B027 AUTOMATED DETERMINATION OF AQUIFER PROPERTIES FROM FIELD PRODUCTION DATA

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1 B027 AUTOMATED DETERMINATION OF AQUIFER PROPERTIES FROM FIELD PRODUCTION DATA Georg M. Mermer, Johaes Pchelbauer 2, Zolá E. Heema Deparme of Peroleum Egeerg, Uversy of Leobe, Ausra 2 RAG, Ausra Absrac Ths paper preses a fully auomaed hsory machg (HM) mehod for average feld ad rego pressures ad for aqufer behavors. Ths meas ha already wh a sgle smulao ru he average pressure ad waer flux s reproduced. The aqufer model, aalycal or grdded wll o be desged as usual a he begg bu a he ed of he HM process. The applcably of he mehod was exesvely esed o may feld cases. Pracce showed ha usual hsory machg me could be reduced by a facor of up o 5. The paper coas a dealed descrpo of he mehod ad suggesos for he mplemeao ay smulao sofware. By preseg hree feld cases he sregh ad applcably of he cocep for black ol ad composoal models wll be show. Iroduco Moder reservor characerzao echques delver more ad more dealed descrpos of he hydrocarbo reservors, bu do o deal wh he aqufers. However, may ol ad gas reservors have a assocaed aqufer, formg a egraed hydrodyamc sysem for whch he pressure decle wll be srogly flueced by he aqufer. I a coveoal hsory machg (HM) process he frs sep mus be o mach he reservor pressure, whch meas o deerme he aqufer properes. Ths ca be a edous work, bu s also he wrog approach. The aqufer parameers are hghly ucera ad should o have a fluece o screeg he geologcal realzaos or o ug he reservor parameers. Ths paper preses a dffere approach. The aqufer model wll be deermed ad roduced a he ed of he HM, well kowg ha mos of he cases o uque soluo exss. The cocep ad workflow s as follows: The grd model s spl a producve area (PA) ad a aqufer grd. The PA s dvded a arbrary umber of volume regos. A arfcal boudary, whch ca be segmeed, s pu ousde he waer ol coac, preferably a he ouer blocks of he PA. Each boudary segme s relaed o oe volume rego. The well raes are defed erm of reservor volumes. Now waer wll be jeced/produced /from he boudary segme o mach he hsorcal rego pressures. The mehod for ha s called Targe Pressure Mehod (TPM). The deermao of he boudary raes ad her dsrbuo bewee he boudary blocks are o rval because he volume regos ca commucae o each oher. Beyod ha rase effecs mus be cosdered oo. Noe ha f he calculaed rego pressure s equal wh he hsorcal oe, he he overall reservor volume produco rae mus be also correc, per defo. Thus he hsory mach ca be sared mmedaely wh machg he ol (or gas) produco, waer cu, GOR, well pressures ad RFT daa. All furher reservor parameer chages fluece he waer flux, bu hs wll be adaped for every ru auomacally. If he HM s compleed, he based o he calculaed waer flux, he parameers of dffere aalycal aqufer models (Va Everdge-Hurs, Fekovch, ec.) are deermed for each boudary segme. Ths sep s smlar o he oe used maeral balace calculaos. Usg he parameers of he aalycal models, he grdded aqufer wll be desged (sze, permeably, effecve compressbly, ec.). Boh, grdded aqufer model ad bes aalycal model ca be used for predcos. 9h Europea Coferece o he Mahemacs of Ol Recovery Caes, Frace, 30 Augus - 2 Sepember 2004

2 2 The paper descrbes he workflow, he arge pressure mehod for predco of he boudary jeco raes, he deermao of he opmal aalycal aqufer model ad a mehod how o creae grdded aqufers. A he ed oe arfcal ad wo real feld examples wll be preseed. The paper s based o wo hess works, hose from Pchelbauer [] ad Mermer [2]. The reader ca fd more deals ad examples relaed o he opc hs documes. Aqufer ad Producve Area Doma Moder reservor smulaors [3] buld he grd model separaely for he reservor ad he aqufer domas, whch wll be merged o oe grd sysem. To avod ay cofuso we use he expresso producve area (PA) sead of reservor. The PA has a greaer exeso ha he hydrocarbo doma, coverg some par of he waer-sauraed formao oo. The orgal dea was o eable he lkage of a PA wh a seres of aqufer models ad vce versa, allowg o ru dffere combaos parallel ad o accelerae he HM process. Older commercal packages cosruc he grd for he whole hydrodyamc sysem oe sep. I such cases he PA grd ca be creaed by deacvag he aqufer grd blocks, ad he aqufer model by elmag he PA blocks. The oly commo par of he wo depede domas s a row of blocks ha lks hese domas ogeher. By defo hs borderle s formed by he ouermos row of PA grd blocks ad s called arfcal boudary. Whe he aqufer ad he PA domas are merged o oe full feld model such a boudary acs as erface for fluxes ad pressures bewee he wo grds. I s mpora o oe ha he arfcal boudary s locaed bewee a edge-waer drve aqufer ad he PA vercally ad o bewee he reservor ad a boom-waer drve aqufer horzoally. Ths boudary ca be also used o operae he domas depedely from each oher by applyg approprae boudary codos. If he waer efflux from he aqufer s kow would be possble o operaed oly he PA grd combao wh he boudary. To mask waer ecroachme waer mus be jeced o he boudary elemes. The jeced waer mus be equal o he amou of waer flowg across he boudary f a aqufer grd was prese. Ths wll resul a pressure performace correspodg o he full feld model. Bu s also possble o operae jus he aqufer grd. The produco of waer mus ake place from he boudary blocks. For HM purposes would be beefcal o have a mehod ha calculaes waer volumes o be jeced o he boudary requred o mach observed feld produco daa, especally feld pressure. Targe Pressure Mehod The PA self may coss of a arbrary umber of volume regos. The dffereao ca be made eher based o geologcal feaures lke fauls ad lhology or ca be doe jus arbrary. The arfcal boudary blocks are o par of ay volume rego. Furher, arfcal boudares do o have o be coues, hey ca be spl up o segmes. Every boudary segme s assged o a dsc volume rego bu hey do o have o border he volume regos hey belog o. The hsorcal average pressure of he volume regos, whch ca be deermed for he pore volume or for he hydrocarbo volume as well, shall be kow as fuco of he me. These average pressure fucos should be reproduced all HM rus as close as possble, herefore hey are called arge pressures. The volume rego o whch a arge pressure ad cosequely a arfcal boudary segme were assged, are called arge area. Noe ha s o ecessary o defe all volume regos as arge areas. The calculao scheme of waer flux requred o mach pressures of a arge area s called arge pressure mehod or shorly TPM. Now le us operae he grd model whou aqufer. To acheve a mach of smulaed ad he arge pressures s ecessary o jec waer o he arfcal boudary. To deerme he rgh waer jeco rae s o a rval ssue. Problems regardg praccal mplemeao are: ) Well boom hole pressures (.e. he perforao block pressures) ca be que dffere ad herefore he deermao of arge produco erms of reservor volume ad so he calculao of he equvale boudary jeco rae s ucera. 2) Compressbly of he arge area ca chage suddely aroud bubble po pressure.

3 3 3) Waer jeced o he boudary does o effec mmedaely he pressure of he hydrocarbo sauraed par. Ths me lag s relaed o he dsace ad he permeably. 4) The arge areas ca commucae wh each oher. 5) I s possble ha arge area ad correspodg boudary are o coeced drecly. 6) Due o he waer jeco he boudary block pressures may exceed he al pressure. The reaso for ha s he low permeably of he boudary area. 7) The jeco rae ca easly oscllae, makg he procedure sable. To make he explaao smple, a frs oe should assume he followg crcumsaces: ) The arge area ad he correspodg boudary segme are drecly coeced. 2) The boudary block volume s eglgble compared o he volume of he arge area. 3) The coducvy bewee he boudary ad he hydrocarbo volume of he arge area (waer sauraed blocks) s hgh. 4) The arge pressure s defed for referece me pos. The me ervals bewee wo referece pos, p, are cosderable loger as he me seps,, used durg he smulao rus. If codos o 3 are vald he rase perod for he arge area wll be shor. Ths meas f oe produces from he arge area ad jecs o he boudary wh cosa pressures he suao wll become seady sae ad he raes cosa afer a me perod s < p. Choosg a me sep s for whch m =, m =,2 he he boudary jeco rae for he acual me sep ca be p deermed assumg seady sae codos. Le be p he acual area pressure ad pressure a he ed of he me sep, he he frs guess for he boudary jeco rae s: p ˆ + he arge + + pˆ p + B wqw = C + Q. () The average produco rae Q + wll be calculaed for a me perod of acual reservor volume as usually, o well-by-well bass. The formao volume facor B w s calculaed for he pressure p ˆ j+. The overall compressbly of he arge area s a black ol formulao: C = ( c + M ) db m m w o g s g C = V V ac Sw So Sg Bwdp = = Bodp Bo dp Bgdp φ, φ φ, + + ) I s possble ha he me sep s oo large, whch meas ha he soluo does o coverge, ad herefore he codo s cao be sasfed. I hs case mus be cu m smaller me seps 9h Europea Coferece o he Mahemacs of Ol Recovery Caes, Frace, 30 Augus - 2 Sepember 2004 db B dr db, (2) where M s he rock compaco facor. The summao s doe over all blocks belogg o he arge area. C wll be calculaed for he pressure p ad p ˆ + as well. For p ˆ + he sauraos ca be ake explc a me or mplc. Expecg ha pressure wll chage from p o p ˆ + durg he me sep, he average compressbly wll be: + + ( C C ) C = (3) Now he soluo ca perform for he me sep usg IMPES or fully mplc scheme, resulg a average area pressure p +, whch wll o be equal o p ˆ +. If he dfferece s small ha he calculaed pressure ca be acceped ad he ex me sep ca perform, f o he me sep mus be repeaed usg a correced boudary jeco rae. The correco ca be calculaed easly from he pressure falure: q w = C + + pˆ p B w +. (4) The me sep ca be repeaed may mes bu expereces show ha mos cases o erao s ecessary, ad also dffcul suaos he soluo coverges maxmum wo seps. The procedure ca fal wo cases:

4 4 wh he cosequece ha o oly he las me sep, bu he las m me seps mus be repeaed wh he correced jeco rae. Noe ha produco ad boudary jeco raes are cosa durg he m small me seps. Naurally, he sage a he sar of he m me seps mus be sored. 2) I s also possble ha s > p, whch meas ha he rase perod s log compared o he referece erval. Theorecally such a case could be hadled based o he prcple of superposo, as roduced by Va Everdge ad Hurs. Ths meas ha a a gve me would be ecessary o modfy he boudary jeco raes over more p ervals backwards. Ths cao be praccal. Such a suao s a dcaor for a mproper seup of he producve area grd ad ca be correced easly by modfyg he grd model. Three possble mprovemes ca be cosdered. The frs s o move he waer-jecg boudary closer o he arge area. Aoher possbly s o revse he permeably of he arge area. Boh modfcaos wll shore he me lag bewee waer jeco a he boudary ad arrval of he pressure supporg effec he arge area. The hrd possbly s o reduce he sze of he arge area by cug he reservor o more volume regos. All hree possbles were successfully esed. The hrd possbly was appled o Example 3 preseed a he ed of he paper. The rae q w has o be dsrbued amog he blocks of he boudary such a way ha he dvdual boudary block pressures p coverge owards he average boudary pressure p. Ths s doe usg a kd of jecvy dex J vald for he boudary block, J = τ p p ) f p p > 0 else J = 0, (5) j ( j j where j s a o-boudary eghbor o ad τ j s he block par rasmssbly of wo blocks. p s a weghed average boudary pressure. The weghg facor ω ca be eher he pore volume of he boudary blocks or he commucao surface o he coeced rego. p = p ω ω (6) If more o-boudary blocks j border a boudary block he jecvy dces have o be summed up. If a o-boudary block j eghbors more ha oe boudary block he jecvy dex has o be reduced accordgly. Fally he waer jeco durg he me sep + for a sgle boudary block follow as, ( J qw ) q = J. (7) w Applyg hs mehodology allows o suppor he dffere volume regos of he producve area doma wh he correc waer flux a way ha he smulaed rego pressures mach he hsorc arge pressures. Ths s guaraeed wh he frs smulao ru already. Sce o uceraes regardg he aqufer rego have o be ake o accou could be mmedaely sared o mach he model o a well bass. For predco rus a aqufer model has o be deermed exhbg he same characerscs as he waer flux deermed by he TPM. Two possbles exs. The frs s o assg a aalycal aqufer model o he arfcal boudares ad he secod s o merge he producve area grd wh a approprae aqufer grd. Idefcao of he opmal aalycal aqufer model The arge pressure mehod calculaes he waer flux across he arfcal boudares ad he accordg boudary pressures as a fuco of me. Ths aalogues a maeral balace calculao ad ca be used a smlar maer o deerme he parameers of a aalycal aqufer model. Such a aalycal aqufer model wll reproduce he waer flux calculaed by he TPM. Thus he already hsory mached producve area grd model s sll vald. To deerme he bes fg aalycal aqufer model followg error fuco should be mmzed: N = [ ] 2 W ( ) W ( ) E = ω, (8) e

5 5 where W( ) s he waer flux from he TPM calculao, W e ( ) s he waer flux from oe of he avalable aalycal models, meoed below, ad ω s a weghg facor based o he elapsed smulao me. I expresses ha waer fluxes akg place a a laer me have more mporace ha early oes. Ths allows reducg rase effecs occurrg mos lkely a early mes. A procedure for deermg a bes fg aalycal aqufer model was developed. For pseudo-seady sae flow codos parameers descrbg he aalycal aqufer model roduced by Fekovch [4] wll be calculaed. For o-seady sae flow codos he parameers of he Va Everdge-Hurs [5] model, respecvely of s modfed verso publshed by Vog ad Wag [6] wll be deermed. For boh flow codos he fg procedure s doe a erave way. Parameer deermao for he Fekovch model For he Fekovch model wo parameers he producvy dex ad he maxmum ecroachable waer mus be deermed. A frs absolue mmum values are calculaed for hem. The calculao of he mmum producvy dex s based o a fe amou of ecroachable waer ad he mmum ecroachable waer s calculaed based o he assumpo of a fe producvy dex. Based o hese al values he waer flux fuco gve by he aalycal model s calculaed. The he cumulave error bewee waer flux of he TPM ad he aalycal model s calculaed. Aferwards he model parameers are creased. Aga waer flux ad error fuco are calculaed. If he error becomes less he parameers are creased aga, f he error becomes larger he parameers are decreased so ha he ew parameers are bewee he al ad he prevous opmzao sep. Ths s doe ul he ermao crero beg eher a maxmum umber of erao seps whou mproveme error or reachg a absolue error lm s reached. Parameer deermao for he Vog-Wag model For he Vog-Wag model he search procedure s dffere. For hs model hree parameers have o be deermed. They are he me coverso facor β, he dmesoless ouer aqufer radus r ed ad a aqufer relaed cosa, C. The umber of dmesoless ouer aqufer rad s lmed. Ths allows esg all of hem. For every radus a dmesoless me coverso facor ad aqufer relaed cosa have o be deermed. Aga he error fuco has o be mmzed. Afer esg all rad he oe wh he smalles error descrbes combao wh s aqufer relaed cosa ad me coverso facor he bes fg Vog-Wag model. The deermao of he me coverso facor ad he aqufer relaed cosa go had had. Sarg from a al β a array of equdsa β values s creaed. Based o hese β vales ad a aqufer relaed cosa of C= a array of waer fluxes wll be calculaed. For every ery he acual C ca be calculaed by C = ( W ) N N 2 ω We, AWe, S ω e, A, (9) = = where W, s he cumulave waer flux a he -h me sep of he aalycal Vog-Wag soluo ad e S e A W, s he cumulave waer flux from he TPM smulao ru. ω s a weghg facor akg o cosderao he elapsed smulao me. Regardg hese C values he real waer flux of he aalycal model ca be calculaed. Based o hs waer flux array he β leadg o he smalles error ca be deermed. If hs β s locaed a he lower ed of he array he β wll be decreased for he ex me sep. If s locaed a he upper ed of he array he β wll be creased. If he β showg he smalles error s locaed somewhere bewee he upper ad he lower ed, hs β wll be ake as a sarg po for he ex erao. Aga a array of β values wll be creaed, bu he cremes wll be smaller ha for he prevous erao sep. Ths cycle wll be performed ul he ermao crero beg eher a maxmum umber of erao seps whou mproveme error or reachg a absolue error lm s reached. 9h Europea Coferece o he Mahemacs of Ol Recovery Caes, Frace, 30 Augus - 2 Sepember 2004

6 6 Coverso of aalycal aqufer parameers o grdded aqufer parameers Eve a aalycal aqufer model, calculaed from he TPM, ca be used ogeher wh arfcal boudares for predco rus may be beefcal o use a grdded aqufer sead. Aalycal models are lmed hadlg sudde chages produco codos ha may lead o rase flow codos. Addoally grdded aqufer models ca be ued easly regoally, by smple chagg he block properes, o ge a beer mach of observed flow behavor. Therefore a procedure o cover aalycal aqufer parameers o grdded parameers was developed. The parameers of a aalycal aqufer model coa formao abou sze, permeably ad compressbly of he aqufer rego. To use hs formao for aqufer grd cosruco s ecessary o cover parameers descrbg a aalycal model o parameers of a grd model. Ths coverso s based o he assumpo ha he producve area s locaed he ceer of a radal symmerc aqufer. Addoally aqufer hckess, porosy ad flow agle mus be esmaed. The esmao ca be doe based o he producve area grd. Coverso of Vog-Wag parameers o grdded aqufer parameers Based o hese assumpos s a smple ask o cover Vog-Wag parameers o grdded aqufer parameers. The areal exe ca be calculaed based o he dmesoless ouer aqufer radus, r ed, ad he radus of he producve area, r w. Thus he ouer aqufer radus s gve by r e = rderw. The calculao of he permeably capacy hk s doe by rearragg he equao descrbg he aqufer relaed cosa C, for radal symmerc cases hk = ( βµ wc) θ, where µ w s he aqufer waer vscosy ad θ s he aqufer flow agle. For a full crcle θ = 2π. Coverso of Fekovch parameers o grdded aqufer parameers The Fekovch model s vald for arbrary aqufer shapes. Neverheless o be able o do a parameer coverso, was decded o assume a radal symmerc aqufer. All assumpos for coverg he Vog-Wag parameers are sll vald. Takg geomery o accou he ouer aqufer radus, r w, ca be smply calculaed from he maxmum ecroachable waer W e by, W e r e = + θc hφp r 2 w. (0) For a full crcleθ = 2π.The aqufer permeably ca be calculaed from he producvy dex ad he Dupu equao. Sce he Fekovch model descrbes pseudo-seady sae flow codos he radus r, where he average aqufer pressure s locaed mus be calculaed accordg o Pedrosa ad Azz [7]. Fally he aqufer permeably k ca be expressed as k J µ r e ( ) r r re l 2 2 r r r 2 r = w w θh 2 e w w e. () Examples The followg hree examples show he applcably of he proposed hsory machg workflow. The workflow for all hree examples was he same. Based o measured reservor pressures a smulao ru usg he arge pressure mehod was doe, whou ay aqufer. Waer ecroachme akes place oly hrough he arfcal boudary. Based o he calculaed waer flux daa a bes fg aalycal aqufer model was deermed. Afer valdag he aalycal model by a addoal smulao ru he aalycal aqufer parameers were covered o properes of a grdded aqufer. The resulg reservor model coag a grdded PA ad a grdded aqufer was valdaed. As wll be show all hree waer ecroachme possbles (TPM, aalycal aqufer ad grdded aqufer) yeld comparable pressure decles.

7 7 Example : Ths es example was doe usg a arfcal reservor havg e ol producg wells. Sarg from a model havg a aqufer ad a PA grd a smulao was ru o calculae he pressure decle. Ths pressure decle was used as arge pressure for subseque smulaos. I a ex sep a ru usg oly he PA grd was doe. As dcaed by Fgure 2 he mssg waer ecroachme leads o a oo fas pressure decle. Usg he TPM he measured pressures (smulaed pressures of he frs ru) could be mached very accuraely. The search procedure for a equvale aalycal aqufer model showed ha he former grdded aqufer exhbed o seady sae flow codos. The smulaed pressure decle of he ru usg he aalycal Vog-Wag aqufer ad he arfcal boudary mached he arge pressures very well. Coverg hese bes fg parameers lead o a descrpo of a grdded aqufer havg a dffere seup ha he al oe, used for calculag he hsory. Ths resuls from he radal symmerc assumpo used for he parameer coverso. Neverheless he calculaed pressure decle correspods o he arge values. The al grd model s show Fgure, he calculaed grd model Fgure 3. The smulaed pressure decles are dsplayed Fgures 2 ad 4. Example 2: For he fauled reservor model dsplayed Fgure 5 a composoal flud characerzao was chose. The reservor has 9 wells, producg ol for 25 years. Fgure 6 shows he measured ad he smulaed pressure decles calculaed by he arge pressure mehod ad calculaed by he bes fg aalycal aqufer ha s of Fekovch ype. Ths example clearly shows ha he proposed workflow does o deped o he used flud descrpo. Example 3: The feld operaed wh 60 wells sce 978 has a raher complcaed geology. The model s buld by layers where he op e layers are fracured. Sealg fauls dvde he producve area o 9 volume regos. Four segmeed, arfcal boudares are assged o he PA border. A large amou of waer-sauraed blocks ca be foud bewee hese boudares ad he ol sauraed reservor area. Therefore he four arge volume regos, where recorded pressure values are avalable do o border he arfcal boudares (Fgure 7). Fgure 8 shows he smulaed pressure decle usg he arge pressure mehod ad he bes fg aalycal aqufer model of he arge area he Wes. For he Eas ad Wes rego he bes fg aalycal model was foud o be of Vog-Wag ype. A Fekovch model descrbes he requred waer flux for he Norh ad Souh rego bes. The grdded aqufer model cosruced based o he calculaed waer flow daa s dsplayed Fgure 9 ad he smulaed pressure decle for he Wes rego Fgure 0. Eve he reservor shows such a complcaed srucure he proposed hsory machg workflow leads o reasoable resuls. Coclusos ) The paper preseed a ew ad praccal mehod for deermg properes of edge waer drve aqufers solely o produco daa. 2) The roduced workflow allows cocerag o he HM of he producve area. The aqufer area does o have o be cosdered. The waer flux of a edge aqufer wll be suppled auomacally ad correcly. 3) A vald aqufer model, eher aalycal or grdded, reproducg surveyed reservor pressures s foud wh a sgle smulao ru. These aqufer models ca be used for predco rus. 4) The preseed mehods ad proposed workflow lead o a bg crease me effcecy for hsory machg sudes. Nomeclaure B formao volume facor [rm 3.sm -3 ] C aqufer relaed cosa [m 3.bar - ] C compressbly [m 3.bar - ] J jecvy dex [m 3.bar] I umber of grd blocks [-] M rock compaco facor [bar - ] N umber of me seps [-] R s soluo gas rao [-] S saurao [-] V volume [m 3 ] W waer coe [m 3 ] c compressbly facor [bar - ] h hckess [m] k permeably [m 2 ] p pressure [bar] q jeco rae [m 3.day - ] r radus [m] me [day] β me coverso facor [-] φ porosy [-] µ vscosy [Pa.s] 9h Europea Coferece o he Mahemacs of Ol Recovery Caes, Frace, 30 Augus - 2 Sepember 2004

8 2 θ aqufer flow agle [rad] τ block par rasmssbly [m 3 ] ω weghg facor [-] Subscrps: A aalycal S smulaed D dmesoless g gas phase grd block dex j o p w s Superscrps: me sep eghborg grd block dex ol phase arge waerphase rase oal Refereces [] Pchelbauer, J.: A ew Mehod of Aqufer Machg Reservor Smulao, Ph.D. hess, Uversy of Leobe, Ausra (Feb. 2003). [2] Mermer G. M.: Ehaced Sraeges for Machg Aqufer Behavor, Maser hess, Uversy of Leobe, Ausra (Aug. 2003). [3] SURE ad SUREGrd Reservor Smulaor Maual, Verso 4., Heema Ol Techology, Leobe, Jue ad Aprl 999. [4] Fekovch, M.J.: A Smplfed Approach o Waer Iflux Calualos-Fe Aqufer Sysems, paper SPE 2603 preseed a he 44h Aual Fall Meeg, held Dever, Colorado, Sep. 28- Oc., 969. [5] Va Everdge, A.F. ad Hurs W.: The Applcao of he Laplace Trasformao o Flow Problems Reservors, Tras., AIME 86, (949) [6] Vog, J.P. ad Wag, B.: Accurae Formulas for Calculag he Waer Iflux Superposo Iegral, paper SPE 7066 preseed a he 987 SPE Easer Regoal Meeg geld Psburgh, Pesylvaa, 2-23 Oc [7] Pedrosa, O.A. Jr., Azz K.: Use of Hybrd Grd Reservor Smulao, paper SPE 5507 preseed a he 985 Symposum o Reservor Smulao, Dallas, Feb Fgures Fgure : Aqufer ad producve area grd coeced va a arfcal boudary (shaded) used for hsory ru of Example. Fgure 3: Aqufer grd ha resuls from he parameer coverso of Example. Fgure 2: Pressure decles for Example Fgure 4: Pressure decles for Example

9 2 Fgure 5: Heavly fauled producve area grd wh arfcal boudary (shaded) of Example 2. Fgure 6: Pressure decles for Example 2. Fgure 7: Heavly fauled producve area grd of Example 3. Arfcal boudary ad relaed arge area are shaded wh he same color. Fgure 8: Pressure decle for he arge area locaed he Wes of Example 3. Fgure 0: Pressure decle for he arge area locaed he Wes of Example Fgure 9: Aqufer grd resulg from he coverso of aalycal o grdded parameers for Example 3.

10 3 9h Europea Coferece o he Mahemacs of Ol Recovery Caes, Frace, 30 Augus - 2 Sepember 2004