Material-Balance-Time During Linear and Radial Flow

Transcription

1 PETROLEUM SOCIETY CANAIAN INSTITUTE OF MINING, METALLURGY & PETROLEUM PAPER 000 MaialBalancTim uing Lina and Radial Flow.M. Andson, L. Maa Fk Associas, Inc. This a is o b snd a h Polum Sociy s Canadian Innaional Polum Confnc 00, Calgay, Alba, Canada, Jun 0, 00. iscussion of his a is invid and may b snd a h ming if fild in wiing wih h chnical ogam chaiman io o h conclusion of h ming. This a and any discussion fild will b considd fo ublicaion in Polum Sociy jounals. Publicaion ighs a svd. This is a in and subjc o cocion. ABSTRACT Poducion daa gnally consiss of vaiabl a and vaiabl flowing ssu. I is convnin o b abl o us svoi modls ha assum a consan flow a, sinc hs soluions hav bn viously divd in h wll sing liau. Thus, i is ncssay o hav a im funcion caabl of conving gnal oducion condiions ino h quiva consan a soluion. Blasingam, and la Agal al hav shown ha MaialBalancTim ovids an xac ansfomaion of consan ssu daa o consan a y cuvs, duing h bounday dominad flow gim. I also yilds a asonabl aoximaion duing adial flow, and whn a and/o ssu vay smoohly. Po has invsigad h ffcivnss of using maialbalancim fo oh ansin flow gims using h consan ssu soluion, ah han h consan a soluion as a bas modl. Th objcivs of his a a wofold. Fisly, i svs o invsiga h alicabiliy of maialbalancim duing h lina flow gim (facu flow), wh h diffnc bwn h consan a and consan ssu soluions is mo onouncd. Fuh o his, maialbalancim cocion facos a quanifid fo boh adial and lina ansin flow gims (his has no viously bn don using h consan a soluion as h bas modl). Scondly, i svs o illusa by synhic and fild xamls, a comaison of maialbalancim agains h logaihmic suosiion im funcion, o dmin und wha cicumsancs maialbalancim os significanly influnc a ansin inaion, in acic. INTROUCTION In ssu ansin ss, h diagnosic lo (loglog lo of ssu and divaiv) is an invaluabl ool fo svoi chaacizaion. Fo vaiabl a dawdown

2 ss and flow and buildu ss, a im suosiion funcion should b usd o conv vaiabl as ino an quiva consan a soluion. Sinc ssu ansin ss a usually dominad by infini acing flow, h widly accd im suosiion funcion is on ha assums adial flow (logaihmic suosiion im). On of h oblms inhn in ssu ansin analysis is ha adial flow is no alys h dominan flow gim. Thus, h is h onial fo misinaion of h diagnosic lo in cain siuaions. In cn yas, diagnosic los of vaious diffn foms hav bn usd o analyz oducion daa. Th nau of oducion daa analysis is diffn han ha of ssu ansin analysis, imaily bcaus of h galy incasd im scal, and bcaus oducion daa nds o b much noisi han ssu ansin daa. Mos of h liau ags ha a im suosiion funcion ha assums bounday dominad flow is mo aoia o us on a diagnosic lo of oducion daa, han any ansin suosiion funcion, bcaus mos oducion daa is und h influnc of som so of svoi dlion. Howv, wih vy low mabiliy svois, his is no ncssaily h cas. Maialbalancim is h im suosiion funcion fo volumic dlion. I is igoous in conving vaiabl a oducion ino quiva consan a oducion, ovidd ha h flow gim is bounday dominad (volumic dlion). Sinc ansins a alys inoducd duing abu changs in a o flowing ssu, only a smoohly vaying a hisoy (such as wha occus a consan boomhol flowing ssu) is valid. Andix A ovids a hoical oof of h validiy of maialbalancim fo conving bounday dominad oil oducion a consan ssu (smoohly vaying a) ino an quiva consan a. Figu (b) shows h coml soluions fo a vical wll in a cylindical svoi oducing a consan a and consan ssu. Figu (b) shows h maialbalancim cocd soluion. Fo gas wlls, h sam hoy holds, ovidd ha sudoim is usd in conjuncion wih maialbalancim. (sudoim ffcivly linaizs h diffusiviy quaion fo gas by including h svoi ssu (im) dndn comssibiliy and viscosiy ms) In his a, h mahmaical dvlomn is limid o consan (slighly) comssibl fluids. Of ins is h alicabiliy of maialbalancim duing h infini acing flow gim, sn (and usually obsvabl) in all oducion daa. uing infini acing flow, maialbalancim is no a igoous soluion. Insad, a im suosiion funcion, which follows h obsvd flow gim (adial, lina, bilina c), ovids h igoous convsion o h quiva consan a soluion. Fo a igoous suosiion funcion, h svoi modl mus b known bfohand, which is usually no h cas. Accodingly, i is ofn imossibl o aly igoous suosiion in im, in a diagnosic lo. In acic, a im suosiion funcion, which follows h dominan flow gim (logaihmic fo adial, squa oo fo lina, h oo fo bilina, maialbalancim fo volumic, c.), is assumd. In h analysis of oducion daa, h suosiion funcion of choic is maialbalancim, bcaus h mhasis is on dlion of h svoi. Thus, i is of aicula ins o dmin h significanc of os ha sul fom h univsal alicaion of maialbalancim, comad o oh suosiion im funcions, and how hy influnc ffciv inaion of h diagnosic lo. THEORETICAL EVELOPMENT In h following scions, wo flow gims a considd, namly a) lina (facu) flow and b) adial flow, assuming a slighly comssibl fluid in an infini svoi. Fo ach flow gim, h consan a and consan ssu soluions a comad (Figus (a) and (b). Addiionally, h maialbalancim funcion is divd and h consan ssu soluion is lod agains i (Figus (a) and (b) Lina Flow (Facud wll in an infini svoi) Th consan a soluion fo u lina flow is as follows:

3 ...() q / Th consan ssu soluion is: q...()...() Fo any givn q, h dimnsionlss im cosonding o h consan ssu soluion is dfind as. Similaly, h cosonding im fo h consan a soluion is. Th aio of hs wo ims a a givn q, is obaind by dawing a hoizonal lin, as shown in Figu (a). This aio is quanifid by solving () and () simulanously: q...() fom which can b sad as a funcion of.6...(5) Equaion (5) shows ha h cocion quid o conv h consan ssu soluion o an quiva consan a is o mulily h masud im by.6. This convsion is xac, bu alis only duing u lina flow. Th dimnsionlss cumulaiv oducion fo h consan ssu, duing lina flow is: Q d Ú 0 Thus, h maialbalancim is: mb Q q...(6)...(7) Equaion (7) indicas ha maialbalancim is doubl h acual im duing u lina flow. Subsiuing (7) ino (5), w g: 8 mb.mb...(8) Equaion (8) shows ha h cocion quid o conv h consan ssu soluion o an quiva consan a is o mulily h maialbalancim by. (Figu (a)). Th abov divaion indicas ha a diagnosic lo ha uss maialbalancim (fo consan ssu lina flow) ducs h im shif fom (6 % o %), bu dos no comlly limina i. Howv, i can b infd fom Figu (a) ha unlss h oducion daa bing analyzd has vy high soluion, h diffnc bwn h wo soluions would cainly b considd insignifican. Radial Flow (Vical wll in an infini svoi) Th consan a and consan ssu soluions fo u adial flow a significanly mo comlx han hos fo lina flow. Thy xis in analyical fom (dvlod by Van Evdingn and Hus) only in Lalac sac, and canno asily b analyically invd o h im domain. Edson 5 ovids a numical cuv fo ach of h soluions, as follows: Consan Ra: log( 0.5 ) < 00 + > 00...(9) Consan Pssu: q < 00

4 q ( ) ( ) >00... (0) Fo any givn q, h aio of o can b dmind by solving h (9) and (0) simulanously. Unlik h lina flow cas, h aio dos no main consan fo adial flow, bu aoachs uniy as im incass (s ins of Figu (b) Fo vy small valus of, h aio of o aoachs ha of lina flow (.6). A a of 5, which is h aoxima sa of acical logaihmic adial flow, h aio is abou.6 (60 % cocion). Alhough his sms significan, Figu (b) shows ha gahically, h wo soluions a acically indisinguishabl fo valus of lag han 5. As Figu (b) indicas, h us of maialbalancim fuh ducs h aio (a 5) o abou.7 (7 % cocion). Again, h magniud of hs cocion facos can b mislading, as Figu (b) shows ha h consan a and maialbalancim cocd consan ssu soluions a ssnially indisinguishabl fo all valus of. CASE STUIES Mahmaically, w hav shown ha h is vy lil obsvabl diffnc bwn a diagnosic lo ha uss maialbalancim, and h consan a soluion. Indd, duing bounday dominad flow, h ansfomaion is xac. I is imoan o no ha his mahmaical dvlomn inhnly assums smoohly vaying a and consan ssu condiions. Wih al daa, his assumion is commonly violad. Thus, h cas sudis will invsiga boh idal and nonidal oaing condiions (disconinuous a / ssu ofils). Th objciv is o dmin o wha xn (and und wha condiions) h us of maialbalancim ngaivly influncs h inaion of h diagnosic lo. An addiional objciv is o coma h suls o hos obaind using a diagnosic lo gnad using logaihmic suosiion im. H, w sn boh synhic and fild xamls. Synhic aa Examls Fo ach synhic xaml, wo cass a invsigad: A) Smoohly vaying a ofil and B) isconinuous a ofil Fo all cass, h following aams a usd; gas svoi (hnc sudossu (Y) is usd insad of ssu): i 0,000 kpa h 0 m oosiy 0 % s w 0 % Fo ach of hs cass, a comaison has bn gnad bwn a diagnosic lo cad using h logaihmic suosiion im funcion and on cad using maialbalancim. Th diagnosic los a snd in nomalizd a foma (q/y). Th divaiv on h los is simly h invs of h sandad ssu divaiv usd in wlls analysis. Vical Wll in Infini Rsvoi k m s 0 Th synhically gnad daa fom his cas is snaiv of u adial flow. Th diagnosic los using logaihmic suosiion im and maialbalancim a shown in Figus a (smooh a dclin) and b (disconinuous as), comad agains h consan a soluion. I is cla fom Figus a and b ha h is acically no diffnc bwn h diagnosic los oducd fom h wo im funcions whn oducion vaiaion is smooh. Boh diagnosic los show a dviaion in hi divaivs fom h u soluion, in alyim. This is a daaavaging hnomnon ha aiss fom no using small imss a h bginning of h flow iod.

5 Fo h disconinuous a cas (Figu (b)), h diagnosic lo divd fom maialbalancim shows a fals bounday dominad nd dvloing in h laim. Th logaihmic suosiion diagnosic lo cocly indicas adial flow fo h duaion of h oducion iod. Hydaulically Facud Wll in Infini Rsvoi k 0.05 m k f w 000 m.m x f 00 m This xaml xhibis h fomaion lina flow iod of a hydaulically facud wll. Th diagnosic los a shown in Figus a(i), a(ii), b(i) and b(ii). Th smooh a cas (a) claly indicas ha boh diagnosic los (a(i) and a(ii)) ovid h coc flow gim idnificaion. Howv, as xcd, h diagnosic lo using maialbalancim is offs by a faco of abou 0%. Th diagnosic lo using logaihmic suosiion im is also offs (his offs could b cocd by using squaoo im). Th disconinuous a ofil shows significan oblms wih h diagnosic lo using maialbalancim (b(i)). Boh h nomalizd a and divaiv daa hav a ngaiv uni slo, suggsing u bounday flow, which is of cous fals. Th suosiion im diagnosic lo (b(ii)) has lil diagnosic valu, as no cla inaion is vidn. Hydaulically Facud Wll in Boundd Rsvoi k m k f w 000 m.m x f 50 m 00 m This cas ovids oducion daa ha should xhibi, in squnc, facu flow, sudoadial flow, followd by bounday dominad flow (volumic dlion). Th diagnosic los a shown in Figus 5a(i), 5a(ii), 5b(i) and 5b(ii). This cas shows ha und smoohly vaying condiions, h maialbalancim diagnosic lo (5a(i)) accualy dics h ons of bounday dominad flow. Alhough no shown, h lo will also yild h coc valu of OGIP (Oiginal GasinPlac), basd on maching o a consan a ycuv. Th logaihmic suosiion im diagnosic lo, howv, dos no cocly dic h ons of bounday dominad flow, and in fac, suggss a much lag dainag aa han is acually sn. Boh los mach h alyim ansins asonably wll (ach is subjcd o h sam alyim avaging o viously mniond). Figus 5b(i) and 5b(ii) show ha und abu a changs, boh diagnosic los xhibi a asonabl mach of q/y. Howv, h maialbalancim lo divaiv (5b(i)) suggss mau bounday flow, whil h logaihmic suosiion im lo divaiv shows h ons of bounday dominad flow occuing oo la. Fild Examls Th a many xamls of al oducion daa ha xhibi boh smooh and disconinuous vaiaions in a and flowing ssu ofils. Th disconinuiis can occu wih a wid ang of fquncis and amliuds. Fo xaml, high fquncy a changs (oducion nois ) may b causd by wllbo and/o sufac dynamics. In conas, low fquncy s changs may occu as a sul of mchanical changs o h wll, o du o a shif in back ssu (fo xaml, imlmning comssion). Th daa a analyzd using an AgalGadn diagnosic lo, which machs h nomalizd a and (invs) ssu divaiv (smoohd using h ssu ingal mhod) o consan a ycuvs basd on dimnsionlss im. Th diagnosic lo uss maial balanc sudoim. To s h validiy of h diagnosic lo, h daa a also hisoy machd using an analyical modl wih igoous suosiion. Examl : Smoohly Vaying Poducion aa wih S Chang (long m oducion) Examl is a hydaulically facud wll wih aoximaly.5 yas of smoohly dclining flow as and boomhol ssus, bu wih a faily abu shif in a na h nd of h fis ya. Figu 6a shows h 5

6 oducion / ssu hisoy fo h wll. Figu 6b is h diagnosic lo and Figu 6c is h hisoy mach using h analyical modl. Th diagnosic lo shows a asonabl mach o on of h facu ycuvs, wih a ansiion ino bounday dominad flow. Th ycuv mach indicas a facu halfgh in h od of 5 ms (assuming infini conduciviy). Th oimum hisoy mach is obaind using a facu halfgh of 60 ms. Th 5% diffnc is likly a sul of h maialbalancim o discussd in h hoy scion. Boh analyss yild an OGIP of aoximaly m, indicaing ha h diagnosic lo aas o dic h ons of bounday dominad flow cocly. Examl : Smoohly Vaying Poducion aa wih S Changs (sho m oducion) Examl conains houly oducion and ssu daa fo a iod of aoximaly monhs, duing which h a sval s changs in ssu. Th wll is no hydaulically facud. Figu 7a shows h oducion hisoy. Figu 7b is h diagnosic lo. Figu 7c is h modl hisoy mach. Th diagnosic lo indicas adial flow, wih a slighly ngaiv skin, followd by a ansiion ino bounday dominad flow in h laim. Th OGIP is simad o b in h od of m. Th modl hisoy mach ags vy wll wih h diagnosic lo, indicaing ha h sviy of h a disconinuiis in his cas is no nough o caus onial fo misinaion. I should b nod ha h laim daa dos no show a wll dvlod bounday dominad nd (his is vidn on h diagnosic lo). Consqunly, h modl hisoy mach is nonuniqu in dmining OGIP. Nvhlss, h modl (cangula singl lay) quis h snc of a las h boundais o adqualy mach h laim ssu bhavio. I is also insing o no ha a diagnosic lo using logaihmic suosiion im (no shown) shows a simila aly im adial flow iod, followd by a ansiion iod in h laim. Howv, h ansiion iod obsvd on h logaihmic suosiion lo suggss a much lag ( ims) minimum OGIP (a PSS (sudosadysa) slo of on h divaiv lo is no qui achd on h logaihmic suosiion diagnosic lo). Fo his cas, i aas ha h diagnosic lo cad using maialbalancim is sufficinly suid fo aoxima svoi chaacizaion and o idnificaion of flow gims. Examl : Noisy Poducion aa (sho m oducion) Examl has fiv monhs of daily oducion and ssu daa. Th wll has bn hydaulically facud. Th oaional condiions fo his wll a such ha h is a ga dal of nois (high fquncy and amliud a changs). This xaml could b considd h fild quiva of synhic cas b (o b). Th inu and suls a shown in Figus 8a, 8b and 8c. Th diagnosic lo (8b) indicas facu lina flow, followd by a laim ansiion o bounday dominad flow. Th lo suggss a faily wll dvlod bounday dominad nd wih an OGIP in h od of m. Howv, h ssu hisoy mach fom h analyical modl (8c) shows ha a saisfacoy mach can b obaind assuming an unboundd svoi. Th ansin analysis (k and x f ) obaind fom h diagnosic lo ags asonably wll wih h ssu hisoy mach. As xcd, h logaihmic suosiion diagnosic lo (no shown) also indicas infini acing flow fo h ni oducion iod. ISCUSSION OF RESULTS Ovall, h sudy has shown ha a diagnosic lo ha uss maialbalancim is usually adqua fo flow gim idnificaion and svoi chaacizaion, vn aly in h oducion lif, whn bounday flow has no bn achd. Th comaison of consan ssu (using maialbalancim) and consan a soluions indicas ha h xcd o is usually minimal. Mo imoanly, h soluions a sn o convg wih im. (Fo insanc, a hydaulically facud wll may hav a % o duing alyim lina flow ha diminishs significanly duing sudoadial flow, and finally convgs o h u soluion duing bounday dominad flow). Th mahmaical comaisons a only valid fo 6

7 consan boomhol ssu condiions (smoohly vaying as), bu also hav acical alicaion fo smoohly dclining as and ssus, and s a changs. Ral oducion daa is aly wihou som dg of disconinuiy (whh is high fquncy nois o occasional s changs in a/ssu). Th synhic and fild xamls hav shown ha sv flucuaions in a/ssu can sn h ossibiliy fo misinaion of flow gims. This is in addiion o h xising ansin o associad wih maialbalancim. Th mos sv xamls a causd by a combinaion of h following wo condiions: High fquncy and high amliud nois Low mabiliy svoi / sho oducing im Ths condiions caus h diagnosic lo o show a mau bounday dominad nd. Thus, h maialbalancim diagnosic lo will almos alys b consvaiv in is simaion of dainag aa and gasinlac. A diagnosic lo using logaihmic suosiion im also ovids significan onial fo misinaion of flow gims. I nds o ovsima h im o ansiion ino bounday dominad flow, vn whn disconinuiis in h a ofil a absn. Indd, his y of lo is fa b suid fo analyzing infini acing flow. CONCLUSIONS ) A diagnosic lo ha uss maialbalancim can b usd wih confidnc in idnificaion of flow gims, ovidd ha h a and ssu vaiaion is smooh wih im. ) Th abov mniond diagnosic lo (smoohly vaying a) can also b usd o quanify svoi ois (mabiliy and skin), bu has alyim os associad wih i. In mos cass, h magniud of h os will b ngligibl in comaison o h soluion of h oducion daa. Nvhlss, o obain a mo accua and coml svoi chaacizaion, h analys should us a modl wih igoous im suosiion, o fin h simas obaind fom h diagnosic lo. (Th diagnosic lo is sufficin fo obaining a ough aoximaion of ansin aams.) Th calculaion of fluidsinlac is igoous fo h diagnosic lo. ) Th is h onial fo misinaion of flow gims whn abu a / ssu flucuaions occu in combinaion wih ansin daa, if maialbalancim is usd. Und hs condiions, h analys should hav accss o boh diagnosic los (maialbalancim basd and logaihmic suosiion im basd). Th maialbalancim lo will nd o unddic h im of h ons of bounday dominad flow, whil h logaihmic suosiion im lo will nd o ovdic i. ) Fo oducion daa ha has sv a / ssu flucuaions, nih h logaihmic suosiion im lo no h maialbalancim lo ovid maningful inaions of ih svoi aams o flow gim idnificaion. NOMENCLATURE h n ay (m) k mabiliy (m) k f w facu conduciviy*widh (m.m) i iniial shuin ssu (kpa) q dimnsionlss a Q dimnsionlss cumulaiv oducion svoi adius (m) aan wllbo adius (m) s wllbo skin a sudoim (days, hous) dimnsionlss im A dimnsionlss im basd on aa dimnsionlss im fom consan ssu soluion dimnsionlss im fom consan a soluion mb dimnsionlss maialbalancim mba dimnsionlss maialbalancim basd on aa mb maialbalancim (days, hous) x f facu halfgh (m) 7

8 REFERENCES. Blasingam, T.A, McCay, T.L, L, W.J: clin Cuv Analysis fo Vaiabl Pssu o/vaiabl Flo Sysms, a SPE 5 snd a h SPE Gas Tchnology Symosium, Januay, 99. Agal, R.G, Gadn,.C, Klinsib, S.W, and Fussll,..: Analyzing Wll Poducion aa Using Combind Ty Cuv and clin Cuv Concs, a SPE 5796 snd a h 998 SPE Annual Tchnical Confnc and Exhibiion, Nw Olans, 70 Smb.. Po J., B..: Effciv Wll and Rsvoi Evaluaion Wihou h Nd fo Wll Pssu Hisoy a SPE 7769 snd a h SPE Annual Confnc and Tchnical Exhibiion, Ocob, 00. Engy Rsoucs Consvaion Boad: Gas Wllsing, Thoy and Pacic, xbook, fouh diion, Eddson, M.J, Gin, H.M, Pakison, H.R, Williams, C., Mahws, C.S: "Calculaion of Fomaion Tmau isubancs Causd by Mud Ciculaion," a SPE snd a h 6 h Annual Fall Ming of SPE, allas, 8 Ocob, EhligEconomids, C., Ramy J., H.: Tansin Ra clin Analysis fo Wlls Poducd a Consan Pssu, a SPE 887 snd a h Annual Fall Tchnical Confnc and Exhibiion of SPE, Smb, 979 8

9 9 APPENIX A Th following divaion ovs ha maialbalancim is a igoous suosiion funcion fo bounday dominad flow: Th consan a soluion fo a vical wll in a cylindical svoi, duing sudosady sa is as follows: Th consan ssu soluion fo h sam condiions as abov is as follows: 0. > A A q Th cumulaiv oducion is calculad as follows: 0 A d Q Ú Î È Î È 0 Wh, A Maialbalancim is now calculad as follows: Raanging h abov, o solv fo, w g Thus, Subsiuing h abov, back ino h consan ssu quaion, w g Î È + mba q Î È + mba Î È + mba c A k and q B kh wh A wf i A A fm m ) ( 0. > + Î È mb q Q Î È + mb Î È + mba A

10 Th invs of h abov is : mba + Thus, w ov ha consan a and consan ssu a quiva und bounday dominad condiions, whn maialbalancim is usd. 0

11 Figu a: Comaison of Consan Pssu and Consan Ra Soluions Facu Lina Flow Figu b: Comaison of Consan Pssu and Consan Ra Soluions Cylindical Rsvoi wih Vical Wll in Cn Raio of o is consan wih im (.6) Raio of o q and / 00 Consan Ra q and / Consan Ra Bginning of "smilog" adial flow (5) 0.00 Raio.6 0 Consan Pssu Consan Pssu E+07 E+08 E+09 E+0 E+ E+ E+ E Figu a: Comaison of Consan Pssu (Maial Balanc Tim Cocd) and Consan Ra Soluions Facu Lina Flow Figu b: Comaison of Consan Pssu (Maial Balanc Tim Cocd) and Consan Ra Soluions Cylindical Rsvoi wih Vical Wll in Cn Raio of o mb 000 Raio of o mb is consan wih im (.) 0 q and / 00 q and / Raio.7 ( 5) singl lin duing bounday dominad flow E+06 E+07 E+08 E+09 E+0 E+ E+ E+ E+

12 Figu a: Cas a Vical Wll in Infini Rsvoi Smooh Ra Pofil (consan ssu oducion) iagnosic Plos Using Radial Suosiion Tim and Maial Balanc Tim Figu b: Cas b Vical Wll in Infini Rsvoi isconinuous Ra Pofil iagnosic Plos Using Radial Suosiion Tim and Maial Balanc Tim Ealyim o is du avaging of q (a) ov h fis ims q/y, ER q/y, inv(er) Using mb causs fals bounday dominad flow nd in h la im suosiion im, maial balanc im mb q/d mb d su q/d su d consan a q/d consan a d suosiion im, maial balanc im mb q/d mb d su q/d su d Cons a q/d Cons a d Figu a(i): Cas a Facud Wll in Infini Rsvoi Smooh Ra Pofil (consan ssu oducion) iagnosic Plo Using Maial Balanc Tim Figu a(ii): Cas a Facud Wll in Infini Rsvoi Smooh Ra Pofil (consan ssu oducion) iagnosic Plos Using Radial Suosiion Tim q/y, ER q/y, ER xf 0 m xf 80 m maial balanc im mbq/d mb d consan a q/d consan a d suosiion im su q/d su d consan a q/d consan a d

13 Figu b(i): Cas b Facud Wll in Infini Rsvoi isconinuous Ra Pofil iagnosic Plo Using Maial Balanc Tim Figu b(ii): Cas b Facud Wll in Infini Rsvoi isconinuous Ra Pofil iagnosic Plos Using Radial Suosiion Tim q/y, ER q/y, ER Using mb causs fals bounday dominad flow nd; lina flow no vidn fom analysis Radial suosiion im dos no yild a usful diagnosic lo maial balanc im mb d/q mb d consan a q/d consan a d suosiion im su d/q su d consan a q/d consan a d Figu 5a(i): Cas a Facud Wll in Boundd Rsvoi Smooh Ra Pofil (consan ssu oducion) iagnosic Plo Using Maial Balanc Tim Figu 5a(ii): Cas a Facud Wll in Boundd Rsvoi Smooh Ra Pofil (consan ssu oducion) iagnosic Plo Using Radial Suosiion Tim q/y, ER q/y, ER Mb diagnosic lo cocly dics h ons of bounday dominad flow Radial suosiion im diagnosic lo ovdics h im of ansiion o bounday dominad flow maial balanc im mb d/q mb d consan a q/d consan a d suosiion im su d/q su d consan a q/d consan a d

14 Figu 5b(i): Cas b Facud Wll in Boundd Rsvoi isconinuous Ra Pofil iagnosic Plo Using Maial Balanc Tim Figu 5b(ii): Cas b Facud Wll in Boundd Rsvoi isconinuous Ra Pofil iagnosic Plo Using Radial Suosiion Tim Mb diagnosic lo dos no hav nough chaac o idnify flow gims; howv, h q/d daa machs h consan a cas vy wll q/y, ER q/y, ER Radial suosiion im diagnosic lo ovdics h im of ansiion o bounday dominad flow; alyim daa machs consan a cas vy wll maial balanc im mb q/d mb d consan a q/d consan a d suosiion im su d/q su d consan a q/d consan a d Figu 6a: Fild Examl Poducion Hisoy Figu 6b: Fild Examl iagnosic Plo (Hydaulic Facu Tycuvs) k 0.5 m xf 7 m OGIP m

15 Figu 6c: Fild Examl Modl Hisoy Mach Figu 7a: Fild Examl Poducion Hisoy k 0. m xf 60 m OGIP m Figu 7b: Fild Examl iagnosic Plo (Radial Flow Tycuvs) Figu 7c: Fild Examl Modl Hisoy Mach k0.6 m s.7 OGIP m k0.5 m s.5 OGIP m 5

16 Figu 8a: Fild Examl Poducion Hisoy Figu 8b: Fild Examl iagnosic Plo k 0. m x f 5 m OGIP m Figu 8c: Fild Examl Modl Hisoy Mach k 0.7 m x f 5 m Unboundd Rsvoi 6