Impedance Analysis as a Tool for Hydraulic Fracture Diagnostics in Unconventional Reservoirs

Transcription

1 Austrlin Journl of Bsic nd Applid Scincs, 7(9): 15-7, 13 ISSN Impdnc Anlysis s Tool for Hydrulic Frctur Dignostics in Unconvntionl Rsrvoirs Amir Rz Rhmni, Mhdy Shirdl Dpt. of Ptrolum & Gosystms Enginring, Univrsity of Txs t Austin, Austin, TX, USA. Abstrct: Production from unconvntionl rsrvoirs plys ky rol in supplying hydrocrbon to th world s incrsing nrgy dmnd. Hydrulic frcturing is on of th most dvncing tchnologis in producing hydrocrbon from low-prmbility rsrvoirs, spcilly in tight nd shl formtions. Th gomtric proprtis of hydrulic frctur ffct significntly th production rt from ths stimultd rsrvoirs. Thrfor, it is of utmost importnc to chrctriz th gomtric fturs of hydruliclly-inducd frcturs during stimultion procsss. Th gomtric proprtis of hydrulic frctur chng with frctur volution du to strss propgtion long th crcks. W hv modld such systm bsd on th nlogy btwn fluidic nd lctric circuits. Th modl tks into ccount th dynmic ltrtion of hydrodynmic impdnc of th frctur. Th hydrodynmic impdnc controls th rltion btwn th frctur fluid flow rt nd prssur drop during stimultion procss. Th proposd modl incorports th ffcts of both hydrodynmic cpcitnc nd rsistnc of th frctur nd th wllbor to comput th frctur impdnc. Downhol msurmnts of prssur nd flow rt cn xplicitly dtrmin th hydrodynmic impdnc of th frctur. Frctur impdnc cn lso b implicitly infrrd from wllhd prssur nd flow rt msurmnts. Th pulstil ntur of hydrulic frcturing trtmnt mks it snsibl to utiliz th wllhd dt for th purpos of impdnc monitoring bsd on th lctromgntic trnsmission lin thory. W hv prformd xtnsiv snsitivity nlyss on th most importnt prmtrs of singl hydrulic frctur for ssssing frctur impdnc. Ths prmtrs includ frctur thicknss, frctur rdil xtnt, nd frctur shp. Through snsitivity nlyss rsults nd rl-tim msurmnt of frctur impdnc, w infr th bsic gomtric proprtis of hydrulic frctur. This rsrch hs ld to n nlyticl pproch for vluting hydrulic frcturs gomtric chrctristics. Th pproch builds n intuition towrd bttr undrstnding how to dsign hydrulic frcturs in mor fficint fshion. Ky words: INTRODUCTION Hydrulic frcturing is on of th most pplicbl topics in ptrolum industry which rquirs furthr dvlopmnt nd invstigtion. This procss, in its most gnrl trm, ncompsss injcting fluid t high rts into th wll. If th prssur xcds th tnsil strngth of th rock, th rock crcks opn, crting vrticl pln propgting wy from th wllbor. Most importnt gomtric chrctristics of th frcturs includ th lngth, hight, nd th width (opning). Lngths cn rng from hundrds to thousnds of ft, hight is usully hundrds of ft, nd th opning is on th ordr of frctions of inch. Frctur gomtry dtrmins th fficincy of hydrulic frctur trtmnt nd hnc, is of utmost importnc to production rt. In this study, our min objctiv is invstigting nd dvloping tool to nlyz th frctur gomtry during th frcturing procss bsd on th concpt of hydrulic impdnc. Th notion of hydrulic impdnc (for dtils, s Tbling 5) cn hlp us chrctriz th complx bhvior of hydrulic frctur systm mor ffctivly through obtining n nlogous lctricl systm. For this purpos w hv tkn th following stps in our study: Introducing impdnc nlysis for frcturd wlls, using th nlogy btwn lctricl circuits nd hydrodynmics. Dvlopmnt of n fficint nd simpl tool to vlut hydrulic frcturs gomtry during typicl hydrulic frcturing prctic. Prforming snsitivity nlysis on frctur gomtry to invstigt th ffct of frctur prmtrs on prssur nd flow rt pulss. Minly frctur gomtris r considrd s rctngulr or llipticl shps. Figur 1 shows th common frctur shps. Figur shows th frctur systm in conjunction to th wllbor. In our nlysis w considr th coupld frctur wllbor systm (Yw nd Ashour, 1996). Corrsponding Author: Amir Rz Rhmni, Mhdy Shirdl, Dpt. of Ptrolum & Gosystms Enginring, Univrsity of Txs t Austin, Austin, TX, USA. E-mil: rhmni@utxs.du 15

2 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 () (b) Fig. 1: Frctur gomtry () rctngulr cross sction (b) llipticl cross sction Fig. : Coupld wllbor/frctur systm Fluid Trnsint Anlysis: In microfluidics, hydrodynmic impdnc is dfind s th rtio of prssur (or prssur hd) to th flow rt through th fluidic lmnt s rsult of th pplid prssur, in th sm wy lctric impdnc is th rtio of lctricl voltg to lctricl currnt. In this sction th frctur systm is trtd s control volum in which instntnous stoppg is occurrd t th downstrm. In fct, whn th frctur is opnd by incrsing th frctur fluid prssur (hd), th frctur growth is continud until it is stoppd. Ths phnomn cn b modld s n instntnous stoppg of flow strm by vlv. Figur 3 () shows th schmtic viw of th systm nd th control volum tht momntum blnc is pplid. In our nlysis w ssum tht friction nd othr minor losss r nglctd in th wllbor. For ccurt modling of wllbor hydrodynmic, comprhnsiv modl (Shirdl nd Sphrnoori, 11) cn b usd. In ddition, w ssum whn th instnt vlv is closd, th fluid immditly djcnt is brought from initil vlocity to rst by impuls of th highr prssur dvlopd t th fc of th vlv. As soon s th first lyr is brought in to rst th sm ction is pplid to th nxt lyr. Figur 3(b) shows th momntum qution control volum nd th corrsponding shock wv trvling towrd upstrm. Th hd chng t th vlv, H, is ccompnid by vlocity chng,. Writing th momntum qution in th z dirction, w obtin 16

3 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 ρa dt ρa dtγ HAdt= Adzρ( ), (1) Whr dz cn b clcultd from coustic vlocity,, nd th tim rquird for th comprssion shock to trvl through th control volum lngth, Eq.. dz = ( ) dt () Substituting th dz from Eq. into Eq.1 w hv ρa dt ρa( ) dt γ HAdt = A( ) dtρ( ). (3) Rrrnging th vribls nd ssuming to b smll w obtin ρ ρ = γ H (4) ρ H = ( ). (5) γ Eq.5 cn b simplifid furthr by ssuming 1. Hnc Eq.5 bcoms ρ H = = = Q. (6) γ g ga Eq.6 shows th rltion btwn th hd chng t th vlv nd th vlocity chng. Ltr on, w dfin s th chrctristic impdnc of th conduit. To dfin th coustic vlocity, w combin Eq.6 with ga mss blnc for th givn control volum in Figur 3(b). () (b) Fig. 3(): Instnt stoppg flow in horizontl frctur (b) Momntum qution pplid to control volum ρa dt = ρl. da ρa. dl Al. dρ, (7) 17

4 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 l l l whr l is dz. Assuming dt = nd dl = ( ), w obtin l ρa = ρlda. Ald. ρ (8) da dρ =. A ρ (9) Using Eq.6, w cn limint from Eq.9. g H =. (1) da dρ ( ) A ρ Assuming P K = s th bulk modulus of lsticity of th fluid, th coustic vlocity bcoms: dρ ρ K / ρ =. (11) K A 1 A P In th following sctions, Eqs.6 nd 11 r frquntly usd for impdnc clcultion. Oscilltory Flow Equtions: Fluid oscilltions in systms my b nlyzd by us of th linr vibrtion thory or lctricl trnsmission-lin thory. In this sction w show th linr vibrtion thory to obtin th prssur hd nd dischrg rt vritions in th systms tht prturbtion is inducd. Using th similr momntum nd mss blnc qutions in th Fluid Trnsint Anlysis sction, w cn find th trnsint diffrntil qution in conduit (piplin). Eqs.1 nd 13 show th systm of qutions for piplin Ag H Q = t z (1) H fq n 1 Q = z gda n Ag t, (13) Assuming H = Hz = Hz h z, z Q =Q z = Qz q z, z H = Ht = Ht ht, t Q =Q t = Qt q t t, Qz= Ht= Qt=, fq n H z = gda n. Eqs.1 nd 13 r convrtd to Cht q z = (14) h z Lqt Rq =, (15) 18

5 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 3ν 1 ga whr R =, L =, C =. Thr is n nlogy btwn Eqs.14 nd 15 nd th lctricl circuit gad ga qutions in n RLC circuit s w will ltr prsnt in impdnc nlysis sction. Solving Eqs.14 nd 15 (Wyli nd Strtr, 198), w obtin Cs st γ ( 1 z γ q = C C z ) = Qz st (16) γ st γ ( 1 γ h = C C z ) = H( z ), (17) whr C1, Cr clcultd from inlt prssur nd disgorg pulss boundry condition: 1 C1 = ( H Z Q ) c (18) 1 C = ( H Z Q ) c. (19) In Eqs.16 nd 17, γ nd s r dfind s γ = iω / () s = σ iω (1) nd Z c is dfind s th chrctristic impdnc 1 Cs =. () Z c γ If w ssum thr is no friction rsistivity ffct, thn s will only hv th imginry trm s s = iω. Substituting C, s nd γ in Eq. w obtin Z c ga =. (3) Rfrring to Eq.6 w rriv in H Q = Z c. (4) In fct, w intrprt th chrctristic impdnc s th rtio of hd chng to dischrg rt chng. It should b notd tht this rtio is not qul to Z c t vry loctions of th conduit. Howvr, From Eqs.16 nd 17, w obsrv tht hydrulic impdnc t ny point is h 1 ( C γz 1 C γz ) Z( z) = = (5) q Z c ( C γz z 1 C γ ) In th impdnc nlysis sction w will xplin th impdnc vlu in ny loction using th lctricl circuit nlysis pproch. W obsrv rsonbly ccurt nlogy btwn ths two mthods. Frctur nd Wllbor Impdnc Anlysis: Figur 4 shows th schmtic of trnsmission lin. Figur 4. shows trnsmission lin, oftn schmticlly rprsntd s two-wir lin (Pozr, 1998). Th short pic of lin of lngth dz in Figur 4. cn b modld s lumpd-lmnt circuit s shown in Figur 4.b, whr R, L, G, C r pr unit lngth quntitis dfind s follows: R =sris rsistnc pr unit lngth, for both conductors in ohm/m. L =sris inductnc pr unit lngth, for both conductors, in H/m. G =shunt conductnc pr unit lngth, in S/m. C = shunt cpcitnc pr unit lngth, in F/m. 19

6 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 A finit lngth of trnsmission lin cn b viwd s cscd of sctions of th form of Figur 4.b. From th circuit of Figur 4.b., Kirchhoff s voltg lw cn b pplid to yild i( zt, ) v( zt, ) Rdzi( zt, ) Ldz v( z dzt, ) = (6.) t nd Kirchhoff s currnt lw lds to v( z dz, t) i( zt, ) Gdzv( z dzt, ) Cdz i( z dzt, ) =. (6.b) t () (b) Fig. 4: oltg nd currnt dfinitions nd quivlnt circuit for n incrmntl lngth of trnsmission lin: ) oltg nd currnt dfinitions. b) Lumpd-lmnt quivlnt circuit. Dividing Eqs.6. nd 6.b by dz nd tking th limit s dz gos to zro givs th following diffrntil qutions: v zt, i zt, = Ri ( z, t) L t t i( zt, ) v zt, = Gv ( z, t) C z t (7.) (7.b) Ths qutions r th tim-domin form of th trnsmission lin, or tlgrphr qutions. Not th similrity of Eqs 7. nd 7.b (stting G = ) with thir hydrodynmic countrprts Eqs.14 nd 15. For th sinusoidl stdy-stt condition with cosin-bsd phsors, Eqs 7. nd 7.b simplify to: d z dz di z dz ( R jωl) I( z) = (8.) ( G jωc) ( z) = (8.b) d z dz Th two qutions cn b solvd simultnously to obtin wv qutions for (z) nd I (z): d I z dz γ z = (9.) γ I z = (9.b) whr γ α jβ ( R jωl)( G jωc) = = is th complx propgtion constnt, which is function of frquncy. Trvling wv solutions to Eqs 9. nd 9.b cn b found s:

7 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 γz γz z = (3.) γz γz I z = I I, (3.b) z γ whr th trm rprsnts wv propgtion in th z dirction, nd th trm rprsnts wv propgtion in th -z dirction. Applying Eq.8. to th voltg of Eq.3. givs th currnt on th lin: z γ γ I( z) = z z γ γ R jωl Comprison with Eq.3.b shows tht chrctristic impdnc, Z c, cn b dfind s: (31) R jωl R jωl Z c = = γ G jωc (3) to rlt th voltg nd currnt on th lin s = Z c = I I (33) Thn Eq.3.b cn b rwrittn in th following form: z γ γz I z = Zc Zc (34) For losslss trnsmission lin (or quivlntly, frictionlss conduit), w cn st R = nd G = : γ = σ jβ = jω LC (35) or β = ω LC (36.) σ = (36.b) As xpctd for th losslss cs, th ttnution constnt, α is zro. Th chrctristic impdnc of Eq.3 rducs to: L Z c = (37) C which is now rl numbr. Th gnrl solutions for voltg nd currnt on losslss trnsmission lin cn thn b writtn s: jβz jβz ( z) = (38.) j z β jβz I z =. (38.b) Zc Zc It should b notd tht th chrctristic impdnc dos not rprsnt ny loss in th systm. W hv ssumd tht th systm is frictionlss. As Eq.33 suggsts, this impdnc rlts th mplitud of th outwrd nd inwrd currnt wv to th mplitud of th outwrd nd inwrd voltg wv. In othr words, no powr is lost in th systm du to th chrctristic impdnc. Th Trmintd Losslss Trnsmission Lin: Figur 5 shows losslss trnsmission lin trmintd in n rbitrry lod impdnc Z L. This problm illustrts wv rflction on trnsmission lins, fundmntl problm in distributd systms. Th sm phnomnon occurs in our hydrodynmic systm. Th wllbor is cting s trnsmission lin nd th hydrulic frcturs r cting s lods s sn in Figur 5. 1

8 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 Z L Fig. 5: Schmtic of trnsmission lin trmintd with n rbitrry lod z jkz Lt s ssum tht voltg wv of th form is gnrtd from sourc t z <. W hv lrdy provn tht th rtio of voltg to currnt for this sourc is th chrctristic impdnc (whn thr is no rflction bck). Howvr, with lod impdnc t th nd of trnsmission lin, th impdnc of th systm t th loction of th lod should b qul to th lod impdnc. Thrfor, thr should b rflction wv xcitd t th lod with consistnt mplitud to stisfy th lod (boundry) condition. Thrfor, th voltg nd currnt cn b writtn s th sum of th incidnt nd rflctd signls s in Eqs 38. nd 38.b. Th voltg nd th currnt t th lod (z = ) r rltd by th lod impdnc: ZL = = Z I c. (39) - Solving for ZLZ Γ= = c ZL Zc (4) Thrfor, th totl voltg nd th currnt r: ( z) = j z j z γ Γ γ (41.) I( z) = j z j z Z c γ Γ γ (41.b) Thrfor, th impdnc t vry loction is: ( z) 1Γ γz 1Γ jβz Z( z) = = Z Z c = I z 1 z c Γ γ 1Γ jβz, (4) whr th lst qution holds whn thr is not ny loss in th systm (R = nd G = ). Pls not tht th voltg nd currnt on th lin consist of th suprposition of th incidnt nd rflctd wvs (stnding wvs). Only whn Γ =, thr is no rflctd wv. To obtin Γ =, th lod impdnc Z L must b qul to th chrctristic impdnc Z c of th trnsmission lin, s sn from Eq.4. Such lod is thn sid to b mtchd to th lin, sinc thr is no rflction of th incidnt wv. Espcil Cs: Opn Circuit Boundry: If th nd of conduit is blockd, thr is no fluid flowing through th nd of th conduit. This cs is similr to n opn circuit, i.., Z L tnding to infinity in th bov trnsmission lin nlogy. For this spcific cs, th rflction cofficint, Γ = 1. Thrfor, bsd on Eq.4, th impdnc for such systm is: ( z) 1 jβ z Z ( z) = = Z I( z) c. (43) 1 jβ z In prctic, w usully hv th prssur nd flow rt dt t th wllhd. Thrfor, bsd on th wllhd dt, w cn clcult th impdnc t th wllhd. Using th bov nlysis, on cn trnsfr th frctur impdnc to th wllhd (Holzhusn nd Gooch, 1985). By compring th wllhd dt with snsitivity nlysis rsults on frctur dimnsions, w cn potntilly vlut th hydrulic frctur. Frctur Impdnc Clcultion: Similr to th trnsmission lin (hr w cll s wllbor), w cn clcult th frctur chrctristic impdnc. As Eq.44 shows (Holzhusn nd Gooch, 1985), th chrctristic impdnc of th frctur conduit is

9 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 Z cf f = (44) gaf whr A f is th totl r of th frctur whr it intrscts th wll nd f is th wv spd in th frctur. Assuming pnny shp frctur r for both frctur wings, w hv Af = π bh. Substituting A f nd Eq.11 for coustic wv spd in Eq.44, th frctur chrctristic impdnc yilds K / ρ 1/ K A 1 A P Z cf =. (45) gπbh K A For th complint conduit, w cn ssum tht 1. Hnc A P Z cf PA 1/ =. 4g π b h ρ A (46) Furthrmor, th cross sction vrition, A, is rltd to th prssur chng s follows (Snddon, 1946): A = Ph (1 ν ) / µ. (47) Z cf Substituting A nd A = πbh in Eq.46 w obtin µ 1/ = 8 ρg π(1 ν) bh 3 (48) In cs th friction loss is nglctd in th frctur, w cn ssum frctur impdnc is idnticl to frctur chrctristic impdnc. Hnc in our nlysis w hv usd frctur nd trnsmission lin (wllbor) chrctristic impdncs to obtin th obsrvd impdnc t th wllhd. Rsults: In this sction, w show th numricl rsults tht w obtind from this study. Figurs 6 nd 7 illustrt th snsitivity of th frctur impdnc to frctur rdius (h) nd frctur width (b). Th plots suggst tht with incrsing th frctur rdius nd width, th frctur impdnc dcrss. In fct, incrsing th frctur volum cuss th fluid mor sily to flow into th frctur. Hnc, th impdnc bcoms smllr in lrgr frctur volums. Eq.48 lso justifis this bhvior mthmticlly. As w xplind in th prvious sction, sinc dt r cquird t th wllhd, th frctur impdnc should b clcultd t this loction. Hnc, w dfind Eq.4 to trnsfr frctur impdnc from bottom-hol to th wllhd. Figur 8 shows th frctur impdnc trnsfr to th wllhd t diffrnt prssur puls frquncis (1, 1, 1). In this grph, w hv ssumd tht th wllbor chrctristic impdnc is 115 (sc/m ). As indictd in this figur, th frctur impdnc blow crtin vlu is trnsfrrd to constnt vlu nd byond tht, dpnding on th frquncy vlu, it dclins or incrss. It is lso intrsting tht in ll of th frquncis, th bsolut impdncs t wllhd r intrsctd t th frctur impdnc qul to wllbor chrctristic impdnc. In ordr to build our intuition on th hydrodynmic impdnc, w hv clcultd th flow rt signl tht w would obsrv t th wllhd for givn input hd (prssur) signl. Figur 9 shows th input hd nd Figur 1 illustrts th rsulting flow rt. Not th phs chng in signl du to th rctiv ntur of our ssumd hydrulic frctur (lod). On cn obtin th rng of frctur hlf-widths nd frctur rdii corrsponding to prticulr impdnc vlu. Morovr, for pnny-shpd frctur, th frctur hlf-width nd frctur rdius r corrltd through (Nolt t l., 1981, Pollrd t l., 1979): ( ν ) Ph 1 b =, (49) πµ whr P is th uniform intrnl prssur in xcss of frctur closur prssur. By solving Eqs. 48 nd 49 simultnously, on cn obtin th frctur hlf-width nd frctur rdius for givn frctur impdnc, xcss prssur, nd formtion shr modulus. Figur 11 shows this procdur grphiclly. In this figur, th solid curvs rprsnt frctur rdius vrsus frctur hlf-widths for frctur impdnc qul to 18 s/m for thr diffrnt formtion shr moduli. Th dottd curvs plot Eq. 49 for two diffrnt vlus of xcss prssur for 3

10 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 formtion shr modulus of 1, psi. Th intrsction of ths curvs with th solid curvs dtrmins th frctur hlf-width nd rdius for givn frctur impdnc (bsd on wllhd dt). 1 Pnny Shp Frctur Impdnc vrsus Frctur Rdius 1-1 bs(zf) (sc/m) Fig. 6: Frctur impdnc vrsus frctur rdius (h) Frctur Rdius (m) Pnny Shp Frctur Impdnc vrsus Frctur width h=1.5m h=3.5m h=6.1m h=15.m h=3.5m bs(zf) (sc/m) Frctur Hlf-width (m) Fig. 7: Frctur impdnc vrsus frctur hlf-width (b) 1 5 bs(zw) vrsus Zf frq=1 frq=1 frq=1 bs(zw) (sc/m) Zf (sc/m) Fig. 8: Frctur impdnc trnsfr to th wllhd frctur impdnc 4

11 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, Hd (m) t(sc) Fig. 9: Prssur (hd) puls induc in th wllhd flow rt (m3/sc) Fig. 1: Th dischrg rt rspons in th wllhd t(sc) Frctur Hight (m) mu=1 psi mu=5 psi mu=5 psi p=5 psi p=5 psi Frctur Hlf-width (m) Fig. 11: Frctur gomtry dignostic using th chrctristic impdnc curv nd frctur closur prssur constrint 5

12 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 Summry nd Conclusions: In this ppr, w discussd fluid trnsint nlysis in hydruliclly frcturd systm. W furthr dvlopd n nlogy btwn lodd trnsmission lin systm nd hydrodynmic frcturd systm. W drivd dtild mthmticl qutions using fundmntl hydrodynmic qutions nd lctricl circuit solutions. W lso modifid som shortcomings of th prcding studis. Utilizing th two diffrnt pprochs, w drivd similr formultions. Morovr, w dvlopd th notion of impdnc trnsfr from downhol to th wllhd nd pplid it for th purpos of vluting frcturd systm using wllhd prssur-flow\rt dt. W finlly prformd snsitivity nlysis for frctur gomtry dignostics to furthr dvlop n intuition on frctur gomtry dignostics using th chrctristic impdnc curv nd frctur closur prssur constrint. Thrfor, wllhd prssur/flow rt dt cn b usd to potntilly infr th frctur gomtris. Nomncltur: w nd w f : Frctur width (m) L nd L f : Frctur lngth (m) h f : Frctur hight (m) R w : Wllbor rdius (m) L d : Lngth long th wllbor from wllhd to frctur loction (m) ρ: Fluid dnsity (kg/m3) H: Mximum prssur hd vrition t th wllhd (m) H : Sttic prssur hd t th wllhd (m) A: Cross-sction (m ) A: Acoustic vlocity (m/s) : Initil fluid flux (m/s) ΔH: Prssur hd vrition (m) γ: Grvity potntil cofficint (ρg) (kg/m s ) z: Longitudinl coordint (m) dt: Tim incrmnt (s) t: tim (s) ΔQ: rition in flow rt (m 3 /s) Δ: rition in fluid flux (m/s) g: Grvity cclrtion (m/s ) K: Bulk modulus of lsticity (P) f: Frquncy (Hz) s: Complx frquncy (rd/s) γ: Propgtion constnt in wll (rd/s) ѡ: Angulr frquncy (rd/s) Z c : Chrctristic impdnc of wll (s/m ) Z f : Frctur impdnc (s/m ) Z w : Frctur impdnc (s/m ) Z: Hydrodynmic impdnc (s/m ) Г: Rflction cofficint (unitlss) Г f : Downhol rflction cofficint (unitlss) REFERENCES Drholt, D.W. nd W.R. McSpddn, Elctromgntic Wv Propgtion, McGrw Hill, Nw York. Holzhusn, C.R. nd R.P. Gooch, Impdnc of Hydrulic Frcturs: Its Msurmnt nd Us for Estimting Frctur Closur Prssur nd Dimnsions, SPE 1389, prsntd t th SPE/DOE Low prmbility Gs Rsrvoirs hld in Dnvr, CO Shirdl, M., K. Sphrnoori, 11. Dvlopmnt of Trnsint Mchnistic Two-Phs Flow Modl for Wllbors, Rsrvoir Simultion Symposium Confrnc, Woodlnds, Txs, USA, Nolt, K.G, nd M.B. Smith, Intrprttion of Frcturing Prssurs, J. Ptr. Tch., pp: Tbling, P., 5. Introduction to Microfluidics, Oxford Univrsity Prss Inc., Nw York. Pozr, D.M., Microwv Enginring, John Wily nd sons, inc., Nw York. Pollrd, D.D. nd G.R. Holzgusn, On th Mchnicl Intrction btwn Fluid-filld Frctur nd th Erth s Surfc, Tctonophysics, 53: Snddon, I.N., Th Distribution of Strss in th Nighborhood of Crck in n Elstic Solid, Proc. Roy. Soc. A., pp:

13 Aust. J. Bsic & Appl. Sci., 7(9): 15-7, 13 Wyli, E.B. nd.l. Strtr, 198. Fluid Trnsints, FEB Prss, Ann Arbor. Yw, C.H., A.A. Ashour, A Study of th Frctur Impdnc Mthod, Annul Tchnicl Mting of th Ptrolum Socity, Cnd, Clgry, Albrt 7