Spatial-Temporal Separation Based on the Dynamic Recurrent Wavelet Neural Network Modelling for ASP Flooding

Transcription

1 Amerca Joural of Appled Mahemacs 7; 5(6: hp://wwwscecepublshggroupcom/j/ajam do: 648/jajam756 ISSN: (Pr; ISSN: 33-6X (Ole Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg Shurog * Yule Ge Auomao School Bejg Uversy of Poss ad elecommucaos Bejg Cha College of Iformao ad Corol Egeerg Cha Uversy of Peroleum (Eas Cha Qgdao Cha Emal address: lshurog@bupeduc (Shurog * Correspodg auhor o ce hs arcle: Shurog Yule Ge Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg Amerca Joural of Appled Mahemacs Vol 5 No 6 7 pp do: 648/jajam756 Receved: December 3 7; Acceped: December 7 7; Publshed: Jauary 8 Absrac: I hs paper a hree-dmesoal spaal-emporal decomposo modellg mehod s proposed o buld he alkal-surfaca-polymer (ASP floodg model whch a ew dyamc recurre wavele eural ework (DRWNN s preseed o defy he emporal coeffces A frs he dealed mahemacal model of ASP floodg s descrbed whch s a complex dsrbued parameer sysem he a hree-dmesoal spaal-emporal modellg mehod s ferred based o Karhue-oeve (K- decomposo o decompose he waer saurao of reservor o a seres of spaal bass fucos ad correspodg emporal coeffces Furhermore he recurre wavele eural ework s used o acqure he defcao model whch he jeco coceraos of ASP floodg ad emporal coeffces are ake as he pu ad oupu formao I order o mprove he capably of dyamc modellg DRWNN s proposed hrough addg feedback layers ad seg he dffere weghs wh me o acheve dyamc memory of he pas formao Cosderg he grade desce mehod for he eural eworks rag easly leads o local mmum ad slow covergece speed he specral cojugae grade mehod s roduced o opmze he weghs of DRWNN A las DRWNN s used o buld he relao bewee he mosure coe of produco wells ad he waer saurao of he correspodg grds hus he fal approxmae model of ASP floodg s fshed he accuracy s proved by model wh four jeco wells ad e produco wells hrough daa from he mechasm model Keywords: ASP Floodg Karhue-oeve Decomposo Dyamc Recurre Wavele Neural Nework Specral Cojugae Grade Mehod Iroduco Wh he old ol felds eerg he laer perod of developme mosure coe of reservor s creasg ad he ol produco s reducg [] How o updae he echcal meas o esure ol recovery s oe of he mos mpora measures o sablze ol produco ASP floodg s a mpora erary ol recovery echque whch s wdely suded [] I ca ehace he ol produco evdely by use of he eraco amog alkal surfaca ad polymer o mprove he physcochemcal propery of reservor However here s lack of a uform mahemacal model descrpo because of he uceray of alkal reaco I s mpora o buld a accurae ad easly o be appled model he research of ASP floodg I addo sce he mechasm model of ASP floodg s a complexed dsrbued parameer sysem s hard o carry ou opmal corol sraeges for he ol produco because of he feaures of fe dmesos spaal-emporal couplg ad complex olear behavor he pus of ASP floodg are he jeco coceraos of alkal surfaca ad polymer; he oupu s he waer cu of produco wells he sae varables of ASP floodg coa waer saurao pressure ad grd coceraos I applcaos dffere mehods are used o search he jeco coceraos of ASP floodg o ge he bes performace dex Oe of

2 Amerca Joural of Appled Mahemacs 7; 5(6: commo ways s o solve he mechasm equaos of ASP floodg such as flud equaos ad seepage equaos ec [3] However he mehod based o mechasm model usually volves a lo of mah operaos ad he process of mahemac reame always eeds a far amou of calculao hough may people have doe researches o modellg of ASP floodg early all he works are abou mprovg or erchg he prmary model he ma work hey have doe s o cosder more fluecg facors or ducve researches; he model s sll complex ad dffcul o be appled So s mpora o cosder usg model defcao mehod o approxmae he ASP floodg sysem so as o smplfy he mahemacal operao process ad wll be helpful o mprove he compuaoal effcecy ad reduce he compuaoal complexy of he algorhm [4] radoal sysem defcao mehods do o cosder spaal formao [5] o model he spaal-emporal dyamcs spaal formao s a very mpora par of sysem Spaal-emporal decomposo echology [6 7] whch comes from Fourer seres expaso s a very useful modellg mehod for dsrbued parameer sysems I ca reflec he formao me doma ad space doma of he sysem by decomposo A spaoemporal varable ca be expaded o a fe umber of spaal bass fucos ad correspodg emporal coeffces Geerally speakg he frs few prmary spaal bass fucos ca reflec he maxmal er formao of he sysem whch provdes a good approxmao because of her separao properes [8] I hs way he spaal-emporal mehod ca oba a fe-dmesoal model A mpora codo of modellg s o guaraee he accuracy of model whch s hghly depede o he choce of spaal bass fucos I parcular Karhue-oeve (K- decomposo whch s called proper orhogoal decomposo ad prcpal compoe aalyss [9 ] s a popular spaal-emporal decomposo approach o fd prcpal spaal srucures ad reduce he dmeso of he daa Amog all lear expasos K- expaso s he mos effce he sese ha for a gve approxmao error he umber of K- bases requred s mmal As a resul K- decomposo ca help o reduce he model dmeso ad he umber of esmaed parameers Oce he spaal bass fucos are desged properly he correspodg saes ca be deermed by projecg he spaoemporal daa oo hese spaal bass fucos o model he pu-sae dyamcs he relaoshp bewee pus ad emporal coeffces should be defed I he process of modellg he dyamc recurre eural ework s a good way for he sysem wh ukow oleary he recurre wavele eural ework (RWNN [ ] combg wavele rasform wh dyamc eural ework ca ge beer modellg capably I s wdely used olear dyamcal modellg problems because of he advaages of eural eworks such as relable heory bass explc praccal sese smple algorhm realzao ad srog adapably ec I order o mprove he dyamc modellg capably of RWNN a ew dyamc recurre wavele eural ework (DRWNN s proposed whch he feedback layers are added for he ework ad dffere weghs are se for feedback layers whch s decreasg wh me o make he ework have he dyamc memory fuco However he grade desce mehod s commoly adoped o ra he parameers of ework whch may cause he local mmum soluos ad slow covergece speed Especally large search space for mulmodal fuco he fuco usually ca fd ou he global opmal value [] May researchers have suded he opmzao mehods whch ca be dvded o fve classes geeral: grade desce mehods Newo mehods cojugae grade mehods heursc opmzao mehods ad agrage mulpler mehods [3] Bu he grade desce mehod has slow coverge speed he Newo mehod eeds o sorage ad compue he Hesse marx whch has huge compug complexy he resuls of heursc opmzao mehod has a cera uceray ad he agrage mulpler mehod s ofe used o deal wh mul-objecve opmzao problems Whle he cojugae grade mehod whch balaces he covergece speed ad compug complexy has good sably ad wh o eed of ay exraeous parameer I s wdely used olear opmzao PRP mehod s vewed as oe of he mos effecve cojugae grade (CG mehods bu he covergece propery s o so good [4] Referece [5] gave a mproved PRP mehod whch ca also be amed as NPRP mehod By modfyg he sep-legh parameer he suffce desce propery ad global covergece are guaraeed o he codo of srog wolf le search o mprove he performace of umercal mehods furher o he bass of [5] a ew specral NPRP (SNPRP cojugae grade mehod was preseed hrough addg he specral parameer ad revsg he sep-legh [6] he auhor also gave he proof abou global covergece ad fas desce O accou of hese advaages he SNPRP cojugae grade mehod s adoped o ra he weghs of DRWNN I hs paper a ew model of ASP floodg s obaed erms of spaal-emporal decomposo K- decomposo s used o model he sae parameers of reservor (waer saurao o a seres of emporal coeffces ad spaal bass fucos o ge he relao bewee he jeco coceraos of ASP floodg ad emporal coeffces he DRWNN s proposed whch dffere weghs decreasg wh me are added for feedback layers o mprove he dyamc modellg capably ad he weghs are raed by he SNPRP cojugae grade mehod Furhermore buld he model bewee he grd waer saurao ad he mosure coe of produco wells wh DRWNN he he fal model of ASP floodg s fshed wh he jeco coceraos as he pus ad he mosure coe as he oupus hs arcle s orgazed as follows: he mechasm model descrpo of ASP floodg s descrbed seco I seco 3 he Karhue-oeve decomposo of hree-dmesoal sae of ASP floodg s roduced I seco 4 a ew dyamc recurre wavele eural ework

3 56 Shurog ad Yule Ge: Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg amed DRWNN s proposed o defy he relaoshps he modellg Besdes ha order o mprove he ably of DRWNN he SNPRP cojugae grade mehod s adoped o ra he weghs Seco 5 gves a llusrave smulao example Fally a few coclusos are preseed seco 6 Mechasm Model Descrpo of ASP Floodg Suppose ha he rego of he reservor s ( x y z Ω 3 Ω R he ma fve compoes of ASP floodg are ol waer polymer surfaca alkal he mahemacal model of ASP floodg ca be descrbed as: [9 ] he flow equao for ol phase: ( S Kkro φ w ( po ρogh + qo Bo µ o Bo he flow equao for waer phase: Kk φs + w ( p ρ gh q rw w w w BwRk µ w Bw he flow equao for polymer phase: ( ( Kk c φ S + + ( ρ rw p p w pw wgh Dw Dwp cp BwRk µ w Bw φpswcp + qc ρr ( φ Crp + B w (3 he flow equao for alkal phase: φsw KkrwcO H DOH coh + ( pw ρwgh Bw BwRk µ w Ka φswc OH + ROH + qe φ SwcOH + + K B bcoh w (4 he flow equao for surfaca phase: Kkrwcws φsso ( pw ρwgh + ( Do + Dos cos BwRk µ w Bo φssw Kkroco s + ( Dw + Dws cws + ( po ρogh Bw BoRk µ o φssocos φsswcws + qd + + ρr ( φ Crs Bo Bw (5 he al codos: Θ Θ w w p( x y z p S ( x y z S c ( x y z c ( x y z Ω he boudary codos: p Sw cθ Ω Ω Ω Polymer ca mprove he vscosy of pore waer so ca be represeed as: 3 sp µ w µ w + apc p + apcp + apc 3 p C sep where µ w s he vscosy of pure waer; ap ap ap 3 sp (6 (7 (8 are emprcal cosas; C sep s he saly of reservor Alkal ca reac wh subsaces he reservor o produce ew subsaces ad cause alkal cosumpo hs process also flueces ol-waer erfacal eso ad he cocree mahemacal descrpo s where ( C wo s σ wo ( C A ( C σ (9 wo s s a σ s ol-waer erfacal eso whe ol dsplaceme age s oly surfaca he dffere ol dsplaceme ages ca affec he sregh of he rock adsorpo ad ca be descrbed as: where ac l l Γ rl ( l a s p + bc l l a ll b are adsorpo parameers (

4 Amerca Joural of Appled Mahemacs 7; 5(6: Because he alkal wll lower adsorpo of polymer ad surfaca o he surface of he pores of rock he adsorpo should be recalculaed whe alkal s jeced o reservor ad he formula for calculao ca be descrbed as: Γ CA C l s p ( rl l ad a where Aad ( C a s adsorpo coeffce he oher parameers ad model explaaos ca be foud he appedx 3 he hree-dmesoal Spaal-emporal Decomposo Modellg Mehod for ASP Floodg ASP floodg sysem s a complex dsrbued parameer sysem whch s very dffcul o be modeled geeral mehod I hs paper a ew hree-dmesoal spaal-emporal model mehod s proposed based o Karhue-oeve decomposo mehod [8] Durg he modellg process for ASP floodg he ma problem s o defy a approprae spaal-emporal model accordg o he pu oupu ad sysem sae daa he whole model for space/me separao of ASP floodg ca be dvded o hree sages: ( Decompose he saes of ASP floodg o spaal bass fucos ad emporal coeffces; ( Idefy he relaoshp bewee he jeco coceraos ad he emporal coeffces; (3 Model he mosure coe of produco wells ad correspodg grd saes 3 hree-dmesoal Spaal-emporal Mehod Based o K Decomposo For he sysem of ASP floodg suppose ha he oupu of x { } y z sysem s Y ( x yj zk N N N j k whch s sampled me ad space where deoes sample pos of oupu me doma N x j N y k N z s Nx Ny Nz sample pos space doma Accordg o he heory of Fourer seres he spaal-emporal oupu Y ( x y z ca be decomposed o a seres of orhoormal spaal bass fucos ad emporal coeffces So he decomposo form ca be expressed as: where ϕ ( x y z ϕ ( x y z ( Y x y ( { } { } bass fucos ad ( coeffces represes he orhoormal spaal represes he emporal I egeerg applcaos esurg he accuracy of he approxmao he model oupu s usually rucaed o he fe form: ϕ ( x y z ( Y x y z (3 where deoes he order of model maly depeds o he sysem spaal bass fucos whch reflec he dyamc characersc of sysem As he descrpo above s clear ha he ma problem of modellg s o ge he doma spaal bass fucos { ϕ ( x y z } { Y ( x } yj zk amog he sysem oupu Nx Ny Nz j k he problem ca be covered o a mmzao problem as he followg objecve fuco: m Y ( x y z ( ( ϕ x y z ϕ x y (4 s ( ϕ ϕ ϕ ( Ω / where f ( x y z f ( x y z f ( x y z ( f ( x y z f x y z deoes orm deoes he mea value of a se Because he spaal bass fucos are u orhogoal hey are submed o he below equao: ( ϕ ( x y z ϕj ( x y z j ϕ ( x y z ϕj ( x y z dxdydz j Ω hus he emporal coeffces ca be descrbed as: ( ϕ (5 x y z Y x y z (6 ϕ ϕ hrough agraga fuco cosruco mehod he cosraed opmzao problem equao (4 ca be srucured he followg form: Cosderg he orhoormal cosra ( ϕ ( x y z ( J Y x y z (( ( x y z ( x y z + λ ϕ ϕ (7 Combe he kow codos ad do he rasformao o equao (7 he ake he varao wh respec o

5 58 Shurog ad Yule Ge: Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg ( ϕ x y z δj Y x y z x y z + ( δϕ ( ( x y z λ ϕ δϕ Ω x y z dxdydz (8 he ecessary codo of exreme value for hs fucoal problem s δ J Sce ϕ ( x y z ca be a arbrary fuco he below equao ca be developed Y x (9 ( yz λ ϕ ( x y z Subsue ( o (9 he Ω λ ϕ (( ( ξ ζ ς ϕ ( ξ ζ ς R x y z ( x y z where ( ξ ζ ς dξdζdς ( ( ( ξ ζ ς ( R x y z Y x y z Y s he correlao fuco bewee wo pos he space Cosderg he muual depedece of value for o make sure ha ( s always equal o zero f ad oly f ( s rue hs s he ecessary codo of exremum for (4 ( ( ξ ζ ς ϕ ( ξ ζ ς ξ ζ ς λϕ R x y z d d d x y z Ω ( ϕ ϕ 3 Sapshos Mehod ( I hs way he orgal opmzao problem s rasformed o he soluo of egral equao ( However ormal codos hs equao s hard o solve Whe he umber of me samples s less ha space samples he sapshos mehod [7] s a good way o solve ( hs mehod ca sgfcaly reduce calculao quaes he sapsho s obaed from samplg for each poso po Y x y z 3 x N dscree me For a example x y N y z N z ad N Nx Ny Nz deoes oe sapsho a 3 Whe me odes are less ha space odes suppose he spaal bass fuco ϕ ( x y z ca be expressed as a lear combao of sapshos as follows: ϕ ( x y z γ Y ( x y z ( For egevalue problem of equao ( below equao ca be go Ω Ω λ ϕ (( ( ξ ζ ς ϕ ( ξ ζ ς ξ ζ ς λϕ ( R x y z d d d x y z ( Y ( x y z Y ( x y z Y ( x y z ( x y z ( ξ ζ ς ( ξ ζ ς Y Y ϕ ( ξ ζ ς dξdζdς Y ( ξ ζ ς (3 Subsue ( o (3 a he same me ad le ( ( ( ( ( (( ξ ζ ς ( ( ξ ζ ς ( ξ ζ ς ( ξ ζ ς Y x y z Y x y z Y x y z Y x y z Y Y Y Y ( γ γ γ γ we ca ge: Ω ( ξ ζ ς ( ξ ζ ς Y γ Y γ Y ( x y z Y ( ξ ζ ς d d d ξ ζ ς Y ( ξ ζ ς γ Y ( x y z Y ( ξ ζ z Y ( ξ ζ z k γdξdζdς Ω λy x y z γ C Y ξ ζ ς Y ξ ζ ς k dξdζdς Defe ( ( Ω (4 so ( λ γ Y x y z Cγ (5 I hs way N N egevalue problem equao ( s covered o he problem: Cγ λ γ (6 where γ s he h egevecor ad λ s he correspodg egevalue he marx C ca be obaed from he oupu samples so he spaal bass fucos ca be compued hrough equao (6 Because C s symmerc ad posve semdefe he compued egefucos are orhogoal he dmeso of he model largely deermes he modellg accuracy ad complexy Whe decdg he umber of he dmeso he egevalues are arraged descedg order he egefucos correspodg o he frs several egevalues are he mos advaced represeave he sysem oupu So he proper umber of spaal bass fucos ca be seleced Geerally speakg he larger he umber of dmeso s he more accurae he accuracy of modellg wll be However he modellg complexy wll crease wh he creasg model order As log as he proper order of model s chose he modellg complexy ad

6 Amerca Joural of Appled Mahemacs 7; 5(6: accuracy ca be guaraeed ad he deal mahemacal model would be acheved Defe ha he oal eergy of sysem s equal o he sum of all egevalues he he bgger he value s he more eergy he bass fuco reflecs For every egevalue λ he proporo ha he eergy reflecs accou for he oal eergy s E λ K j λ j (7 Geerally speakg whe he proporo s more ha 99% we hk hese bases ca reflec mos eergy he correspodg s he eeded dmeso he he spaal bass fucos ad emporal coeffces are obaed 4 Dyamc Recurre Wavele Neural Nework Modellg Based o SNPRP Cojugae Grade Mehod I order o esablsh he relaoshp bewee he emporal u suppose ha he coeffces ( ad he pu dyamc bewee ( ad u ca be expressed by a olear auoregressve wh exogeous pu (NARX model [8]: ( ( ( ( ˆ ˆ ˆ ˆ F u u u (8 where u ad are he maxmum pu ad oupu lags respecvely he modellg process of NARX model whch s black-box model does eed o kow s eral mechasm ad he mahemacal model ca be esablshed based o s pu ad oupu daa Because he eural ework has good olear approxmao capably powerful operao ably srog faul olerace ad robusess so s wdely used he feld of olear sysem defcao Dyamc eural eworks ca drecly reflec he sysem dyamc characersc ad s wdely used he dyamc sysem modellg o mprove he capably of dyamc modellg hs paper he feedback layers for dyamc recurre wavele eural ework s added ad dffere weghs for hem chroologcal order are se A ew algorhm whch called dyamc memory feedback wavele eural ework (DRWNN based o he above-meoed dea s proposed Cosderg he weakess of he radoal grade desce mehod for he eural eworks rag he SNPRP cojugae grade mehod s used o opmze he weghs of DRWNN 4 he Nework Srucure Cosderg he coveoal dyamc eural eworks ca oly memory he oupu of las hdde layers by addg he mul-layer of feedback o DRWNN ca memory several dyamc feedback of hdde layers I he srucure of DRWNN he umber of feedback layers reflecs he memory of he hsorcal daa as show Fgure Fgure he srucure char of DRWNN he srucure of DRWNN cosss of four layers: he pu layer he hdde layer he oupu layer ad he feedback layer he ework coas pu euros a oupu euros m hdde layer euros Q feedback layers x R s he pu wh dmesos for he me ; y ( R s he

7 6 Shurog ad Yule Ge: Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg m oupu; H ( R x ( R c s he oupu of hdde layers; s he oupu of feedback layers hrough addg Q feedback layers o he hdde layers o dyamcally memory he oupu of hdde layers Suppose he daa of he qh feedback layer s H( k q ad s wegh s Wb [ wb wb wb ] wb ( I order o make q q q mq he ework acheve dyamc oblvo fuco as q wb Because of he weghs ragg from o wb q s se q wb wll be smaller wh me whch ca gradually forge furher formao Suppose α s he feedback ga he oupu of feedback layer ca be compued by he followg formula: x c Q ( q (9 α wb H ( q q he dyamc equaos of DRWNN ca be expressed as: m y( W H ( h ( b ( H ( ϕ a ( m Q h ( Wjxj ( + α vkwbqhk ( q j k q where (3 W s he lked wegh bewee he hdde layer of euro ad oupu layer of euro; W j s he lked wegh bewee he pu layer of euro j ad he hdde layer of euro ; v s he lked wegh bewee he k feedback layer k ad he hdde layer of euro ; H ( s he oupu of he hdde layer of euro ϕ ( s Morle wavele fuco whch ca be descrbed as: where ( ϕ cos 75 e (3 ( a ( h b b s raslao coeffce a s dlaao coeffce ad 4 he SNPRP Cojugae Grade Mehod he SNPRP cojugae grade mehod s a ew cojugae grade mehod whch s preseed [6] I has he suffce desce propery ad global covergece propery uder he codo of srog le search Furhermore eeds less sored formao whch esures he hgh compuaoal effcecy he a bref roduco o hs mehod s gve as follows Cosderg a geeral ucosraed opmzao problem { f ( x x R } m where f : R R s a oe order couously dffereable olear fuco ad s grade vecor s g( x f ( x gk g( xk For a geeral specral cojugae grade mehod [3] below equao s sasfed x + x + d (3 k k ηk k d k gk k gk + βkdk k δk (33 where d k s he search dreco β k s he parameer ad η k s he sep-legh facor he specral coeffce s defed as sk yk δk y k gk gk sk (34 I SNPRP cojugae grade mehod he parameer defed as β SNPRP k δ k k gk gkg k gk δ g k gk β k s (35 he correspodg covergece ad sably has bee proved [6] he algorhm process s as follows Gve he x R ε d g k ; If g k ε he ed he whole calculao; Compue he sep-legh facor η k ; Execue compuao as (3 ad le g g ( x ; k + k + SNPRP Compue he specral parameer β k ad d k+ as equaos (34 (35 ad (33 separaely e k k + he go o sep he he SNPRP cojugae grade mehod s used o ra he weghs of DRWNN 43 he rag Algorhm of DRWNN Suppose he y( ad y deoe he real oupu ad he expeced oupu a me sep he error s he cos fuco s e e y y (36 e E ( ( ye ( y ( ( e( (37 he oal error fuco from he me sep o N s

8 Amerca Joural of Appled Mahemacs 7; 5(6: N E y y e ( e ( (38 he rag goal of DRWNN s o adjus he parameers o decrease E gradually From he cha rule of eural ework o he bass of SNPRP cojugae mehod he weghs ca be ferred as follows N ( + η + β d δ W W W E W ( + η e( H + β d δ { } Defe Θ Θ j k Θ W v a b he ( ( s s s + η + β s d δ Θ ( η ( H + + Θ s s s e W β s d δ Θ E (39 (4 where k m j s 345 correspods o he varables W j v k a b η s s he learg coeffces of W W v ab d ( E ( Ψ ( where Ψ Θ varable E Ψ s s β d 3 s + δ { } (4 W s correspods o he W For he equaos above ( H H Wj h (4 m Q Hk ( q xj ( + α vk ( wbq k q wj ( q H H vk h Q (43 m Hk ( q αhk ( + α vk ( wbq v k ( q k q ( H ϕ ( a Q (44 m ( Hk ( q + α vk ( wbq a ( a ( q k q ( H ϕ ( b Q (45 m ( Hk ( q + α vk ( wbq b ( b ( q k q where ϕ ( 75s ( 75 e cos( 75 e ( ( ad a ( b ( h ( b ( a ( H ( he al rag value s xc ( Wj ( H ( H ( H ( v ( a ( b ( k ca be calculaed by j k Wh he mehod above he fal DRWNN s deermed 44 Expermeal Resuls ad Aalyss I order o verfy he modellg ably of he proposed DRWNN A sgle pu sgle oupu dyamcal olear sysem s used for he esg he dyamc equao s descrbed as: ω y + f + g + y + f ( 3y ( / ( 6 + y ( + y ( (46 3 g ( u ( + u ( u ( + 6s(kπ / 5 + 4s ( kπ / 75 ω s a where u ( s he pu y( s he oupu zero mea whe ose I hs seco he eural ework used for esg coas 6 hdde layers pu layer oupu layer ad he umber y of he y of feedback layers of DRWNN s 3 u ( ad frs me seps are ake as he rag samples he rag sgal s show Fgure I order o compare he performace he RWNN s roduced o model wh he same rag way [] he rag umber of boh RWNN ad DRWNN s mes

9 6 Shurog ad Yule Ge: Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg u( y( me sep Ipu sgal Desred resul DRWNN resul RWNN resul me sep Oupu sgal Fgure rag sgals he smulao resuls are show Fgure o expla he resuls beer he maxmal absolue error E m s defed he mea error E ad he roo-mea-square error RMSE as follows able he error of rag samples RMSE E m E RWNN DRWNN I ca be show ha o maer for he maxmal absolue error he mea error or he roo-mea-square error he resuls of DRWNN s always less ha ha of RWNN ad he model oupu curve of DRWNN s closer ha ha of RWNN So DRWNN has beer modellg ably able he error of esg samples RMSE E m E RWNN DRWNN I order o verfy he geeralzao ably of DRWNN he u from he me sep o 4 s served as pu sgal he esg sgal ɶu ( ad he oupu ɶy s obaed he rag sgal s show Fgure 3 ad he correspodg errors are show able hese resuls demosrae ha he oupu of DRWNN s much closer o he real oupu whch verfes he DRWNN has beer model geeralzao ably Usg he SNPRP cojugae grade mehod o opmze he weghs of DRWNN s effecve u( me sep Ipu sgal * m E max y y N (47 E RMSE N * y y (48 N N * ( y y (49 N y( Desred resul DRWNN resul RWNN resul * where y s he real value y s he calculaed value ad N s he umber of samples he rag resuls are show as able me sep Oupu sgal Fgure 3 esg sgals

10 Amerca Joural of Appled Mahemacs 7; 5(6: Modellg for ASP Floodg wh Proposed Spaal-emporal Separao Mehod 5 Reservor Descrpo Suppose ha he reservor of ASP floodg cosss of four jecos ad e produco wells All wells dsrbue uformly; here s oe jeco well a he ceer of every four produco wells he dsrbuo of wells s show Fgure 4 he waer jeco rae of each jeco well s 83 m 3 /d he rae of produco wells s as defed able 3 Usg reservor smulao sofware CMG he whole ol produco process ca be obaed by smulao he whole produco me lass for 96 mohs ad sampled oupu ca be go by he sapshos mehod he jecg me of ASP s served as he al me so he oal me odes are 97 ad he oal space odes are N 7 able 3 he lqud volume quay of produco wells wells S- S- S-3 S- S- S-3 S3- S3- S3-3 m 3 /day Modellg ad Verfcao for ASP Floodg Fgure 4 he dsrbuo dagram of well poso As for hs reservor he legh s 63m; he wdh s 63m ad he hckess s 999m here are 7 layers all; he hckess of each layer s 857m; ad he e hckess s 486m; he deph of upper surface s 4m; he porosy of every layer s 3 ad he pore volume s 97 6 m 3 he al grd cocerao of ASP s g/ he al waer saurao s show Fgure 5 he grds of reservor are dvded hree drecos x y z he grds x ad y are dvded o us respecvely ad he grds z are dvded o 7 us he oal umber s 7 Fgure 5 he dsrbuo dagram of al waer saurao Gve he ASP floodg above he approxmae model s bul wh he mehod proposed hs paper I order o suffcely movae sysem he smulao s ru o he sofware CMG for 5 mes he sofware radomly geeraes dffere jeco sraeges for every moh ad he process of jeco oally lass for 48 mohs he he hree-dmesoal spaal-emporal modellg mehod s used o ge he spaal bass fucos ad he emporal coeffces of he 5 ses of grd saurao So as o ge he bes srucure o represe he ASP floodg sysem use oe group of he spaal bass fucos o recosue he sysem wh oher 49 groups of emporal coeffces ad calculae he average mea square errors of all samplg pos for hese 49 mes Afer execug hs process for 5 mes a group of spaal bass fucos wh he smalles mea square error s served as he spaal bass fucos for modellg he relaoshp bewee jeco coceraos of ASP ad emporal coeffces s ge by DRWNN he combe he spaal bass fucos o buld he model bewee grd waer saurao ad jeco coceraos of ASP A he same me buld he model bewee he grd waer saurao ad mosure coe of produco wells wh DRWNN he mahemacal form ca be descrbed as follows { ˆ w ( s } ˆ ( fˆ ( Sˆ ( f F ˆ f w w w w w S (5 where w s he lag of mosure coe s s he lag of grd waer saurao he he model bewee he jeco cocerao of ASP ad mosure coe of produco wells ca be bul afer egrag he models above whch ca be expressed as equaos (3 (8 ad (5 I order o compare he fluece of he umber of he spaal bass fucos he performace dcor s defed as follows: RMSE ( e x y z dxdydz dxdydz (5

11 64 Shurog ad Yule Ge: Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg where e( x y z Sˆ ( x y z S ( x y z w Geerally speakg he more he umber of he spaal bass fuco s he more formao of sysem he model reflecs ad he hgher accuracy he model has Bu a he same me he model wll be sesve o he exeral dsurbace ad he geeralzao ably wll decrease Besdes he dmeso of he model creases O he oher had he less quay he spaal bass fuco s; he model wll reflec less formao ad he error ca be very bg I s mpora o choose he proper amous of he spaal bass fucos I order o es he fluece of dffere umbers spaal bass fuco o he accuracy of modellg dffere umbers of spaal bass fucos s chose o recosruc he sysem wh he emporal coeffces ad calculae he average RMSE whch s show able 4 able 4 RMSE for dffere umber of spaal bass fucos RMSE From able 4 he accuracy of model crease wh he umber of spaal bass fucos However whe he umber s over 4 he RMSE of model wll o crease whch dcae ha he spaal bass fucos ca reflec he whole reservor so he umber of spaal bass fucos adoped hs paper s 4 Besdes ha he umber of feedback layers of he DRWNN eworks for jeco coceraos ad emporal coeffces s 3; he correspodg hdde layers s 6; he lags of emporal coeffces s 4 ; he lags of jeco coceraos s u 3 he umber of feedback layers of DRWNN eworks for mosure coe of produco wells ad he correspodg grd waer saurao s ; he hdde layers s 8; he lags of mosure coe s w 3; he lags of grd waer saurao s s Model for he ASP floodg wh he spaal-emporal separao mehod based o DRWNN whch s proposed hs paper he errors for hs model are ha he mea RMSE for grd waer saurao s 85 ad he mearmse for mosure coe of produco wells s 378% hs demosraes he good modellg ably I order o verfy he geeralzao ably he jeco coceraos of ASP floodg are gve radomly: 74 u kg/m 3 u P kg/m A ( 93 6 u kg/m 3 he dsplacg age jeco S oally lass for 48 mohs whch are dvded o 3 slugs uformly ad he res me s waer floodg he specfc jeco ca be foud Fgure 6-a Because he whole sysem s fve-dmesoal cosderg he me ad value of waer saurao ca be ploed he pcure drecly I 3 order o llusrae clearly he grds ragg from o ( 3 a he hrd layer s chose he resul ad error of modellg are show Fgures 6 ad 7 w he jeco mass cocerao (kg/m 3 Waer Saurao Waer Saurao me/moh me/moh 5 me/moh he jeco mass cocerao he oupu of real sysem he oupu of modellg Fgure 6 he comparso dagram of waer saurao Alkal Surfaca Polymer o aalyze he geeralzao ably furher we compue he error wh equao (48 he mea absolue errors for he mosure coe of produco wells ad for he grd waer saurao are 97% ad 38% respecvely Fgure 8 compares he mosure coe of modellg ad ha of he 5 5 Grd x Grd x

12 Amerca Joural of Appled Mahemacs 7; 5(6: smulao sofware I ca be kow ha he error s very small whch verfes ha he whole model has beer geeralzao ably From he above he hree-dmesoal spaal-emporal separao modellg mehod based o DRWNN ca model for ASP floodg well I has good modellg accuracy ad geeralzao ably Error of waer saurao Comparso of mosure coe/% me/moh Fgure 7 he error dagram of waer saurao Fgure 8 he comparso dagram of mosure coe of produco wells 6 Coclusos I hs paper a hree-dmesoal spaal-emporal separao modellg approach s proposed o esablsh he defcao model for ASP floodg A frs spaal sae (he waer saurao s expaded oo a seres of doma spaal bass fucos ad emporal coeffces he a ew dyamc recurre wavele eural ework whch feedback layers are added ad dffere weghs are se wh me o acheve dyamc memory of he pas formao s proposed o buld he relaoshp bewee he emporal coeffces ad he jeco coceraos of ASP floodg o avod he local mmum ad low covergece he SNPRP cojugae grade mehod s adoped o ra he weghs of DRWNN Besdes ha DRWNN s also used o 5 Grd x 5 Orgal curve Modelg curve me/moh 5 esablsh he model of he mosure coe of produco wells ad he correspodg grd waer saurao he smulao of ASP floodg s carred ou o show he effecveess of hs spaal-emporal modellg mehod he resul shows ha he mehod proposed hs paper has good accuracy ad geeralzao ably I s suable o be used for modellg for complex dsrbued parameer sysems lke ASP floodg Ackowledgemes Auhors would lke o hak he assocae edor ad he aoymous referees for her valuable commes ad suggesos hs work s suppored by Naoal Naural Scece Foudao uder Gra No Naoal Naural Scece Foudao uder Gra No Naural Scece Foudao of Shadog provce uder Gra No ZRFM he Fudameal Research Fuds for he Ceral Uverses uder Gra No 5CX664A Coflcs of Ieres he auhors declare ha here s o coflc of eres regardg he publcao of hs paper Appedx For he mechasm model of ASP floodg he specfc meag of parameers s as follows below [9 ]: K s he absolue permeably of rock; kro k rw are he relave permeably of waer ad ol; po p w are he pressure of waer ad ol; So S w are he ol saurao ad waer saurao; Bo B w deoe he volume facors of ol ad waer; pcow ( x y z s he capllary force; c Θ p soh are he coceraos of polymer Θ { } surfaca ad alkal; c j s he mass cocerao of compoe soluo j ; µ o µ w are he vscosy of waer ad ol; φ φp φ s are he rock porosy reachable porosy of polymer ad reachable porosy of surfaca; φ φ f f f are he reachable porosy facors; R ps as a s s descedg coeffce of relave permeably; k K a s he speed facor of he o exchage ad adsorpo capacy; s he adsorpo cosa of surfaca; v w s he seepage velocy; R OH s he alkal cosumpo; Crp C rs are he adsorpo qualy of u mass of rock of polymer surfaca; ro r w are he flow coeffces of ol ad waer; q o q w are he flow rae of ol ad waer he sadard sae; qc qd q e are he raspor velocy of shaf floodg ages; D wooh s he dffuso coeffce; { } { } { } K b D wo j s p s he dffuso coeffce of j compoe j soluo

13 66 Shurog ad Yule Ge: Spaal-emporal Separao Based o he Dyamc Recurre Wavele Neural Nework Modellg for ASP Floodg he flow erms are defed as follows: q o he waer cu s ( fw qou ( x y z ψp ( x y z ψp fwqou ( x y z ψp qw q ( x y z ψw ( x y z ψp ψw qwcp ( x y z ψp qc qwcp ( x y z ψw ( x y z ψp ψw qwcws + qocos ( x y z ψp qd qwcs ( x y z ψw ( x y z ψp ψw qwcoh ( x y z ψp qe qwcoh ( x y z ψw ( x y z ψp ψw f w rw r + r w o he cocerao of surfaca s defed as he alkal cosumpo s c s qwcws + q c q + q w o o os Ra φsw r + r + ( where r deoes he alkal cosumpo per u volume of reaco he ol ad waer relave permeably k ro k rw ca be descrbed as B rw ro w w K A ( S ( S C where ABCD are he defcao coeffces he specfc mehod s show [] he descedg of permeably s caused by he adsorpo of polymer ca be descrbed as R k max ( Rk qp + q max p D Refereces [] E Verheye Ol exraco mperls Afrca's Grea akes Scece ( [] Y Zhu Q Hou W u e al Rece Progress ad Effecs Aalyss of ASP Floodg Feld ess Caada Psycharc Assocao Joural (5 ( [3] F Abadl Smulao Sudy of Ehaced Ol Recovery by ASP (Alkale Surfaca ad Polymer Floodg for Nore Feld C-segme Deparme of Peroleum Egeerg & Appled Geophyscs ( [4] Y Ge S R S u e al Spaal-emporal ARX Modellg ad Opmzao for Polymer Floodg Mahemacal Problems Egeerg (4 [5] G Prado A Chuso G Plloeo Maxmum Eropy vecor kerels for MIMO sysem defcao Auomaca 79 ( [6] D Coca S A Bllgs Idefcao of fe dmesoal models of fe dmesoal dyamcal sysems Auomaca 38 ( ( [7] H Deg H X G Che Specral-approxmao-based ellge modellg for dsrbued hermal processes IEEE rasacos o Corol Sysems echology 3 (5 ( [8] C K Q H X A me/space separao-based Hammerse modellg approach for olear dsrbued parameer processes Compuers & Chemcal Egeerg 33 (7 ( [9] C Wa M Pa e al Performace mproveme of magec aomaly deecor usg Karhue oeve expaso Ie Scece Measureme & echology (5 ( [] N Hamlo M uku R B Cal ow-order represeaos of he caocal wd urbe array boudary layer va double proper orhogoal decomposo Physcs of Fluds 8 ( ( [] F Y Zhao Z Y Ma Nolear dyamcal sysem smulao based o recurre wavele eural ework Joural of Sysem Smulao 9 (7 ( [] N Yu Uversal grade mehods for covex opmzao problems Mahemacal Programmg 5 (- ( [3] F Rohlauf Opmzao Mehods Desg of Moder Heurscs Sprger Berl Hedelberg 45- [4] Y H Da Y X Yua he Nolear Cojugae Grade Mehod Shagha: Shagha Scece ad echology Press [5] Zhag A mproved We-Yao-u olear cojugae grade mehod for opmzao compuao Appled Mahemacs & Compuao 5 (6 ( where q p deoes he adsorpo quay of polymer [6] Z Q P R Z X We A New Specral Cojugae Grade Mehod for Solvg Ucosras Mmzao Problem Joural of Souhwes Uversy (Naural Scece Edo 38 (7 (6 5-

14 Amerca Joural of Appled Mahemacs 7; 5(6: [7] C Hua N S Y me-space ARX modellg ad predcve corol for dsrbued parameer sysem Corol heory & Applcaos 8 ( ( 7-76 [8] C K Q H X Nolear dmeso reduco based eural modellg for dsrbued parameer processes Chemcal Egeerg Scece 64 (9 ( [9] S R X D Zhag Opmal corol of polymer floodg for ehaced ol recovery Dogyg: Cha uversy of peroleum press [] C Z Yag e al Ehaced ol recovery for chemcal floodg Bejg: Peroleum Idusry Press [] Y Ge S R K X Qu A Novel Emprcal Equao for Relave Permeably ow Permeably Reservors Chese Joural of Chemcal Egeerg ( ( [] F M F El-Sousy K A Abuhasel Adapve Nolear Dsurbace Observer Usg Double oop Self-Orgazg Recurre Wavele-Neural-Nework for wo-axs Moo Corol Sysem IEEE rasacos o Idusry Applcaos 99 (7 [3] Z Wa C Hu Z Yag A specral PRP cojugae grade mehods for ocovex opmzao problem based o modfed le search Dscree ad Couous Dyamcal Sysems - Seres B (DCDS-B 6 (4 (