Chapter 8 - Transient Laminar Flow of Homogeneous Fluids

Transcription

1 Cha 8 - Tansn Lamna Flow of Homognous Fluds 8. Tansn Flow Th ansn condon s only alcabl fo a lavly sho od af som ssu dsubanc has bn cad n h svo. In accal ms, f ssu s ducd a h wllbo, svo fluds wll bgn o flow na h vcny of h wll. Th ssu do of h xandng flud wll ovok flow fom fuh, undsubd gons n h svo. Th ssu dsubanc and flud movmn wll connu o oaga adally away fom h wllbo. Th gadually xndng gon affcd by flow s sn n Fgu 8.. In h m fo whch h ansn condon s alcabl s assumd ha h ssu sons n h svo s no affcd by h snc of h ou bounday, hus h svo aas nfn n xn. Fgu 8. Concual vw of ansn flow o o achng ou bounday Th condon s manly ald o h analyss of wll ss n whch h wll's oducon a s dlbaly changd and h sulng ssu sons n h 8.

2 Cha 8 - Tansn Lamna Flow of Homognous Fluds wllbo s masud and analyzd dung a bf od of a fw hous af h a chang has occud. Thn, unlss h svo s xmly small, h bounday ffcs wll no b fl and h svo s, mahmacally, nfn. In hs cas, boh h ssu and ssu dvav, wh sc o m, a funcons of boh oson and m,.., = f,. Two mods of ansn oducon a consan oducon a and consan flowng wllbo ssu. Boh a shown n Fgu 8.. Th fgu llusas ha consan a mls a dcln n wllbo ssu and ha consan ssu mls a dcln n oducon a. Mahmacal soluons of boh condons a wll documnd n h lau. Th connuy uaon fo ansn flow, n Casan coodnas, can b xssd as: v 8. As dscbd n Scon 6., combnng acy s Law wh v k 6.60 d gz 6.6 o wh an uaon of sa, fo xaml, assum an dal lud wh comssbly s consan, c P 6.66 suls n a dffusvy uaon. Fo adal gomy, sooc and homognous oous mda, h dffusvy uaon s, c k 8. To solv hs oblm us nal and bounday condons. Fo an nfn-acng svo, wh consan a oducon a h wllbo, h followng condons aly. 8.

3 Cha 8 - Tansn Lamna Flow of Homognous Fluds P,0 P P, P Th wllbo s assumd o b nfnsmally small,.. 0, o ln souc. A mahmacal chnu o solv h ansn, adal flow of a sngl-has, slghly comssbl flud can b accomlshd alyng h Bolzmann ansfom. Th soluon s: c s 8.4 4k, E s s wh E-s s known as h xonnal ngal, E x ds and x s dfnd as: x s c x. A wdly ald aoxmaon occus fo x < 0.0, such ha 4k E x ln x ln.78x 8.6 Fgu 8. Comason of E funcon wh naual log aoxmaon 8.3

4 Cha 8 - Tansn Lamna Flow of Homognous Fluds 8. Suoson Th ncl of suoson sas h oal ssu do a any on n h svo s h sum of h ssu dos a any on causd by flow n ach of h wlls n h svo. As an xaml, consd a h wll sysm llusad n Fgu 8.3. ab B A ac C Fgu 8.3. Alcaon of suoson n sac fo a h wll sysm Th oal ssu do n wll A s h summaon of h ssu dos causd by ach wll. 8.7 oal A A B C Subsung n E. 8.7 h ln souc soluons, and convng o fld uns, suls n h followng uaon fo oal ssu do n ll A. 4. a B o k oal A ln 688c wa 4. b B c o 948 E ab k 4. B c c o 948 E ac k Common alcaons of hs mhod nclud: valuang mulwll ssu ansn ss;.g., aculaly fo nfnc o uls ss, smulang ssu bhavo n svos wh boundas, and fo dnfyng wll locaons. S a

5 Cha 8 - Tansn Lamna Flow of Homognous Fluds 8.5 In comac dmnsonlss fom: n j j P j B, 4., 8.9 Suoson of vaabl a and m Th objcv of suoson of vaabl a and m s o modl vaabl a bhavo n wlls o cocly analyz ssu s masumns. Fgu 8.4 llusas a sml sngl wll, wo-a s. Th oblm can b dfnd as a som m,, >, wha s wf? Th wll oducs a a a, fom = 0 o a whch m h oducon a changs o fom o. o m - Fgu 8.4. Two-a s Th soluon usng h ncl of suoson s: S w c k B S w c o k B T 688 ln ln Exssd n comac fom, 688 ln 4. j j n j S w c j k B T 8. Th gnal fom of uaon fo suoson n sac and m/a s gvn by,

6 Cha 8 - Tansn Lamna Flow of Homognous Fluds 4.B m n, P j j, j j 8. wh m s h numb of wlls and n s h numb of as fo ach wll. A scal cas of E. 8.0 occus f = 0 shu n. Th sul s h Hon Aoxmaon o ssu buldu sng. Mhod of Imags Emloyng a chnu calld h mhod of mags dvs h ssu bhavo of a wll na a salng lna faul o oh flow ba. In hs fomulaon h ffc of a faul s smulad by assumng h snc of anoh dncal wll oducng a a symmcal oson acoss h faul, as shown n Fgu 8.5. Th mag wll nacs wh h acual wll so ha no flow occus acoss h faul. Th sulng ssu do a h al wll du o s own oducon and h "nfnc do" fom h mag wll add ogh o smula cocly h ssu bhavo of h al wll as hough w n h oxmy of h faul. faul al mag d d Fgu 8.5. Schmac of a no flow bounday wh a al and mag wll locad a dsanc d fom h bounday. By alyng suoson w can calcula h ssu a any on n h xy lan. Fo h scal cas a h wllbo and consan oducon a, 8.6

7 Cha 8 - Tansn Lamna Flow of Homognous Fluds P wf P wf 4.B w d P E x E x o f assum ln aoxmaon, 8.3 P 948c 4.B k ln S E k 688c w d h h vous dfnon of x, h scond a of E. 8.3 s abbvad noaon wh h subsc fs o h m funcon and h susc h saal dsanc. Examl Consd a sngl salng faul locad 50 f away fom h acv wll. ha s h boom hol flowng ssu af flowng 350 sbd fo 8 days, gvn h followng nfomaon. k = 5 md c = x 0-5 sa - = 0.50 c = 6.0% B o =.3 s bbl/stb h = 50 f = 3000 s w = f Skn = +5. Chck f h logahmc aoxmaon s vald 948c x k x x0 584 Fo = w, hn x w = 3.5x0-8 << 0.0, hus ln aoxmaon als Fo = L hn x L = >> 0.0, hus E funcon als. Calcula h ssu do 948 c d 4. B k P ln S E k 688c w ln x * s 8.7

8 Cha 8 - Tansn Lamna Flow of Homognous Fluds 3. Calcula h boom hol flowng ssu wf s If a ssu buldu s s un n h wll hn boh suoson n sac and m mus b ald o dvlo a soluon. In abbvad fom, h sulng soluon can b xssd as, ws 4.B w w E x E x d d E x E x 8.4 As an aoxmaon, consd m suffcnly lag such ha x <<0.0 and hus a h wllbo h logahmc aoxmaon s asonabl. Also, aly n h buldu whn s small and ovdd d s lag, hn h agumns of h E funcon bcoms lag and hus E0. Subsunly h las wo ms n E. 8.4 aoach zo, and h uaon ducs o; wf 4.B ln 8.5 Thus aly n h ssu buldu h slo of h buldu lo wll man unchangd fom h convnonal analyss. As h shun m ncass, h logahmc aoxmaon bcoms asonabl fo h mag wll ms, hus wf 4.B ln 4.B ln ln

9 Cha 8 - Tansn Lamna Flow of Homognous Fluds Th sul s a doublng of h slo. An xaml s shown n Fgu 8.6. Fgu 8.6 Examl of Hon ssu buldu lo wh lna dsconnuy Ealough, 977 To sma h dsanc o h lna dsconnuy fo a ssu buldu s us h nscon on of h wo sagh lns as shown n Fgu 8.6. If w ua Es. 8.5 and 8.6, h sul s an xsson fo h dsanc o h faul k d 8.7 c x wh g s h xonnal of Eul s consan =.78. E. 8.7 s vald fo lag. Examl Th followng wll and svo daa was acud fo h s daa of Fgu 8.6. = 530 hs = 0.0 c k = 40.6 md = 0.09 c =.6 x 0-6 s - Fom h fgu h nscon on = 85, hfo fom E. 8.5 h dsanc o h faul, d = 66 f. 8.9

10 Cha 8 - Tansn Lamna Flow of Homognous Fluds 8.3 Unsady sa wa nflux modl Th ansn nau of many aufs suggss a m dndn m s ncssay o calcula h wa nflux no a svo. Consd a ccula svo of adus b, as shown n fgu 8.7, n a hozonal, ccula auf of adus,, whch s sooc and homognous. b Fgu 8.7 Schmac of a adal svo/auf sysm In hs cas h nn bounday s dfnd as h nfac bwn h svo and h auf. I hus bcoms usful o u ssu a h nn bounday o b consan and obsv h flow a as ns h svo fom h auf. Th soluon o hs oblm s known as h consan mnal ssu soluon CTPS and was solvd by Van Evdngn and Hus, 949. Th condons ncssay a:,0, o 0 acy s Law gvs h a of flud nflux, b 8.8 Infn boundd fn dmnsonlss adus, = / b, hn a h nfac,

11 Cha 8 - Tansn Lamna Flow of Homognous Fluds Th cumulav nflux s gvn by, d d fn dmnsonlss m,, as d b c k d 8. wh k,,, and c a os of h auf. Subsu no E. 8., b hc o d b hc 8.3 wh s h dmnsonlss cumulav wa nflux un ssu do mosd a h svo/auf bounday. fn h wa nflux consan, B, as: f b hc B 8.4 wh f s facon of h auf whch subnds h svo ccumfnc;.., fo a full ccl as shown n Fg. 8.7, f = /360 = 360/360 =. Thfo fo any ssu do, = a h bounday, h wa nflux can b xssd as: B 8.5 Th goous soluon fo was dvlod hough Lalac ansfoms, u o Y u o J u du u 8.6

12 Cha 8 - Tansn Lamna Flow of Homognous Fluds wh h ngal mus b solvd numcally. In acc,, s snd n abula o gahcal fom. Fgu 8.8 s an xaml fo boundd and nfn adal aufs. Fgu 8.8 mnsonlss wa nflux, consan mnal ssu cas, adal flow I s ossbl o xnd hs hoy o calcula h cumulav wa nflux cosondng o a connuous ssu dcln a h svo/auf nfac. To do so, h ssu hsoy s dvdd no a ss of dsc ssu ss. Fo ach ssu do,, h cosondng wa nflux can b calculad usng h suoson of h saa nfluxs wh sc o m o gv h cumulav wa nflux. Fgu 8.9 llusas h ocdu fo aoxmang h connuous ssu dcln. Fo xaml, h cumulav flud oducd a m,, by h ssu do s: Lkws, fo h nx s, B o 8.7 B

13 ssu Cha 8 - Tansn Lamna Flow of Homognous Fluds o m Fgu 8.9 vlomn of dsc ssu ss fo connuous svo/auf bounday ssu. Summng h ms, n B j j 8.9 j0 o f h ssu laaus a akn as nfnsmally small, Examl B d mn h wa nflux dung h fs ya fo h followng svo/auf sysm. = 0.09 c = 6x0-6 s - A = 6 acs = 0.5 c k = 75 md A a = 50,000 acs h = 9. f = 80 dg 8.3

14 Cha 8 - Tansn Lamna Flow of Homognous Fluds b Faul Soluon Calcula wa nflux consan B.9hc b f 455 bbl / s 8.4 Calcula k c b.643 { days} 8. Pssu hsoy Tm od P, s P, s, s Calcula fom gah o abls Tm od Tm,days, s

15 Cha 8 - Tansn Lamna Flow of Homognous Fluds Calcula Tm od Summaon m bbls 0.50 =, = 6, = 89, = 43,