Decline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.

Transcription

1 Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing. Mainly usd baus hy ar asy o s u and o us in h fild. Thy ar no basd on any of h hysis of h flow of oil and gas hrough h rok formaions, mirial in naur. Th mos ommon forms ar daily flow ras vs. h monh. War and gas ras ar ommonly lod along wih h oil ra, or GOR and WOR. Cumulaiv roduion vs. h monhs is also vry ommon, boh oil and war an b lod. Ths los ar lod boh on linar los and smi-log los wih h on h log sal. Exonnial dlin (onsan fraional dlin) Harmoni dlin, and Hyrboli dlin. d d b d b & d mirial onsans basd on roduion daa d= xonnial dlin d= harmoni dlin <d< hyrboli dlin Exonnial Dlin Th rlaiv dlin ra and roduion ra dlin uaions for h xonnial dlin modl an b drivd from volumri rsrvoir modl. Cumulaiv roduion xrssion is obaind by ingraing h roduion ra dlin uaion. Rlaiv Dlin Ra Considr an oil wll drilld in a volumri oil rsrvoir. Suos h wll s roduion ra sars o dlin whn a riial (lows rmissibl) boom hol rssur is rahd. Undr h sudo-sady sa flow ondiion, h roduion ra a a givn dlin im an b xrssd as:

2 kh( ).472r 4.2B ln rw s (2) whr = avrag rsrvoir rssur a dlin im, = h riial boom hol rssur mainaind during h roduion dlin. Th umulaiv oil roduion of h wll afr h roduion dlin im an b xrssd as: kh( ). 472r 4.2Boln rw d s (3) Th umulaiv oil roduion afr h roduion dlin uon dlin im an also b valuad basd on h oal rsrvoir omrssibiliy: B i (4) o whr = oal rsrvoir omrssibiliy, i = iniial oil in la in h wll drainag ara, = avrag rsrvoir rssur a dlin im zro. Subsiuing E (3) ino E (4) yilds: kh( i d B. 472r o 4.2Boln s r w ) (5) Taking drivaiv on boh sids of his uaion wih rs o im givs h diffrnial uaion for rsrvoir rssur:

3 kh(.472r 4.2ln rw ) s i d d (6) Sin h lf-hand-sid of his uaion is and E (2) givs E (6) boms d d or h rlaiv dlin ra uaion of whr Produion Ra Dlin kh.472r 4.2B ln rw.472r 4.2 iln rw kh d b d b Euaion (6) an b xrssd as: kh.472r 4.2 i ln rw b ) d d d d s s By saraion of variabls, E (8.) an b ingrad s d d (7) (8) (9). () ( () bd d ( o yild an uaion for rsrvoir rssur dlin: ) b (2) (3) Subsiuing E (3) ino E (2) givs wll roduion ra dlin uaion:

4 or kh( ).472r 4.2Boln rw b ) B s b (4) i b ( (5) o whih is h xonnial dlin modl ommonly usd for roduion dlin analysis of soluion-gas-driv rsrvoirs. In rai, h following form of E (5) is usd: whr i is h roduion ra a =. I an b shown ha onsan for xonnial dlin. Cumulaiv Produion b i (6) 2 3 n b.... Tha is, h fraional dlin is 2 n Ingraion of E (6) ovr im givs an xrssion for h umulaiv oil roduion sin dlin of i.., Sin i b, E (8) boms b d i b d (7) i b. (8) 8 Drminaion of Dlin Ra. (9) b i Th onsan b is alld h oninuous dlin ra. Is valu an b drmind from roduion hisory daa. If roduion ra and im daa ar availabl, h b- valu an b obaind basd on h slo of h sraigh lin on a smi-log lo. In fa, aking logarihm of E (8.6) givs: lni b ln (2)

5 whih imlis ha h daa should form a sraigh lin wih a slo of -b on h log() vrsus lo, if xonnial dlin is h righ modl. Piking u any wo oins, (, ) and ( 2, 2 ), on h sraigh lin will allow analyial drminaion of b-valu baus and giv ln ln lni b (2) 2 lni b2 (22) b ln. (23) 2 2 If roduion ra and umulaiv roduion daa ar availabl, h b-valu an b obaind basd on h slo of h sraigh lin on an vrsus lo. In fa, rarranging E (9) yilds: (24) i b Piking u any wo oins, (, ) and ( 2, 2 ), on h sraigh lin will allow analyial drminaion of b-valu baus and giv i b (25) i b (26) 2 2 b. (27) 2 2 Dnding on h uni of im, h b an hav diffrn unis suh as monh - and yar -. Th following rlaion an b drivd: b 2b 365b. (28) a m whr b a, b m, and b d ar annual, monhly, and daily dlin ras. d

6 Effiv Dlin Ra Baus h xonnial funion is no asy o us in hand alulaions, radiionally h ffiv dlin ra has bn usd. Sin x x for small x- valus basd on Taylor s xansion, b b holds ru for small valus of b. Th b is subsiud by b ', h ffiv dlin ra, in fild aliaions. Thus E (6) boms Again, i an b shown ha i b' (29) 2 3 n... b' 2 Dnding on h uni of im, h b ' an hav diffrn unis suh as monh - and yar -. Th following rlaion an b drivd: n b' b' 2 b' 365 a m d.. (3) whr b' a, b' m, and b' d ar annual, monhly, and daily ffiv dlin ras. Examl Problm Givn ha a wll has dlind from sb/day o 96 sb/day during a on-monh riod, us h xonnial dlin modl o rform h following ass: Soluion: a) Prdi h roduion ra afr mor monhs b) Calula h amoun of oil rodud during h firs yar ) Proj h yarly roduion for h wll for h nx 5 yars. a) Produion ra afr mor monhs: b m ln m m m m ln.482/monh 96

7 Ra a nd of on yar bm.4822 m m 6.27 sb/day If h ffiv dlin ra b is usd, m m 96 b ' m.4/monh. From b' y m 2 b'.4 m 2 on gs b' y.3875/yar Ra a nd of on yar b' sb/day y b) Th amoun of oil rodud during h firs yar: b y /yar 6.27, ,858 b y sb or b d ln day ,858 sb,

8 ) Yarly roduion for h nx 5 yars: ,68 sb 6 365, 2 b.4822 (2) 2 i sb/day ,834 sb , 3 b.4822 (3) 3 i 23. sb/day ,639 sb , 4 b.4822 (4) 4 i 4.9 sb/day ,6 sb 4 365, 5

9 In summary, Go do, Yar Ra a End of Yar (sb/day) Yarly Produion (sb) - 28,858 7,68,834 6,639 4,6 68,73 Cli #3 Am Souhas San Andrs, Pondrosa Prolum

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