PRESSURE DERIVATIVE ANALYSIS FOR HORIZONTAL WELLS IN SHALE RESERVOIRS UNDER TRILINEAR FLOW CONDITIONS

Transcription

1 VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. PRESSURE ERVATVE ANALYSS OR HORZONTAL WELLS N SHALE RESERVORS UNER TRLNEAR LOW CONTONS rddy Hubro Escobar, María Aljandra Cabrra and Asrid Juliana Oriz Univrsidad Surcolobiana/CENGAA, Avnida Pasrana, Niva, Huila, Colobia fscobar@usco.du.co ABSTRACT Unconvnional shal rsrvoirs appar as a soluion o h dplion of convnional rsrvs, howvr, hir ulra low prabiliy, rquirs hydraulic fracuring ha hlps iproving h fluid flow owards h wll. Th dsign and craion of hs fracurs is copl. Knowing hir propris, and h rsrvoir s, as wll, is of gra iporanc for fild anagn. This sudy prsns a pracical hodology for wll s inrpraion in shal rsrvoirs using h analyical rilinar flow odl, which dscribs a sys consising of a horizonal wll wih ulipl fracurs in rly low prabiliy rsrvoirs. Analyical prssions wr dvlopd basd upon uniqu faurs found on h prssur and prssur drivaiv curvs for h drinaion of fracur conduciviy ( w ), half-fracur lngh ( ), ari prabiliy ( ) and inrnal rsrvoir prabiliy ( ). inally, synhic apls for boh oil rsrvoirs and gas foraions wr dvlopd o succssfully vrify h accuracy of h dvlopd quaions. Kywords: unconvnional rsourcs, shal rsrvoirs, hydraulic fracurs, rilinar flow odl, TS chniqu.. NTROUCTON n h coninuous sarch o incras hydrocarbon producion undr h high global nrgy dand, h world has currnly urnd is anion o nonconvnional dposis. Thr svral yps of unconvnional rsourcs; on of h rfrs o shals. Wha as shals hir unconvnional characr is hir ulra-low prabiliy, ha can oscilla bwn 0-6 and 0 - darcis, bu gnrally is givn in h ordr of nanodarcis (0-9 darcis) and possss sall por diars ha oscilla bwn and 0 μ, plaining why hy coonly ar considrd as iprabl. Eploiaion of hs yps of rsrvoirs rquirs long horizonal wlls and h iplnaion of a good hydraulic fracuring job, wih h solid purpos of craing nsiv nwors of arificial fracurs, o incras h wll produciviy ind and o opiiz h rcovry of rsrvs. Pras (96) and Pras, Hazbro and Sriclr (96) bgan sudying h ffc of vrical fracurs on h bhavior of h rsrvoirs. Ray and Raghavan (974) prsnd a soluion for a vrical fracur of infini conduciviy in a vrical wll. Basd on h abov, Cinco-Ly, Saanigo and oinguz (978) did h sa for a fini-conduciviy fracur. Cinco-Ly and Saanigo (98) analyzd h ransin prssur bhavior of fini-conduciviy fracurd wlls and found a nw flow rgi calld bilinar. Caacho (984), Bnn al. (985), Caacho, Raghavan and Rynolds (987) who obaind ransin prssur analyical soluions for a vrical wll inrcpd by a fini-conduciviy vrical fracur in naurally fracurd rsrvoirs. Cinco and Mng (988) prsnd si-analyical odl for finiconduciviy fracurd wlls in naurally occurring foraions wih boh ransin ari flow and psudosady sa ari flow. Thy niond a rilinar flow ford by a ari linar flow supriposd o h bilinar flow in h foraion-fracur sys. Th rilinar flow odl, Brown al. (009), divids h rsrvoir ino hr zons wih avrag propris: inrnal rsrvoir, rnal rsrvoir and hydraulic fracurs. Thir soluion of h diffusiviy quaions was obaind in h Laplac doain spac sparaly for ach zon and h calculad rsuls in h Laplac doain wr rvrsd again in h i doain using h Shfs algorih (970). Al-Hussainy and Ray (966) and Al-Hussainy, Ray and Crawford (966) inroducd h ida of gas psudoprssur in ordr o giv an analyical soluion o gas dposis. Bcaus h inrnal rsrvoir is a naurally fracurd diu, h dual porosiy odl of Kazi (969), -Swaan (976) and Srra, Rynolds and Raghavan (983) for h basis of h rilinar flow odl. n addiion, h odl assus ha h produciv lif of a fracurd horizonal wll dpnds on h volu of hydraulic fracurs, Raghavan, Chn and Agarwal (997). Th bhavior of fluids wihin a rsrvoir coposd of vry igh ari, hydraulic fracurs and naural fracurs (usually inducd a h i of hydraulic fracuring), in a horizonal wll, can b inrprd using an analyical odl nown as rilinar flow odl, forulad and vrifid by Brown (009) and Brown al. (009). As iplid by is na, h rilinar flow odl assus hr linar flows during h produciv lif of h wll and is inrpraion was basd on a sris of prvious wor on h bhavior of fluids in porous dia, as wll as in vrical and horizonal wlls in naurally and/or hydraulically fracurd rsrvoirs. This rsarch prsns an nsion of h TS Tchniqu, Tiab (995), for wll prssur s inrpraion basd upon h rilinar flow odl prsnd by Brown al. (009), wih which nw quaions wr dvlopd o characriz foraions wih ulipl fracurs in horizonal wlls for boh oil and gas rsrvoirs. Th quaions hav bn forulad for hr diffrn cass dpnding of h flow rgi yps. Th 346

2 VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. dvlopd quaions wr succssfully sd wih synhic apls. (P) is a gas psudoprssur funcion, Al-Hussainy al. (966); Al-Hussainy and Ray (966); givn by: y=d/ Lin of syry No-low Lin No-low Boundary P P ' P ( ) dp' (6) z Pb HYRAULC RACTURE Φ, C, K OUTER RESERVOR ΦO, CO, KO Φ, C, K Th dinsionlss i is: ( c ) (7) NNER RESERVOR NATURALLY RACTURE Φ, C, K Φf, Cf, Kf y So dinsionlss variabls rlad o rsrvoir gory and fracurs ar: (8) igur-. Schaic rprsnaion of h rilinar flow odl wih hr coninuous flow rgions for a horizonal uli-fracurd wll, Brown, M. (009).. MATHEMATCAL MOEL Th rilinar flow odl forulad by Brown (009) and Brown al. (009) assus linar flow in hr diffrn rsrvoir zons: inrnal dposi, rnal dposi and hydraulic fracurs, as shown in igur-. Th raining odl assupions can b found in Brown al. (009). Th analyical dinsionlss prssur soluion for a horizonal uli-fracurd wll, Brown al. (009), is givn by: P ( P ) () sc anh 0 Ohr odl-rlad parars ar providd in Appndi A. Th dinsionlss oil prssur and prssur drivaiv ar givn by: h P ( P ) i Pwf 4.q B () sc, h * P ' ( * P') 4.q B (3) or gas Wlls, sc, h P Pi Pwf 4q T Msc, HORZONTAL WELL h * ( P) ' * ( P)' 4.5q T (5) Msc, (4) y y (9) w w (0) 3. WELL PRESSURE BEHAVOR As niond bfor, hr cass can a plac: Cas considrs hr characrisic flow rgis (igur-). A bilinar flow rgi is obsrvd a arly i. This occurs onc h ransin wav dcs wo siulanous linar flow rgis in h fracurs and innr rsrvoir, rspcivly. A scond linar flow (firs linar) appars in rspons h fluid flow fro h naural fracur nwor o h hydraulic fracurs. inally, a (firs) psudosady-sa priod is bcaus h aris do no fd h naural fracurs. Gnrally, h firs linar flow can b sn whn h produc is ls or qual o, or siply whn and a vry low valus which nsurs h flow incapabiliy fro ari o inducd naural fracurs. Bcaus is ari prabiliy indpndn, his cas norally as plac whn h niond prabiliy valu is ulra-low nough (<0-9 ) so h prssur in h s changs linarly wih i. Cas involvs h acing of fiv flow rgis (igur-): Bilinar, firs linar and firs psudosadysa priod of cas ; hn, scond linar and scond psudosady-sa. ollowd by h firs psudosadysa priod, and afr a considrabl rspons i, h prssur changs linarly wih i and h scond linar flow can b obsrvd; onc h ransin wav rachs h hydraulic fracur liis or h inrnal rsrvoir boundary, h scond psudosady-sa priod is dvlopd. Thr iss a crain rlaionship bwn - and h scond linar flow rgi. rqunly, whn has larg valus, h scond linar flow bcos visibl in h prssur drivaiv vrsus dinsionlss i log-log plo; whn grows, h duraion of h scond linar flow also dos so, and his, in urn, occurs la, as bcos sall. 347

3 * P ' VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. n Cas 3 is possibl o diffrnia hr flow rgis (igur-): a arly is, a bilinar flow is obsrvd, lar a linar flow, h sa ha w hav calld h scond linar flow in cas, followd by a psudosady or scond psudosady sa. This cas is givn du o h siulanous ovn of ari fluids o naural fracur and naural fracur o hydraulic fracur, undr condiions of larg and valus..e+08.e+07.e+06.e+05.e+04.e+03.e+0.e+0.e+00.e-0 Bilinar flow = Cas s linar = s psudosady = nd linar = Cas Cas 3 nd psudosady =.E-0.E-04.E-03.E-0.E-0.E+00.E+0.E+0.E+03.E+04.E+05.E+06.E+07.E+08 igur-. insionlss prssur drivaiv bhavior for a rsrvoir undr rilinar flow condiions wih cas, cas or cas TS TECHNQUE OR OL SHALE RESERVORS Bilinar flow rgi. Th govrning quaion dvlopd for his flow rgi is: hf f ( * P ') ().689 w Rplacing h dinsionlss quaniis, Equaions 3 and 7 ino Equaion and solving for h fracur conduciviy, w :.5.057hf f ( c) w qscb, h (* P') () irs linar flow rgi: Th govrning quaion for h firs linar flow is: * P' L (3).360 L Rplacing dinsionlss variabls and solving for h inrnal rsrvoir prabiliy, : irs psudosady-sa priod: Th govrning quaion for h firs psudosady-sa is: pss (5) y P ' pss Again rplacing h dinsionlss parar and solving for half-fracur lngh, : qsc, Bpss yh c * P' pss Scond linar flow: s govrning quaion is: f * P' L h ( c ) L.783 y ( c ) (6) (7) Th ari prabiliy,, is solvd for onc h dinsionlss quaniis ar rplacd in h abov prssion: L sc, f.08 c y ' h P L q Bh (8) Th Scond psudosady-sa priod has h following govrning Equaion: c pss c hf P '.55 (9) pss yh Which allows finding an prssion for drining h half-fracur lngh,, onc h dinsionlss variabls ar usd: q, scbhfpss y hh ( c ) * P' pss (0) 3. TS TECHNQUE OR GAS SHALE RESERVORS Th Bilinar flow rgi govrning quaion is: h f f * ( ) '.689 w P () Rplacing h dinsionlss quaniis givn by Equaions 5 and 7 in Equaion and solving and solving for h fracur conduciviy, w : L sc, c h* P' q B L (4) 348

4 VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. w hf f c qmsc, T h * P ' () Th govrning quaion for h irs linar flow rgi is: * ( P) ' L L (3).360 Afr rplacing h dinsionlss variabls lads o find h inrnal rsrvoir prabiliy, : q Msc, T L c h ( P)' L (4) irs psudosady-sa priod: Th govrning quaion for h firs psudosady-sa is: * ( P) ' pss (5) y pss Rplacing h dinsionlss parar and solving for half-fracur lngh, : qmsc, Tps s.697 yh c P ' pss (6) Th scond linar flow has h following govrning quaion: f * ( P) ' L h ( c ) L. 783 y ( c) (7) Rplacing h dinsionlss rs in h abov prssion lads o solv for h ari prabiliy, : Th L Msc, f c yh ( P)' L q (8) Th govrning quaion of h Scond psudosady-sa priod is givn by: ' c pss c h P (9) f * ( ).55 pss yh Rplacing h dinsionlss rs and, hn, solving for h half-fracur lngh,, will rsul: qmsc, Thfpss.787 yhhc ' P pss (30) nrscion poins Th quaions rsuling fro h poins of inrscion of h diffrn flows ar h sa for boh gas and oil rsrvoirs. nrscion of firs psudosady-sa and bilinar flow: 3 4 hff c w pss i y (3) flow: nrscion of firs psudosady sa and linar pss Li y c nrscion of bilinar-firs linar: w c h Bl Li f f (3) (33) nrscion of scond psudosady-sa and bilinar flow: f w h yh f pssi c c 3 4 (34) nrscion of scond psudo sady sa and scond linar flow: h c (35) pssli nrscion of bilinar-scond linar: w y f ( c) hf ( c) Li (36) nrscion of scond psudosady sa and firs linar: hf c pssli y h c (37) 349

5 VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. nrscion of scond linar wih firs psudosady sa h c (38) f ( c) L pssi 4. SYNTHETC EXAMPLES Th word apls usd daa fro Bown al. (009). 4.. Eapl A prssur s was siulad for an oil rsrvoir draind by a horizonally fracurd hydraulic wll (igur- 3) and daa prsnd in Tabl-. Esia: h halffracur lngh, hydraulic fracur conduciviy and prabiliy of h naural fracurs. Soluion: Th following copuaions ar prford using Equaions A-9, A- and A-: λ =.0758Χ0-0, ω = , Ƞ O = 6.786Χ0 - dpsi/cp, Ƞ = d-psi/cp and Ƞ = d-psi/cp. Cas is confird fro igur-3 fro which h following characrisic poins wr rad: = 0.09 hr ΔP = 4.8Χ0-3 psi (*ΔP ) =.Χ0-3 psi pss = 4760 hr ΔP pss = psi (*ΔP ) pss = 0.8 psi L = 3. hr ΔP L = psi (*ΔP ) L = 0.0 psi pss- Li = hr pss- Li = hr -Li =.6404 hr ind h fracur conduciviy, w, wih Equaion, naural fracurd nwor prabiliy,, for Equaion 4 and half-fracur lngh,, wih Equaion 6, so ha:, psi P and *P ' *.35* 4.76* *380* 0.7* 0.006*8.*0.E+0.E+00.E-0.E-0 P psi P psi * P ' 00psi. L L pss 0.09 hr ( * P ') 0. 8 psi L 3.hr.E-03 * P' 3. 0 psi.e-0.e-0.e+00.e+0.e+0.e+03.e+04.e+05, hr pss 3 pss hr igur-3. Prssur and prssur drivaiv vs. Ti loglog plo for apl (oil rsrvoir). Tabl-. npu daa for apls and Brown, M. (009). Parar Eapl Eapl h, f r w, f y, f , f μ, cp , d 5Χ0-6 Χ0-6, fracion c, psi h, f.5 f, d f, fracion c f, psi h f, f , fracion , d 500 *05 c, psi w, f , f q, STB/ or Mscf/ B o ρ f, f/f f T, R *0 * * * *380*.0* d.5.9*0 w.057*0.005** * 0.7* * d-f 3 380*.*0 ind naural fracur prabiliy,, wih Equaion 3, rsrvoir lngh, y, using Equaion 3 and fracur conduciviy, w, wih Equaion 33. y 50* *0.7* f 3430

6 ndndvol. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd *.79*0 *93*08.5 * 0.3*0.7* d 5 0.3*0.7*0.006*50 w *93 *0.005**.640 w 55.0 d-f Tabl-. Coparison of rsuls for apl. Equaion Parar npu This wor Error, % w, d-f , f , f , d y, f w, d-f Eapl A gas prssur s was siulad wih daa fro Tabl- and h prssur and prssur drivaiv daa ar rpord in igur-4. ind h sa parars as for apl. Soluion. Th following daa wr rad fro igur-4. = 9.8Χ0-7 hr Δ(P) = 5.8 psi /cp (*Δ(P) ) = 6.6 psi /cp pss = 3.9Χ0 hr Δ(P) pss = 8790 psi /cp L =.46Χ0 - hr (*Δ(P) ) L = 08 psi /cp pss- Li = 4.8 hr pss- i =. hr -Li = hr (*Δ(P) ) pss = 770 psi /cp Δ(P) l = 553 psi /cp (P), psi /cp *(P) 'a(p) a.e+06.e+05.e+04.e+03 * ( P )' 08 psi / cp L.E+0 P ( ) 58psi..E+0.E+00 3 P ( ) psi /cp pss P ( ) 553psi / cp L * ( P )' 6. 6 psi hr 3 * ( P )' psi / cp L 0.46 hr pss 39.hr.E-07.E-06.E-05.E-04.E-03.E-0.E-0.E+00.E+0.E+0.E+03, hr igur 4. Prssur and prssur drivaiv vs. i log-log plo for apl (gas rsrvoir). Using Equaions and 36, 8 and 35, 30 and 34, fracur conduciviy, w, ari prabiliy,, halffracur lngh, and naural fracurd nwor prabiliy,, wr siad and rpord in Tabl-3. pss Tabl-3. Coparison of rsuls for apl. Equaion Parar npu This wor Error, % w, d-f , d , f , f , d w, d-f CONCLUSONS a) Nw quaions ar prsnd o characriz syss consising of a horizonal wll wih ulipl fracurs in ulra-low prabiliy rsrvoirs using characrisic poins found on h prssur drivaiv so naural fracur prabiliy, ari prabiliy, half-fracur lngh and hydraulic fracur conduciviy can b sia and vrifid. b) Th rilinar odl allows hr possibl cass according h dvlopd flow rgis. n cas : bilinar flow, firs linar flow and psudosady-sa priod ar prsnd. or cas, h dvlopd flow 343

7 VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. rgis ar: bilinar, firs linar, firs psudosadysa, scond linar and scond psudosady-sa. inally, for cas 3, h obsrvd flow rgis ar: bilinar, followd by a linar and psudosady-sa - sa calld as scond linar and scond psudosady-sa in cas, which can b idnifid fro such propris as ari prabiliy and h rsuling valus of h inrporosiy flow parar and h dinsionlss soraiviy raio. Nonclaur B oraion volu facor, rb/stb, f 3 /SC C Hydraulic fracur conduciviy, dinsionlss C R Rsrvoir conduciviy, dinsionlss c Toal coprssibiliy, /psi d isanc bwn wo adjacn fracurs, f h Rsrvoir hicnss, f h f Thicnss of naural fracurs, f h f Toal hicnss of naural fracurs, f h Thicnss of ari slabs, f Prabiliy, d f Naural fracur inrinsic prabiliy, d Hydraulic fracur prabiliy, d Prabiliy of h innr rsrvoir, d O Prabiliy of h our rsrvoir, d Mari inrinsic prabiliy, d (P) Psudoprssur, psi /cp n Nubr of hydraulic fracurs n f Nubr of naural fracurs P Prssur, psia P insionlss prssur P i niial rsrvoir prssur, psi q low ra, STB/d q,sc low ra for a hydraulic fracur, oil STB/d, gas Mscf/d S Laplac parar Ti, hr T Rsrvoir Tpraur, R insionlss i *P insionlss prssur drivaiv *(P) insionlss psudoprssur drivaiv *Δ(P) Psudoprssur drivaiv, d/mscf w Hydraulic fracur widh, f Rsrvoir siz, -dircion, f Hydraulic half-fracur lngh, f y Rsrvoir siz, y-dircion, f z Ral gas- coprssibiliy facor Gr α β Δ ф ƞ λ μ π ρ f ω Parar dfind in rilinar flow odl Parar dfind in rilinar flow odl iffrnc opraor Porosiy, fracion iffusiviy, f/hr Transissiviy raio, ransin dual porosiy odl Viscosiy, cp Pi Consan dnsiy of naural fracurs, fracurs/f Soraiviy raio, ransin dual porosiy odl Subscrips Bilinar flow insionlss Ernal boundary f Naural fracur Hydraulic fracur i niial nnr Rsrvoir Mari L irs linar flow L Scond linar flow O Our Rsrvoir pss irs psudosady sa pss Scond psudosady sa R Rsrvoir Sc Sandard condiions Toal wf lowing wllbor ξ yp of diu: L,,O REERENCES [] Al-Hussainy R. & Ray H. J. 966, May. Applicaion of Ral Gas low Thory o Wll Tsing and livrabiliy orcasing. Sociy of Prolu Enginrs. doi: 0.8/43-B-PA. [] Al-Hussainy R., Ray H. J. & Crawford P. B. 966, May. Th low of Ral Gass hrough Porous Mdia. Sociy of Prolu Enginrs. doi: 0.8/43-A-PA. [3] Bnn C. O., Caacho-V., R. G., Rynolds A. C. & Raghavan R. 985, Ocobr. Approia Soluions for racurd Wlls Producing Layrd Rsrvoirs. Sociy of Prolu Enginrs. doi: 0.8/599- PA. [4] Brown M Analyical Trilinar Prssur Transin Modl for Muliply racurd Horizonal Wlls in Tigh Rsrvoirs. MS hsis, Colorado School of Mins, Goldn, Colorado, USA. [5] Brown M. L., Ozan E., Raghavan R. S. & Kazi H. 009, January. Pracical Soluions for Prssur Transin Rsponss of racurd Horizonal Wlls in Unconvnional Rsrvoirs. Sociy of Prolu Enginrs. doi: 0.8/5043-MS. [6] Caacho-V. R. G Rspons of Wlls Producing Coingld Rsrvoirs: Unqual racur Lngh. MSc Thsis. Univrsiy of Tulsa. Tulsa, OK. [7] Caacho-V. R. G., Raghavan R. & Rynolds A. C. 987, March. Rspons of Wlls Producing Layrd 343

8 VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. Rsrvoirs: Unqual racur Lngh. Sociy of Prolu Enginrs. doi:0.8/844-pa [8] Cinco L., H., Saanigo V.,. & oinguz A. N. 978, Augus. Transin Prssur Bhavior for a Wll wih a ini-conduciviy Vrical racur. Sociy of Prolu Enginrs. doi: 0.8/604- PA. [9] Cinco-Ly H. & Saanigo-V.. 98, Spbr. Transin Prssur Analysis for racurd Wlls. Sociy of Prolu Enginrs. doi:0.8/7490- PA [0] Cinco-Ly H., & Mng H.-Z. 988, January. Prssur Transin Analysis of Wlls wih ini Conduciviy Vrical racurs in oubl Porosiy Rsrvoirs. Sociy of Prolu Enginrs. doi: 0.8/87-MS. [] Swaan O., A. 976, Jun. Analyic Soluions for rining Naurally racurd Rsrvoir Propris by Wll Tsing. Sociy of Prolu Enginrs. doi: 0.8/5346-PA. [] Escobar.H. Rcn Advancs in Pracical Applid Wll Ts Analysis. 05. Nova publishrs Nw Yor. Publishd by Nova Scinc Publishrs, nc. Nw Yor. 45p. Nov. 05. [3] Gringarn A. C., Ray H. J. & Raghavan R. 974, Augus. Unsady-Sa Prssur isribuions Crad by a Wll wih a Singl nfini-conduciviy Vrical racur. Sociy of Prolu Enginrs. doi: 0.8/405-PA. [4] Kazi H. 969, cbr. Prssur Transin Analysis of Naurally racurd Rsrvoirs wih Unifor racur isribuion. Sociy of Prolu Enginrs. doi: 0.8/56-A. [5] Ozan E., Brown M. L., Raghavan R. & Kazi H. 0, April. Coparison of racurd-horizonal- Wll Prforanc in Tigh Sand and Shal Rsrvoirs. Sociy of Prolu Enginrs. doi: 0.8/90-PA. [6] Pras M. 96, Jun. Effc of Vrical racurs on Rsrvoir Bhavior-ncoprssibl luid Cas. Sociy of Prolu Enginrs. doi: 0.8/575-G. [7] Pras M., Hazbro P. & Sriclr W. R. 96, Jun. Effc of Vrical racurs on Rsrvoir Bhavior- -Coprssibl-luid Cas. Sociy of Prolu Enginrs. doi: 0.8/98-PA. [8] Raghavan R. S., Chn C.-C. & Agarwal B. 997, Spbr. An Analysis of Horizonal Wlls nrcpd by Mulipl racurs. Sociy of Prolu Enginrs. doi: 0.8/765-PA. [9] Srra K., Rynolds A. C. & Raghavan R. 983, cbr. Nw Prssur Transin Analysis Mhods for Naurally racurd Rsrvoirs (includs associad paprs 940 and 304). Sociy of Prolu Enginrs. doi:0.8/0780-pa. [0] Shfs H Nurical nvrsion of Laplac Transfors. Counicaions. ACM. 3(): [] Tiab Analysis of Prssur and Prssur rivaiv wihou Typ-Curv Maching: - Sin and Wllbor Sorag. Journal of Prolu Scinc and Enginring. : 7-8. S ric convrsion facor Bbl E-0 = 3 cp.0* E-03 = Pa-s f 3.048* E-0 = f * E-0 = psi E+00 = Pa Appndi A Addiional quaions for h rilinar flow odl, (Ozan al. 0) ar givn blow: C w (A-) S (A-) C (A-3) anh y R R w (A-4) R R u (A-5) CR y C R (A-6) Oy 3433

9 VOL. 3, NO. 0, MAY 08 SSN Asian Rsarch Publishing Nwor (ARPN). All righs rsrvd. R S R S anh ( ) R (A-7) O R (A-8) n Equaions A-3 and A-8, ξ =,, O. ( c ) (A-9) n Equaion A-5, u sf( S) nrnal naurally fracurd rsrvoir: 3S f( S) anh 3S (A-0) ual porosiy parar for h inrnal naurally fracurd rsrvoir: ( ch ) (A-) ( ch) f (A-) hh f f 3434