History Matching in Two-Phase

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THE SPE I MAGE LI BRARY SPE 8250

Hi st ory Matchi ng i n Two- PhasePet r ol eum Reser voi r s

8250

Wat son, A. T. , * SPE,Cal i f or ni a I nst . of Technol ogySei nf el d, J . H. , SPE,Cal i f or ni a I nst . of Technol ogyGaval as, G. R. , SPE,Cal i f or ni a I nst . of Technol ogyWoo, P. T. , SPE,Chevr on Oi l Fi el d Resear ch Co.

DECEMBER 1980

Abst r act

An aut omat i c hi st or y- mat chi ng al gor i t hm based onan opt i mal cont r ol appr oach has been f ormul at ed f orj oi nt est i mat i on of spat i al l y var yi ng per meabi l i t y andpor osi t y and coef f i ci ent s of r el at i ve per meabi l i t yf unct i ons i n t wo- phase r eser voi r s. The al gor i t hm usespr essure and pr oduct i on r at e dat a si mul t aneousl y.The per f or mance of t he al gor i t hm f or t hewat er f l oodi ng of one- and t wo- di mensi onal hypot het i calr eser voi r s i s exami ned, and pr oper t i es associ at edwi t h t he par amet er est i mat i on pr obl em ar e di scussed.

I nt r oducti on

Ther e has been consi der abl e i nt er est i n t hedevel opment of aut omat i c hi st ory- mat chi ngal gor i t hms. Most of t he publ i shed wor k t o dat e onaut omat i c hi st ory matchi ng has been devoted t osi ngl e- phase reservoi r s i n whi ch t he unknownpar amet er s t o be est i mat ed ar e of t en the reservoi rpor osi t y ( or st or age) and absol ut e per meabi l i t y ( ortr ansmi ssi bi l i t y) . I n t he si ngl e- phase pr obl em, t heobj ect i ve f unct i on usual l y consi st s of t he devi at i ons

bet ween t he pr edi ct ed and measur ed r eservoi rpr essur es at t he wel l s. Par amet er est i mat i on, orhi st or y mat chi ng, i n mul t i phase r eser voi r s i sf undament al l y mor e di f f i cul t t han i n si ngl e- phaser eser voi r s. The mul t i phase equat i ons ar e nonl i near ,and i n addi t i on t o t he por osi t y and absol ut eper meabi l i t y, t he r el at i ve per meabi l i t i es of each phasemay be unknown and subj ect t o est i mat i on.Measur ement s of t he r el at i ve r at es of f l ow of oi l ,wat er , and gas at t he wel l s al so may be avai l abl e f ort he obj ect i ve f unct i on.

The aspect of t he r eser voi r hi st or y- mat chi ng

pr obl em t hat di st i ngui shes i t f r om ot her par amet er


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THE SPE I MAGE LI BRARY SPE 8250

est i mat i on pr obl ems i n sci ence and engi neer i ng i s t hel ar ge di mensi onal i t y of bot h t he syst em st at e and t heunknown par amet er s. As a r esul t of t hi s l ar gedi mensi onal i t y, comput at i onal ef f i ci ency becomes apr i me consi der at i on i n t he i mpl ement at i on of anaut omat i c hi st or y- mat chi ng met hod. I n al l paramet erest i mat i on met hods, a t r ade- of f exi st s bet ween t heamount of comput at i on per f ormed per i t erat i on and

t he speed of convergence of t he met hod. Ani mpor t ant savi ng i n comput i ng t i me was r eal i zed i nsi ngl e- phase aut omat i c hi st or y mat chi ng t hr ough t hei nt r oduct i on of opt i mal cont r ol t heor y as a met hodf or cal cul at i ng t he gr adi ent of t he obj ect i ve f unct i onwi t h r espect t o t he unknown parameters. Thi st echni que cur r ent l y i s l i mi t ed t o f i r st - or der gr adi entmet hods. Fi r st - or der gradi ent met hods gener al l yconver ge mor e sl owl y t han t hose of hi gher order .Never t hel ess, t he amount of comput at i on r equi r edper i t er at i on i s si gni f i cant l y l ess than t hat r equi r edf or hi gher - or der opt i mi zat i on met hods; t hus,

f i r st - or der met hods ar e at t r act i ve f or aut omat i c hi st or ymat chi ng. The opt i mal cont r ol al gor i t hm f oraut omat i c hi st ory matchi ng has been shown t opr oduce excel l ent r esul t s when appl i ed t o f i el dpr obl ems. Ther ef or e, t he f i r st appr oach t o t hedevel opment of a general aut omat i c hi st or y- matchi ngal gor i t hm f or mul t i phase r eser voi r s woul dseem t o pr oceed t hr ough the devel opment of anopt i mal cont r ol appr oach f or cal cul at i ng t he gr adi entof t he obj ect i ve f unct i on wi t h r espect t o t hepar amet er s f or use i n a f i r st - or der met hod.

I n t hi s wor k an al gor i t hm based on opt i malcont r ol t heor y i s pr esent ed f or t he est i mat i on ofabsol ut e per meabi l i t y, por osi t y, and r el at i veper meabi l i t y i n t wo- phase reser voi r s usi ng pr essur e andpr oduct i on r at e dat a. Det ai l ed r esul t s ar e pr esent edf or t est pr obl ems f or hypot het i cal one- andt wo- di mensi onal wat er f l oodi ng si t uat i ons.

SPEJ

P. 521


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History Matching in Two-Phase

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