A physical basis for Hubbert s decline from the midpoint empirical model of oil production

Transcription

1 Enrgy and Sustainability 377 A physical basis for Hubbrt s dclin from th midpoint mpirical modl of oil production R. J. Winr 1 & D. M. Abrams 2 1 Rsarch Corporation, USA 2 Dpartmnt of Earth, Atmosphric, and Plantary Scinc, MIT, USA Abstract Just ovr fifty yars ago, Hubbrt corrctly prdictd that US oil production in th lowr-48 stats would soon pak, dspit substantially rising production at th tim. Hubbrt basd his prdiction on th mpirical obsrvation that th production rat at oil rsrvoirs typically follows a bll-shapd curv that paks nar th midpoint of production. W prsnt a highly simplifid physical modl of oil production that givs insight into Hubbrt s succss. This toy modl rlats th total ara of activ wlls to futur production and allows for svral intrsting analytical conclusions. W show analytically, contrary to intuition but consistnt with historical data, th numbr of activ wlls at any givn fild must pak aftr th production rat paks. Sinc oil production is prssurdrivn, th toy modl trats an oil rsrvoir as a sald containr filld with liquid oil and highly prssurizd gas. A pip xtnds into th containr with a cross-sctional ara, which rprsnts th total ara of wlls and changs continuously ovr tim. As gas xpands, it forcs liquid out of th containr through th pip but also drops in prssur, vntually limiting production. W assum th flow obys Brnoulli s principl and th gas xpands isothrmally, which lads to a nonlinar ordinary diffrntial quation for th vlocity of fluid xiting th containr. For most rasonabl choics of th ara function, th diffrntial quation rquirs a numrical solution. Howvr, it is possibl to spcify a plausibl ara function for which th modl yilds an analytical xprssion for th production rat that corrsponds to th bll-shapd curv usd by Hubbrt to prdict th pak in US oil production. Kywords: Hubbrt s pak, oil production, logistic growth, prssur-drivn flow. WIT Transactions on Ecology and th Environmnt, Vol 105, 2007 WIT Prss ISSN (on-lin) doi: /esus070371

2 378 Enrgy and Sustainability 1 Introduction In 1956, Hubbrt corrctly prdictd US oil production in th lowr-48 stats would pak around 1970 [1]. Many oil rcovry xprts grtd Hubbrt s prdiction with scpticism [2], sinc it cam at a tim whn oil production in th lowr-48 stats was rising substantially. Hubbrt basd his prdiction on th simpl mpirical obsrvation that th pak production rat at oilfilds (.g. in units of barrls pr day), or mor broadly in oil producing rgions, typically occurs at about th midpoint in production, i.. at th tim whn half th rcovrabl oil has bn producd. Hubbrt obsrvd that typically a sigmoidshapd function is a good approximation to data for th total volum of oil rcovrd from an oil rsrvoir. Th solution to th logistic diffrntial quation is an xampl of such a function. Th drivativ of this function with rspct to tim thn corrsponds to th oil production rat and is a bll-shapd curv. Thus, by fitting data for th oil production rat in th lowr-48 stats to a function that has th form of th drivativ of th solution to th logistic diffrntial quation, Hubbrt was abl to xtrapolat th curv and corrctly prdict its pak would occur around In this papr, w dvlop a highly simplifid physical modl of oil production, in ffct a toy modl, which lucidats undrlying physical aspcts of oil production and hlps xplain Hubbrt s succss. Th toy modl rlats th total cross-sctional ara of activ wlls to futur production. W show analytically th modl is quivalnt to th logistic diffrntial quation undr plausibl assumptions about how th ara of producing wlls typically varis ovr tim. W also show analytically, contrary to intuition, th numbr of activ wlls at any givn fild must pak aftr th production rat paks. W confirm this countr-intuitiv consqunc of th modl with historical data from major oilfilds on four continnts. 2 Logistic growth applid to oil production Hubbrt pionrd th ida of using logistic growth to modl oil production [1 3]. Th logistic growth function satisfis th logistic diffrntial quation Q = r(1 Q/ Qtot ) Q, whr Q is th quantity that is growing, Q is th drivativ of Q with rspct to tim, r is th initial rat of growth, and Q tot is th valu to which Q is asymptotically growing. Logistic growth dscribs any growth procss in which th pr capita growth rat, Q / Q, dcrass linarly as Q incrass. In th cas of oil production, Q rprsnts th cumulativ oil producd (.g. in barrls), Q rprsnts th production rat (.g. in barrls pr day), and Q tot rprsnts th total rcovrabl oil that ultimatly will b producd from a rsrvoir or, mor broadly, from an oil producing rgion. Q = Q tot /(1 + xp(r(t m t))) is th solution to logistic diffrntial quation, whr t m is th midpoint tim (i.. th tim at which Q has grown to Q tot /2). Th WIT Transactions on Ecology and th Environmnt, Vol 105, 2007 WIT Prss ISSN (on-lin)

3 2 drivativ of this function is givn by Q = rqtot xp( r( tm t)) /(1 + xp( r( tm t))), which is a bll-shapd function that paks at th midpoint tim t m. In 1956, Hubbrt fit data for th U.S. oil production rat in th lowr-48 stats with a curv having th functional form of Q and corrctly xtrapolatd th production rat to its pak in 1970 [1]. A ky implication of Hubbrt s us of logistic growth to mpirically modl oil production is that th 1970 pak occurrd whn about half th total rcovrabl oil in th lowr-48 stats had bn producd. In othr words, according to th logistic growth modl, th pak in th oil production rat signifis th midpoint of production. Is thr an undrlying physical justification for this dclin from th midpoint modl byond th mpirical obsrvation that production in many oilfilds and rgions appars to oby logistic growth? In th nxt sction w dvlop a highly simplifid modl that offrs a physical basis for th obsrvd dclin from th midpoint bhaviour. 3 Th toy modl Our toy modl is intndd to captur ssntial physical aspcts of oil rcovry, which is a prssur-drivn procss ordinarily limitd by a drop in prssur and infiltration by impuritis such as watr or gas. W trat an oil rsrvoir as a sald containr filld with oil (liquid ptrolum), which is assumd to b incomprssibl, and an idal gas at prssur P g (t) and volum V g (t). W assum th initial prssur of th gas P g 0 is much gratr than atmosphric prssur P atm. A pip, which has a small volum compard to th volum of oil, xtnds into th containr with a cross-sctional ara A(t), which rprsnts th total ara of all wlls and, for simplicity, is assumd to b a continuous function of tim. Expanding gas forcs liquid oil out of th containr through th pip. W assum th flow of oil obys Brnoulli s principl, v 2 = 2(P g (P atm + ρgh))/ρ, whr v(t) is th vlocity of xiting liquid oil, ρ is th dnsity of oil, and h is th hight of th column of oil in th pip. W also assum th gas xpands slowly and thus isothrmally, which implis P g0 V g 0 = P g V g = P g (V g0 + V ), whr V (t) is th volum of oil that has xitd th pip (i.. th oil producd). Combining ths quations and rarranging givs V = * P atm P g 0 V g 0 + ρv 2 /2 V, (1) g 0 * whr P atm = P atm + ρgh. Diffrntiating qn. (1) yilds th following xprssion for th volum pr unit tim of liquid oil xiting th pip: ρp V vv V =, (2) ( P /2) * 2 2 atm + ρv Enrgy and Sustainability 379 WIT Transactions on Ecology and th Environmnt, Vol 105, 2007 WIT Prss ISSN (on-lin)

4 380 Enrgy and Sustainability whr a dot indicats diffrntiation with rspct to tim. Combining qn. (2) with consrvation of mass, which rquirs that V = Av, lads to a nonlinar ordinary diffrntial quation for th vlocity: * 2 2 AP ( atm + ρv /2) v =. (3) ρpv Equations (1-3) constitut th toy modl. In a ral oilfild, th total ara of wlls and th wllhad prssurs ar th critical dtrminants of th production rat. Similarly, in th toy modl on must spcify th initial prssur and volum of gas and th tim-varying ara A to solv qn. (3) for th vlocity v, which thn can b substitutd into qn. (1) and qn. (2) in ordr to find th volum V and flow rat V of xiting oil. Th prssur that drivs th flow of oil at a ral oilfild is oftn du to a varity of fluids (.g. watr and gas). Th assumption in th toy modl of a singl xpanding gas driving th flow is a simplification intndd to approximat th prssur-drivn aspct of primary oil rcovry. For most plausibl choics of A, th toy modl nds to b solvd numrically. Howvr, it is possibl to solv th modl analytically by making th ansatz v = v m xp(r(t m t)/2), whr v m is th vlocity of oil xiting th pip at th midpoint tim. Thn qn. (3) is solvd if PV ρrv At () = (2 ) r( tm t)/2 m * 2 r( tm t) 2 Patm + ρvm. (4) Substituting th ansatz for v into qn. (2) yilds PV r V t. (5) r( tm t) () = * 2 r( tm t) 2 (2 Patm / ρvm + ) V as givn by qn. (5) has th sam functional form as th drivativ of th solution to th logistic diffrntial quation, which is th bll-shapd curv that Hubbrt usd to prdict U.S. pak oil production. Compar V in qn. (5) to th xprssion for Q in th prvious sction to s that thir functional forms ar th sam. Slf-consistncy rquirs that consrvation of mass V = Av is still satisfid and this is asily vrifid. Altrnativly, on can substitut th ansatz into qn. (1) and diffrntiat to arriv at qn. (2). Figur 1 shows th rsulting gnric curvs for A and V whn th sam valu of paramtrs in qn. (4) and qn. (5) ar usd. Th functions A and V ar similar bll-shapd curvs with th pak for A occurring aftr th pak for V. In ral oilfilds, th numbr of activ wlls, to which A is roughly proportional, incrass as production ramps up and typically rachs a platau as WIT Transactions on Ecology and th Environmnt, Vol 105, 2007 WIT Prss ISSN (on-lin)

5 Enrgy and Sustainability 381 Figur 1: A plot of production rat V (solid curv) and ara of activ wlls A (dashd curv) vrsus tim for typical paramtrs, showing that th two functions ar bll-shapd and A paks aftr V. a fild maturs. Howvr, th conomics of diminishing rturns dictats that th numbr of activ wlls must vntually dcras to zro as production tails off and wlls with low production ar shut down. Thus, a bll-shapd curv for th function A is plausibl. Morovr, a considration of historical oil production data also maks it plausibl that th pak for th numbr of wlls occurs aftr th pak for th production rat in ral oilfilds (i.. A paks aftr V ). W hav compild historical oil production data from issus of th Oil & Gas Journal dating back to 1973 for a larg numbr of oilfilds around th world [4]. Such data consistntly indicats th numbr of activ wlls paks aftr th production rat paks, as can b sn in fig. 2, which shows th production rat and numbr of activ wlls for oilfilds on four continnts. Indd, it turns out th toy modl prdicts this countr-intuitiv rsult for any bll-shapd A. This prdiction follows from consrvation of mass, V = Av, and th simpl physical assumption in th toy modl that th vlocity of oil xiting a wll is monotonically dcrasing ovr tim as oilfild prssur drops, i.. v < 0. At Hubbrt s pak, WIT Transactions on Ecology and th Environmnt, Vol 105, 2007 WIT Prss ISSN (on-lin)

6 382 Enrgy and Sustainability which occurs whn V = Av + Av = 0, A = Av / v which implis A > 0 (sinc A and v ar intrinsically positiv) and thrfor A has not yt pakd. This gnral rsult, basd on simpl physical rasoning, xplains th data in fig. 2 and othr historical data for which th numbr of activ wlls is incrasing vn as th production rat is dclining. Figur 2: Th oil production rat (squars) and th numbr of activ wlls (triangls) vrsus tim for oilfilds from four continnts. 4 Conclusion W hav dvlopd a toy modl, basd on ssntial physical aspcts of oil production, which hlps xplain Hubbrt s pak. W hav shown analytically, undr plausibl assumptions about how th cross-sctional ara of activ wlls at ral oilfilds typically varis ovr tim, th modl is quivalnt to th logistic diffrntial quation. W also hav shown analytically, contrary to intuition, th numbr of activ wlls at an oilfild always paks aftr th production rat paks. W prsntd historical data from major oilfilds on four continnts that WIT Transactions on Ecology and th Environmnt, Vol 105, 2007 WIT Prss ISSN (on-lin)

7 Enrgy and Sustainability 383 confirm this countr-intuitiv consqunc of th modl. Thus, our highly simplifid physical modl quit naturally givs ris to Hubbrt s pak, i.. Hubbrt s dclin from th midpoint mpirical modl of oil production. Acknowldgmnts Th authors would lik to thank D. d Graaf for suggsting this approach to modlling oil production, Stv Strogatz and Michal Boardman for usful consultations, and Rogr Blanchard for suggsting a possibl sourc of data. Rfrncs [1] Hubbrt, M.K. Nuclar nrgy and fossil fuls. Am. Pt. Inst. Div. of Prod. Spring Mting, Publication No. 95, Shll Dvlopmnt Company: Houston, pp. 1-40, [2] Dffys K.S., Hubbrt s Pak: th Impnding World Oil Shortag, Princton Univrsity Prss: Princton, [3] Hubbrt, M.K., Th world s volving nrgy systm. Am. J. Phys. 49, pp , [4] Abrams, D.M. & Winr, R.J., in prparation. WIT Transactions on Ecology and th Environmnt, Vol 105, 2007 WIT Prss ISSN (on-lin)