C-Curves. An alternative to the use of hyperbolic decline curves S E R A F I M. Prepared by: Serafim Ltd. P. +44 (0)
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1 An ltntiv to th us of hypolic dclin cuvs Ppd y: Sfim Ltd S E R A F I M info@sfimltd.com P. +44 (
2 Contnts Contnts... i Intoduction... Initil ssumptions... Solving fo cumultiv... 3 i
3 Intoduction In svoi ngining, it is oftn usful to mnipult o ct ppoximt poduction pofils. To listic, ths pofils should honou th totl mount of covl svs nd thi shps should mtch ppoximtly whtv is typicl fo th typ of svoi considd. Fo filds showing xponntil dclin, it is stight-fowd to constuct sonl dclin cuv, using th ltionship Expctd dclin t = (Poduction t / (Rmining tchniclly covl svs Fo filds showing hypolic dclin, it is much mo difficult to constuct sonl ltionship. Th son fo this is tht, in going fom xponntil to hypolic dclin, th implicit stong ltionship twn pofils nd covl svs hs n lost. An ltntiv xtnsion of xponntil dclin is givn low, clld th C-cuv. Divd duing wok on th Al fild (hvy, with undlying wt, in high pmility tuidit snd, this is sd on xtnding th cumultiv poduction qution, th thn th poduction ts qution, nd yilds th following ltionship = R... wh + liquid R = cummultiv poduction R = tchniclly covl svs liquid = cummultiv liquid poduction = constnt (0.23 fo Al Extm South = constnt (5.62 fo Al Extm South Initil ssumptions Th justifiction fo xponntil dclin is usully quotd s th mpiicl osvtion tht dq q wh = q = poduction t t = tim = constnt. This is thn xtndd to hypolic dclin in th fom
4 dq q =. q Howv, xmintion of th physics of simpl systms tht xhiit xponntil dclin (such s th dcy of dioctiv pticls, o poduction of gs fild und pu dpltion suggsts tht th fundmntl div is tht th dcy/poduction t is popotionl to th mining popultion/svs. In ou cs, this would d wh = ( R = cummultiv poduction R = ultimt covy Th C-cuv mthod is to xtnd this ltionship to th mo gnl fom d = ( + ( ( R R β o, in dimnsionlss fom d = ( + β wh = / R nd x is ny msu of fild ging vy liquid / R Th plot low illustts how this ltionship is sufficintly gnl to giv good mtch to typicl individul wll dclins in Al, s pdictd in svoi simultion. 2
5 Fitting Eclips pdictions of individul wll dclins (SME P0 clips sults P50 clips sults P90 clips sults C-cuv - =0.263; = 8 C-cuv - =0.08; =20 Hypolic with fixd svs - = 3.7 d/ Th quivlnt ltionship in hypolic dclin is d = β It cn sn fom th plot tht hypolic dclin dos not giv good fit, t lst not with th sm figu fo ultimt svs. On Al, it ws found tht hypolic dclin gv sonl fits only with high xponnts (i.. clos to hmonic dclin nd nomously high svs figus (.g. 00 million ls fo Al Extm South, compd to STOIIP of 320 million ls. Th ky id hind this ppoch is tht it is th d/ ltionship tht mtts fo cting lif-of-wll o lif-of-fild poduction pofils. Th xct fom of th ltionship chosn dos not mtt much, poviding it is sufficintly gnl to fit th shp of dclin s osvd in lity o s pdictd in Eclips. Th C-cuv ltionship ws chosn so s to sily solvl to yild fomul tht cn sily usd nd mnipultd. Solving fo cumultiv Stting fom th initil qution d = ( + β (A th vils cn split s follows 3
6 ( + β d = (B Intgting oth sids givs ln + β = ( x + α (C whα is constnt. [ Poof of intgtion of lft hnd sid d d ln + β = d d ln β + =. ] n ( + β..( ( + β Solving qution (C fo givs = ( x +α β = ( + β 2 n Th usul oundy conditions includ poduction stts with dy i.. d/ = whn = At th stt of poduction (i.. whn = x = 0 Ths oundy conditions llow us to xpss th α nd β in tms of oth vils, s follows Condition ( implis (fom qution (A - = -( + β i.. β = Applying condition ( to qution (B givs ln + ( = (0 +α i.. 4
7 α = 0 Applying ths vlus of α nd β to qution (C givs = x + Chnging fom nd x to R, liquid nd givs = R... + liquid R Intstingly, it ws found tht th, x fomultion could lso usd in th diffnt wy. If x ws tkn to th num of wlls dilld on th fild, nd to th fction of movl mining t th nd of fild lif ft th x wlls hd n poducd to vy high wt-cuts, thn d = β 2 ws found to giv vy good mtch to st of simulto uns with vying nums of wlls, s is illusttd in th two plots low. AXS - Effcts of wll dnsity on covy Ultimt covy in 2020 (mm UR Clition to simultion Vliion to dditionl simultion uns Additionl c p wll Additionl covy p 000 ft (mm Wll footg (000 ft 5
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