo fom aquife he pessue ends o seady sae which causes deivaive o dop o zeo. In some cases leakage faco may be oo lage ha hee is no sable deivaive poion fo adial flow analysis. ISCUSSIONS Type cuve analyses have seveal disadvanages fo esimaion of accuae esevoi popeies. We demonsae he uiliy of TS and he applicaion of he second deivaive fo esimaion of accuae esevoi paamees especially leakage Faco fo a leaky aquife model in a homogenous layeed esevoi wih help of wo synheic examples. Example 1 demonsaes an impoan poin abou uiliy of ou mehod which uses inesecion poin of fis and second deivaive fo obaining ciical esevoi paamees in cases whee adial flow is paially masked by wellboe soage. MATHEMATICAL FORMULATION Mahemaical model Onsage and Cox (00) povided he soluion of he diffusiviy equaion wih wellboe soage and skin in Laplace space fo a esevoi wih an undelying leaky aquife, P in which; K ( u) s uk ( u) uk u C K u s uk( u) 1 o 1 1( ) o( ) 1 u b () (1) b b k v, conf (3) hconf k vconf, w (4) khhconf The dimensionless quaniies ae defined below as: P 10 1 b E+01 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 P Figue-. imensionless pessue vesus ime log-log behavio fo seveal values of dimensionless leakage facos k (5) c w khp (6) 141.qB kh(* P ') * P' (7) 141.qB kh( * P") * P" (8) 141.qB Convenional analysis Figue- pesens he dimensionless pessue behavio obained fom Equaion (1). In his log-log plo i is obseved ha he seady sae is eached a a diffeen ime depending on he leakage faco value. Acually, hese values ae coelaed as obseved in Figue-3. A song dependency of he pessue a which seady sae akes places on leakage faco is clealy obseved and esablished as given below: b khp ss qb e (9) 4765

4 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. Theefoe, fom a log-log plo of dimensionless pessue dop vesus dimensionless ime is easy o obseve ha he pessue dop is consan once he seady sae is fully developed. Howeve, any convenional plo, fo example, semilog o Caesian, can be used o find he seady-sae pessue by dawing a hoizonal line on he lae seady-sae peiod and finding he inecep on he y- axis. In any case, his is obseved by a fla behavio of eihe pessue o pessue dop. This value is ead and eplaced ino Equaion (10) o easily obain he leakage faco. I is woh o emind ha unlike pessue deivaive, pessue dop is sensiive o skin effec; hen, P ss in Equaion (9) has o be fee of skin effecs. This means ha: Pss Pi Pwf Ps (10) Addiionally, he equivalen ime poposed by Agawal (1980) is ecommended o be used in buildup ess. TS echnique This echnique was inoduced by Tiab (1995) and is based upon specific feaues found on he pessue and pessue deivaive vesus ime log-log plo. Fo he case deal in his pape efe o Figue-4 o obseve he dimensionless pessue and pessue deivaive behavio. I is seen ha duing adial flow egime he pessue deivaive is govened by a fla saigh line (zeo slope) wih an inecep of 0.5. Tiab (1995) demonsaed ha afe Equaion (7) is equalized o 0.5, an expession o find pemeabiliy is given: 1.E+01 b 1.E-0 b e khp ss qb P, *P ' & *P " 1.E-0 P * P " * P ' Inecep poin Maximum poin 1.E-05 1.E P Figue-3. Behavio of he leakage faco as a funcion of he pessue dop duing seady-sae. 1.E+01 1.E-05 1.E+01 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 Figue-5. imensionless pessue, pessue deivaive and second pessue deivaive vesus ime log-log plo fo b = * P ' 0.5 P & *P ' 1.E-0 b E-05 1.E+01 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Figue-4. imensionless pessue and pessue deivaive vesus ime log-log behavio fo seveal values of dimensionless leakage facos. Pss Pws Pwf Ps (11) Equaion (10) is applied o dawdown and Equaion (11) is applied o buildup ess. P, *P ' & *P " 1.E-0 * P ' b E+01 1.E+0 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Figue-6. imensionless pessue, pessue deivaive and second pessue deivaive vesus ime log-log plo fo seveal values of dimensionless leakage facos. 70.6qB k h (* P ) (1) 4766

5 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. Tiab (1995) also povided an expession o find he skin faco by eading he pessue dop a any abiay ime duing adial flow: P k s 0.5 ln 7.43 (* P') c w (13) Howeve, fo he inees of his wok, he pessue deivaive does no help much since i has a decaying behavio as seen in Figue-5. Then, besides pessue and pessue deivaive given in Figue-5, Figue-6 also includes he second pessue deivaive behavio. in boh plos wo chaaceisic feaues ae clealy seen: (1) a poin of inesecion beween deivaive and second deivaive, and () a maximum poin displayed by he pessue deivaive which is given in Figue-6 fo ohe cases sudies. kh( * P") P * " (14) 141.qB Solving fo pemeabiliy: 6.63qB k h ( * P") max (15) Equaion (15) may be he pacical use wheneve he adial flow egime is obscued by wellboe soage. The second obsevaion is bee explained in Figue-7 whee a pefec coelaion beween ime and leakage is obained and given below: ( ) nsi b b k max 1.005log c w *P ' 1.E-0 b * P' ( * b ) log max Figue-7. Coelaion beween he ime a which he maximum value of he second deivaive akes place and he leakage faco. 1.E-06 1.E-05 1.E-0 1.E+01 *b Figue-9. Log-log plo imensionless pessue deivaive a vesus dimensionless ime muliplied by dimensionless leakage faco. b b kin p log c w b k max 1.005log c w 10 (16) Thid, fo cases whee he maximum poin is difficul o be seen, pobably due o much noise, and he inecep beween he wo deivaives is seen, a pefec coelaion is epoed in Figue-8. The obained coelaion is povided below: log inp Figue-8. Coelaion beween he ime a which he pessue and second pessue deivaives inecep and he leakage faco. Seveal obsevaions can be dawn fom Figue-6. The fis one is ha he maximum poins ae funcion of ime and second deivaive is pacically consan a a dimensionless value of Then: b kin p log c w 10 (17) The fouh and final obsevaion in Figue-6 allows seeing a dependency of leakage faco on boh ime and second deivaive, alhough he las one has a weak dependency. Equaion (16) is a final coelaion of he leakage faco as a funcion of pessue deivaive and ime ead a he maximum poin. Howeve, fo easie manipulaions, he second pessue deivaive is divided by he pessue deivaive duing adial flow egime. If 4767

6 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. his is unclea and pemeabiliy is known, hen, pessue deivaive can be solved fom Equaion (1). Z F F whee, k F1 1/log max c w F (* P') ( * P'') max (18) (19) (0) Pessue, pessue deivaive and second pessue deivaive vesus ime ae povided in Figue-10. I is equied o chaaceize his es wih he sole pupose of esimaing he leakage faco. Soluion by convenional analysis Acually, his example canno be solved by convenional analysis since boh wellboe soage and aquife effecs mask he adial flow egime. Theefoe, neihe skin faco no pemeabiliy can be deemined. Howeve, assuming boh pemeabiliy and skin faco is known, hen, fom he log-log plo of P vs., Figue-10, he value a which he hoizonal line is dawn duing he lae seady-sae peiod ineceps he y-axis is: P s psi ss 1/ b 10 Z (1) A final obsevaion comes ou fom Figue-9 which is a log-log plo of dimensionless pessue deivaive vesus he poduc of dimensionless ime muliplied by he dimensionless leakage faco. As seen in he plo he lae seady-sae peiod unifies ino a single line. So, dawing a saigh line of a slope of - will yield o he following fiing: * P ' () ( * b ) The inecep of Equaion () wih he adial flow egime pessue deivaive ( *P =0.5) povides he following expession: b c w (3) k nsi being nsi he poin of inesec beween he adial flow and he negaive wo-slope lines. EXAMPLES Synheic example 1 A simulaed es was pefomed using Onsage and Cox (00) model wih he below infomaion: B = bbl/stb q = 30 STB/ h = 00 f = 1 cp w = 0.4 f c = 1x10-4 psi -1 P i = 3500 psi = 30 % k = 10 md b = C = 10 s = 1 Thee is a need of finding he semilog slope fom he classic pemeabiliy equaion in ode o find he pessue dop due o skin faco: P, *P ', & *P ", psi 1.E+0 1.E+01 1.E-0 inecep P s psi ( * P") max psi ss nsi 4h 1.E-0 1.E+01 1.E+0, h in p 4.4 h max ( * P ) psi (heoeical) 4.84 h Figue-10. Log-log plo of pessue, pessue deivaive and second pessue deivaive vesus ime fo example qB k (4) hm Solving fo m; 16.6qB 16.6(30)(1)(1.005) m.451 psi/cycle hk (00)(10) Since: P 0.87ms s( * P ') (5) s The pessue dop due o a skin faco of one is.133. Then, P ss= 8.7 psi which used in Equaion (9) will povide: b (10)(00)(8.7) (30)(1)(1.005) e

7 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. Soluion by TS Technique The following infomaion was ead fom Figue- 10: nsi = 4 h max = 4.84 h inp = 4.4 h ( *P ) max = psi This is a vey common case whee he adial flow is obscued by wellboe soage. Besides, he seady-sae peiod also conibues o no having a well-defined adial flow egime line. Then, he second deivaive a he maximum poin can also be used o esimae pemeabiliy fom Equaion (15): 6.63(30)(1)(1.005) k md (00)(0.434) Since we need he value of pessue deivaive duing adial flow egime -acually i can be obained fom Equaion (5)-, hen, i is esimaed fom Equaion (1): 70.6qB 70.6(30)(1)(1.005) ( * P ) psi hk (00)(10) The leakage faco can be found fom Equaions (16) and (17), especively: The poin of inesecion beween he adial flow egime and he negaive wo- slope lines is also used o find leakage faco fom Equaion (3); (0.3)(1)(0.0001)(0.4 ) b (10)(4) Synheic example Anohe pessue es was simulaed wih he below infomaion: B = 1.00 bbl/stb q = 550 STB/ h = 150 f = 0.5 cp w = 0.5 f c = 5x10-5 psi -1 P i = 3850 psi = 0 % k = 60 md b = C = 0 s = 0 Figue-11 pesens he semilog plo of pessue vesus ime and Figue-1 conains a log-log plo of pessue dop, pessue deivaive and second pessue deivaive vesus ime. Find pemeabiliy, skin faco and he leakage faco using boh convenional analysis and TS echnique b b (10)(4.85) 1.005log (0.3)(1)(0.0001)(0.4 ) (10)(4.4) log (0.3)(1)(0.0001)(0.4 ) Finally, using he coodinaes of he maximum poin of he second deivaive in Equaions (18) hough (1), he leakage faco is also obained: (0.1)(198.48) F1 1/ log 0.17 (0.3)(1)(0.0001)(0.4 ) F Z (0.17) b (6.5) / P wf, psi P, *P ', & *P ", psi P1 h 3830 psi 1h m = -4.9 psi/cycle Inesec P psi P.46 psi ss 1.E-05 1.E-0 1.E+01 1.E+0 1.E+0 1.E+01 1.E-0, h Figue-11. Semilog plo of pessue vesus ime example. P ' psi ( * P').11 psi nsi 1.6 h 1.E-0 1.E+01 1.E+0, h in p 1.95 h max ( * P") max psi.1 h Figue-1. Log-log plo of pessue, pessue deivaive and second pessue deivaive vesus ime fo example. 4769

8 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. Soluion by convenional analysis The following infomaion is ead fom he semilog plo epoed in Figue-10. m = -4.9 psi/cycle P 1h = 3830 psi P ss=.46 psi Pemeabiliy and skin faco ae found wih Equaions (4) and (5), especively. 16.6(550)(0.5)(1) k md (150) (0.04) s 0.5 ln (0.)(0.5)( )(0.5 ) s 0.06 Making use of he ime a which he maximum second pessue deivaive akes place find he leakage faco fom Equaion (16); b (61.83)(.11) 1.005log (0.)(0.5)( )(0.5 ) Using he ime of inesecion beween he deivaives find he leakage faco wih Equaion (17); s P P k 1h i log 3.3 m c w (5) b (61.83)(1.95) log (0.)(0.5)( )(0.5 ) s log (0.)(0.5)( )(0.5 ) s 0.43 The seady-sae peiod is easily obseved in boh plos, Figues-11 and -1. Recalling he consan pessue value ead fom Figue-11 (P ss=.46 psi): he leakage faco is found fom he applicaion of Equaion (9); b 1: (60.84)(150)(.46) (550)(0.5)(1) e Soluion by TS Technique The following infomaion was ead fom Figue- inp = 1.95 h max =.1 h ( *P ) max = psi = 0.04 h (*P ) =.11 psi P = psi nsi = 1.6 h Using he value of he pessue deivaive duing adial flow egime, pemeabiliy is easily found fom Equaion (1); 70.6(550)(0.5)(1) k 61.3 md (150)(.1) Taking an abiay poin duing he adial flow egime, skin faco is esimaed wih Equaion (13); Find he leakage faco fom he applicaion of he coodinaes of he maximum poin of he second deivaive wih Equaions (18) hough (1) (61.83)(.1) F1 1/ log (0.)(0.5)( )(0.5 ).11 F Z ( ) b (7.407) / The poin of inesecion beween he adial flow egime line and he negaive wo-slope line is used o find anohe value of leakage faco fom Equaion (3); (0.)(0.5)( )(0.5 ) b (1.6) COMMENTS ON THE RESULTS The esuls obained fom boh echniques ae quie saisfacoy. Alhough, he convenional analysis povides a single finding of he leakage faco, he esuls wee excellen in boh execises, even hough, he fis poblem canno be compleely caied ou by convenional echnique due o he missing of he adial flow egime. As fa as he use of he TS echnique is concened, fou diffeen foms fo esimaing he leakage faco ae povided. The bes esuls wee obained fom 4770

9 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. he use of he ime a which he maximum poin displayed by he second deivaive akes place. The expession ha uses he inesec poin fomed beween he wo deivaives povided less accuae esuls, due o he fac ha i is difficul o exacly define such inecep. The soluion povided by Equaion (18) hough (1) povided bee esuls in he second poblem. I may be due o he fac ha wellboe soage could affec he soluion. In boh execises he soluion found wih Equaion (3) which uses he inecep beween he adial line and he negaive wo-slope line (aificially added) povided excellen esuls. I is woh o emak ha he maximum poin of he second pessue deivaive was also used o find an appoximaion fo he deeminaion of pemeabiliy. The fis example is a ypical case of is use since he adial flow was undefined. CONCLUSIONS a) New expessions wee inoduced fo he deeminaion of he leakage faco by using boh convenional analysis and he TS echnique. A oal of five expessions wee inoduced fou of which belong o he TS echnique. The expessions wee saisfacoy applied o synheic examples. b) The TS echnique used he poenial of he second deivaive fo he esimaion of he leakage faco. The maximum poin of he second pessue deivaive was also used o povide an appoximaion o find esevoi pemeabiliy. c) The value of leakage faco found ou by pesened new mehods lead o bee esimaion and is much moe obus han using ype-cuve analysis which may lead o inaccuae esimaions. d) Fuhe analysis and hisoical daa is needed o show ha he calculaed leakage faco values can be used in classic diffusiviy equaion o esimae accuae amoun of wae influx which can be compaed wih VEH wae influx model o es is validiy. Nomenclaue B b C Volume faco, b/stb Leakage faco, f Wellboe soage coefficien, bbl/psi c Toal sysem compessibiliy, psi -1 h k P P i P wf q w Resevoi hickness, f Laplace paamee Resevoi pemeabiliy, md Pessue, psi Iniial esevoi pessue, psi Wellboe flowing pessue, psi Wae flow ae, BP Wellboe adius, f s *P *P (*P ) Skin faco Time, days imensionless ime coodinae imensionless pessue deivaive imensionless second pessue deivaive Pessue deivaive ( *P ) Second pessue deivaive e) Geeks f) g) Poosiy, facion Viscosiy, cp h) Suffices i) max conf i inp nsi ss Maximum of he second pessue deivaive Confining laye imensionless Iniial Inecep of pessue deivaive and second pessue deivaive Radial Inesec of he adial flow line and he negaive wo-slope line Seady sae v, conf Veical in confining laye ACKNOWLEGEMENTS The auhos hank Univesidad Sucolombiana and he Univesiy of Oklahoma fo poviding financial suppo fo he compleion of his sudy. The second auho would also like o hank M. Anan Kuma who gave him chance o wok on CBM esevoi. REFERENCES Allad. R. and Chen S. M Calculaion of Wae Influx fo Boomwae ive Resevois. Sociey of Peoleum Enginees. doi:10.118/13170-pa. Al-ghanim Jassim Abdulaziz, Nashawi I. S. and Malallah, A. 01. Januay 1. Pedicion of Wae Influx of Edge- Wae ive Resevois Using Nonpaameic Opimal Tansfomaions. Sociey of Peoleum Enginees. doi:10.118/15066-ms. Agawal R. G A New Mehod o Accoun fo Poducing Time Effecs When awdown Type Cuves 4771

10 VOL. 10, NO. 11, JUNE 015 ISSN Asian Reseach Publishing Newok (ARPN). All ighs eseved. Ae Used To Analyze Pessue Buildup And Ohe Tes aa. Sociey of Peoleum Enginees. doi:10.118/989- MS. Cae R.. and Tacy G. W An Impoved Mehod fo Calculaing Wae Influx. Sociey of Peoleum Enginees. Coas K. H A Mahemaical Model Wae Movemen abou Boom-Wae-ive Resevois. Sociey of Peoleum Enginees. doi:10.118/160-pa. Tiab Analysis of Pessue and Pessue eivaive wihou Type-Cuve Maching: 1- Skin and Wellboe Soage. Jounal of Peoleum Science and Engineeing, Vol. 1, pp Also Pape SPE 543 (1993) Poducion Opeaions Symposium held in Oklahoma Ciy, OK. Van Evedingen, A. F. and Hus W The Applicaion of he Laplace Tansfomaion o Flow Poblems in Resevois. Sociey of Peoleum Enginees. doi :10.118/ G. Cox.O. and Onsage P. R. 00. Applicaion of Leaky Aquife Type Cuves fo Coalbed Mehane Chaaceizaion. Pape SPE pesened a he SPE Annual Technical Confeence and Exhibiion, San Anonio, Texas, 9 Sepembe- Ocobe. oi: /77333-MS. Fekovich M. J A Simplified Appoach o Wae Influx Calculaions-Finie Aquife Sysems. Sociey of Peoleum Enginees. doi:10.118/603-pa. Guo B., Sewa G. and Too M. 00. Linealy Suppoed Radial Flow-A Flow Regime in Layeed Resevois. Sociey of Peoleum Enginees. doi:10.118/7769-pa. Hanush M.S. and Jacob C.E Non-Seady Radial Flow in an Infinie Leaky Aquife. Tansacions of Ameican Geophysics. U. 36, 95. Hanush M.S Analysis of daa fom pumping ess in leaky Aquifes. Tansacions of Ameican Geophysics. U. 36, 70. Hanush M.S Modificaion of he Theoy of Leaky Aquifes. Jounal Geophysical Reseach. 65, Neuman S.P. and Wihespoon P.A Field deeminaion of he hydaulic popeies of leaky muliple aquife sysems. Wae Resouces Reseach. 8(5): Onsage P. R. and Cox. O Aquife Conols on Coalbed Mehane evelopmen in he Powde Rive Basin, Wyoming. Sociey of Peoleum Enginees. doi :10.118/63090-MS. Schafe PS., Howe T. and Ownes R.W Managing Wae-ive Gas Resevois. Gas Reseach Insiue. p Sehfes H Algoihm 368: Numeical invesion of Laplace ansfom, Communicaion of he ACM. 13(1):