Estimation of Aquifer Parameters Using the Numerical Inversion of Laplace Transform Noaman A.F. El-Khatib, SPE, King Saud University

Transcription

1 SPE 848 Emaon of qufe Paamee Un he umeal nveon of Laplae Tanfom oaman.f. El-hab, SPE, n Saud Unvey Copyh, Soey of Peoleum Ennee n. Th pape wa pepaed fo peenaon a he SPE h Mddle Ea Ol Show & Confeene o be held n Bahan 5-8 pl. Th pape wa eleed fo peenaon by an SPE Poam Commee follown evew of nfomaon onaned n an aba ubmed by he auho. Conen of he pape, a peened, have no been evewed by he Soey of Peoleum Ennee and ae ube o oeon by he auho. The maeal, a peened, doe no neealy efle any poon of he Soey of Peoleum Ennee, offe, o membe. Pape peened a SPE meen ae ube o publaon evew by Edoal Commee of he Soey of Peoleum Ennee. Eleon epoduon, dbuon, o oae of any pa of h pape fo ommeal pupoe whou he wen onen of he Soey of Peoleum Ennee pohbed. Pemon o epodue n pn eed o an aba of no moe han wod; lluaon may no be oped. The aba mu onan onpuou anowledmen of whee and by whom he pape wa peened. We Lbaan, SPE, P.O. Box 886, hadon, TX , U.S.., fax ba new mehod peened fo he emaon of paamee fo a ula aqufe by nonlnea eeon analy un numeal nveon of Laplae anfom. The paamee emaed ae he elave aqufe e e, he oavy hφc and he anmbly h/µ. Thee paamee ae neeay o alulae wae nflux needed n pefomane pedon of ol and/o a eevo by maeal balane baed mehod. Wae nflux daa ae fed o he van-evednen and Hu VEH uneady ae oluon o oban he equed aqufe paamee by nonlnea eeon analy un he mehod of lea quae. Beaue he oluon n Laplae pae mple han he oluon n he eal me doman, numeal nveon of Laplae anfom wa ued o oban he paal devave of he VEH oluon wh epe o aqufe paamee needed fo lea quae mehod. The Levenbe mehod wa ued fo paamee emaon o uaanee onveene. Th poedue poved o be effen and fee of ompuaonal and onveene poblem enouneed when un eal me oluon. Two appoahe ae ued o epeen he vaable peue hoy: he ep peue SP and he lnea peue LP mehod.the wo appoahe ae ued o eneaed wae nflux daa and he value obaned by boh mehod ae ompaed wh he aual aumed daa. The LP mehod found o yeld moe auae eul and ued n he paamee emaon alohm. The developed alohm an be appled fo pefomane pedon of ol and a eevo unde wae dve and fo he mulaneou emaon of onal hydoabon n plae OHP and aqufe paamee baed on maeal balane equaon. noduon The emaon of nal hydoabon ol and/o a n plae OHP of ea mpoane fo fuue developmen of hee eevo. Volume mehod baed on eoloal and em.daa an be ued a ealy ae of developmen. Wh a eaonable emae of OHP, he maeal balane equaon MBE an be ued o ped fuue eevo pefomane. f enouh poduon daa ae avalable fo a ven eevo, he MBE an be ued o emae he OHP. Fo volume eevo no wae nflux, he MBE lnea he paamee and G. n h ae, he MBE epeen an equaon of a plane. Havlena and Odeh howed how an be aaned a an equaon of a ah lne by oupn poduon and peue dependen em. Tehan, howeve, ndaed ha eeon hould be pefomed on he onal non-ouped MBE o peeve he phyal meann of eeon vaable. n hee ae, lnea o mulple eeon analy by he mehod of lea quae ued o emae he onal ol n plae and he a ap ao m fo ol eevo o he onal a n plae G fo a eevo. Fo non-volume wae dve eevo he maeal balane equaon an be ued o emae boh OHP and aqufe paamee. n aqufe model debn wae nflux fom he aqufe no he eevo needed. n mo feld ae uh model nonlnea. The Van-Evednen and Hu VEH uneady ae model an exa analyal oluon fo ula aqufe wh homoeneou popee. The model lnea wh epe o he wae nflux onan B, bu nonlnea wh epe he dmenonle aqufe e e and me adumen fao whh anfom eal me no dmenonle me. heefoe obvou ha lnea eeon an no be ued dely o emae boh OHP and aqufe paamee B,, and e. To oveome he nonlneay poblem, mo nveao ued ome nd of a al and eo appoah. n uh ae value fo aqufe paamee ae aumed and lnea eeon pefomed o emae and m fo ol o G fo a eevo. The andad devaon o he um of quae of eo alulaed. The value of aqufe paamee ae haned and he poe epeaed and he value of paamee ha yeld he malle um of quae of eo ae eleed. Mo nveao alo ued he abulaed value of ven by Van-Evednen and Hu o alulae he wae nflux. ouhey e al. 4 ued abulaed value of he oluon obaned by Waenbae and amey 5 o e he eo em. By pefomn dffeen un whn he lm of paamee, hey expeed he um of quae of eo a a eond ode

2 SPE 848 polynomal n he paamee and pefomed he mnmaon o oban he opmum value of he paamee. Chen, Chen and ln 6 ued polynomal appoxmaon of by ln e al. 7 o evaluae he wae nflux and ued he mplex eah mehod fo paamee emaon. dffeen appoah ued by ohe nveao 8-9 he aqufe nfluene funon F whh he eevo peue epone o a un ae of wae nflux. dely popoonal o he dmenonle peue oluon fo onan ae. howeve eaed a a eneal funon eadle of aqufe eomey and homoeney. an hu povde a funon fo eman wae nflux fom peue hoy. Howeve, he exapolaon oude he poduon daa ane needed fo pedon of fuue pefomane. Th exapolaon queonable unle he F develop a lea end whn he avalable me whh hould be lae enouh o eah he end. ffeen ype of aqufe develop dffeen end and o nowlede of he hape of he aqufe needed whh lm he eneal applably of h appoah ne mean o handle ae of unnown o eula hape. Chaa and Malefan popoed he applaon of nonlnea eeon o he Van-Evednen and Hu oluon n eal pae o emae aqufe paamee. They oulned he poedue and evaluaed he paal devave bu dd no peen aual oluon. The VEH oluon n eal me pae vey omplex and ompuaonal and onveene poblem may be enouneed. The oluon n Laplae pae mple and he ue of he numeal nveon of he Laplae anfom mae eae o evaluae he f and eond devave wh epe o he aqufe paamee. Th appoah wll be appled n h pape. Theoal Condeaon The onvenonal fom of he MBE an be follown n he follown fom p β β Gpβ Wpβ w o βo βo β G β β We... Equaon an be ued fo undeauaed ol eevo wh G o fo a eevo wh. n mple model, he wae nflux fom he aqufe no he eevo eaed un Shlhu eady ae model. We p p d Th mae he maeal balane equaon lnea n he paamee, G and an he follown fom. y X G X X.. Equaon epeen a hype plane and mulple eeon analy an be ued o emae he hee paamee, G and un poduon and PVT daa. The eady ae wae nflux model howeve aely debe he aual behavo of aqufe. n exa mahemaal oluon of he dffuvy equaon ven by van Evednen and Hu fo a adal flow yem a onan emnal peue. The oluon ven by We B p 4 whee fo We n Bbl, h and w n f, n md and n day B.9 hϕ C w µ Cϕ w The dmenonle funon ven by e n α n J α n e J α J α α n e n n e... 7 whee α n ae oo of he equaon J α n e Y α n -Y α n e J α n.. 8 Fo vaable peue ae, he pnple of upepoon onvoluon appled o emae he umulave wae nflux no he eevo. The oluon ven by We B τ P τ dτ.. 9 qufe Paamee Fom Eq. 4-7, he paamee needed o evaluae he wae nflux We a a ven me ae he aqufe onan B, he elave aqufe adu e e / w and he onan n Eq. 6 whh anfom eal me no dmenonle me. The paamee B and an be expeed n em of he ommonly ued paamee of anmbly T h/µ and oavy S hφ a follow B.9hφ C.9S... w w T µφ Cw Sw So he anmbly T and oavy S an be obaned fom paamee B and C a follow T 4. B.... B S w Moeove, he paamee B elaed o he eevo poe volume V p a follow B C V p.. 4 Sll elaed he paamee B o he onal ol n plae βo B CV p C S w Smla elaon an alo be wen fo he ae of a eevo o ol eevo wh a a ap.th elaon edue he numbe of paamee o be emaed by one paamee by man B and, G, o m dependen. lhouh h may ound aave mahemaally, phyal valdy queonable. The em V p n Eq. 4, 5 epeen all he poe volume peen n he eevo whn he adud w and hne h nludn any hale and nononneed poe volume. The em n he maeal balane equaon

3 SPE 848 epeen he ol ubeed o expanon due o peue hane whh nlude only ol peen n neonneed poe pae. heefoe mpoan o have B and o G a wo ndependen vaable when applyn eeon analy fo paamee emaon o he maeal balane equaon. umeal nveon of Laplae Tanfom The omplexy of un equaon 7 n ompun he wae nflux We appaen. F Eq. 8 mu be olved eavely fo enouh numbe of ueve oo α n. The ummaon n Eq.7 o be onnued unl onveene of he nfne ee aheved. The poblem fuhe omplaed n he paamee emaon poblem whee n addon o evaluan We, he devave of mu alo be evaluaed. Thee dffule pomp ed he nveaon of he pobly of pefomn evaluaon and opmaon n Laplae pae un he Sehfe alohm fo he numeal nveon of Laplae anfom. n Laplae pae, he expeon fo he anfom of he dmenonle wae nflux / The nvee of e e e e. 6 by he ehfe alohm ln ln l V 7 To e he valdy of Eq.7, value of wee alulaed fo value of e of,, 4, 5 and a dffeen value of. The eul ae hown n Table and ploed n F..Compaon wh able ven by van-evednen and Hu how vey ood aeemen o whn %. nown ha he maxmum ulmae value of.5 e. Thee value ae.5, 4, 7.5,, 49.5fo he epoed e value epevely. Value n able how an ollaon abou hee value of ±.5 a lae value of. Calulaon of We Un SP and LP mehod Fo vaable eevo peue,he peue hoy appoxmaed no a numbe of onan peue ep wh donnuou ump a he daa pon a hown n f.. The neaon n Eq. 9 hen appoxmaed by a ummaon a follow We B P 8 Th mehod alled he ep peue SP mehod. Table : Geneaed m. Wae nflux m. m. Wae nflux me e e e 4 e 5 e m. Wae nflux e 5 E- E- E E E m enonl e Tm e F. : menonle Wae nflux fo Lmed qufe Vo and Wan ueed appoxman he peue behavo by a ee of lnea emen onnen ueve daa pon. Th expeed o ve a moe auae epeenaon of he peue hoy. The ba fo h mehod o eplae P n Eq. 9 by he lope m and nean by pa o oban dm We B τ τ dτ.. 9 dτ whee τ dτ.. Equaon 9 hen appoxmaed by a ummaon o We B m 4

4 4 SPE 848 Th mehod alled h lnea peue LP mehod. o be noed ha he lope m alulaed w.. he dmenonle me n Laplae pae, un ule fo anfom of he neal, l. whee ven by Eq. 6 To e and ompae he SP and LP mehod, he peue dop wa eneaed fo he ae of a onan wae nflux ae.e.we q wh q onan. The follown daa wee ued o eneae he peue behavo: Pemeably md. Thne h f. Pooy φ % Voy µ p. eevo adu w f qfe ad. ao e 5. Compebly C 5 E -6 p - Wae nflux ae e. Bbl/day Fom h daa, he follown value ae obaned Tanmbly T h/µ md. f./p. Soavy S hφ E -5 f. / p B.9 w.8 Bbl/p.64 h -.66 day - Tme day Table : Geneaed aa fo Conan ae P p We e. Bbl LP SP Tue E5 The dmenonle peue obaned fom van Evednen and Hu oluon fo he ae of a loed fne eevo wh onan emnal ae P w P w / e e e e.. l P 4 w Wae nflux We Bbl 8E4 6E4 4E4 E4 We P Peue op P P.eul fo he ven eevo and flud daa ae evaluaed un he numeal nveon alohm of Laplae anfom and ompaed wh hoe ven n able of efeene. The aual peue dop obaned fom he dmenonle peue un he elaon T p P w 4. q The obaned peue dop v. me hown n f.. The peue ep fo he SP mehod and he lnea emen fo he LP mehod ae hown n F,. Wae nflux wa alulaed by he wo mehod un Eq. 8 and and evaluan and by he ehfe alohm applyn Eq. 6, 7 and. The eul ae ven n Table and hown n F. 4 oehe wh he aual aumed lnea daa. The abolue eo n We fo boh mehod ae hown n F. 5. The eul ndae ha he LP mehod moe auae han he SP mehod and wll be ued exluvely fo paamee emaon. Peue op P P E Tme day 5 4 F. : Peue op fo Conan Wae nflux ae Tme day F. : Peue ppoxmaon by SP & LP Mehod

5 SPE Calulaed Wae nflux We Bbl bolue Eo n We BBl E5 E4 E LP SP eal E E E E 4 E Obeved Wae nflux Bbl F. 4: Geneaed Wae nflux By SP & LP Mehod Tme day F. 5 : bolue Eo n We by SP and LP Mehod qufe Paamee Emaon The maeal balane equaon an be wen n he follown fom β β G β W β p o LP β β β o o p ln ln B P V.. 6 onlnea eeon an be appled o Eq. 6 fo he mulaneou emaon of eevo onal hydoabon n plae and G n addon o he aqufe paamee B, and e. Howeve, he obeve of h wo o e he poedue of pefomn eeon analy un he numeal nveon of Laplae anfom of he dmenonle wae nflux funon, Theefo, we lmed h wo o he emaon of he aqufe paamee fom We daa. The wae nflux We an be alulaed fom poduon and PVT daa un Eq. f auae emae of and G ae avalable. lo, We an be alulaed f an emae of he volume of he nvaded one, V Pnv, obaned fom he p w SP G β β movemen of he wae/ol ona and uual and pooy map of he eevo un he equaon We W P w V P nv Sw So β ϕ.. 7 Gven value of wae nflux We and aveae peue dop n he eevo P a dffeen me, he mehod of lea quae appled o emae aqufe paamee. The mehod ued o fnd he value of he hee aqufe paamee B, C and e whh mnme he um of quae of he dffeene beween obeved wae nflux value and hoe alulaed un he hee paamee fo all daa pon. So equed o mnme he follown obeve funon ln ln SSE We B m V... 8 n h equaon he lope m evaluaed w.. he eal me whh me he lope w.. he dmenonle me.th equaon an be wen a SSE B,, e 9 whee he edual eo ven by We B m ln V ln The mnamly ondon fo he obeve funon ae: SSE θ θ. Whee θ he veo of paamee B,, and e The e of Equaon ae he nomal equaon of he yem and mu be olved mulaneouly fo paamee θ. Sne he equaon ae nonlnea, an eave poedue mu be ued. The applaon of he mul-vaable ewon apon mehod o he nomal equaon eul n he well nown Gau mehod of opmaon. The oluon a any eave ep ven n a max fom a Hδθ.. Whee H ae he elemen of he Hean max whh ae he paal devave of he nomal funon wh epe o paamee θ and he veo δθ he hane n he value of he paamee n he eaon ep. Fom Eq, H θ θ θ θθ... Expeon fo he elemen of he max H and veo ae deved n he ppendx. The Gau mehod ue he wo em n Eq. fo he Hean max and hu eque evaluaon of eond devave. modfed fom whh doe no eque evaluaon of he eond devave he Gau-ewon mehod. n h mehod, only he f em ued whh nvolve podu of f devave only. The ufaon fo doppn he eond em n he Hean max ha he eond devave n ha em ae mulpled by he edual whh uppoed o be vey mall loe o he oluon.

6 6 SPE 848 Th howeve mh no be he ae a pon fa fom he oluon. n h wo we wll onde he full fom of he Hean max ne eond devave an be evaluaed analyally when he Sehfe alohm ued fo he numeal nveon of Laplae anfom a hown n he appendx. lhouh nown ha he Gau mehod and he modfed Gau-ewon mehod have quada onveene nea he oluon, may dvee f he nal ue fa fom he oluon. On he ohe hand, he mehod of eepe deen uaanee deean he obeve funon n eah eaon bu onveene low. Mehod peened by Levenbe and Maquad ae ombnaon of Gau and he eepe deen mehod. To uanee ha he obeve funon deean, he Hean max H mu be pove defne. an be made o by addn lae pove numbe o daonal elemen. Thu, he max H n Eq. eplaed by HλV whee λ a pove numbe and V a daonal max. Levenbe 4 oo V a he deny max whle a bee hoe ueed by Maquad 5 o ae he daonal elemen of V equal o he abolue value of he daonal elemen of H. So, he yem of equaon o be olved a eah eaon beome H λ V δθ 4 vey lae value of λ λ equvalen o he eepe deen mehod whle λ epeen o he Gau mehod. The yem of he hee lnea equaon epeened by Eq. 4 an be olved by he Gau elmnaon mehod o by Came ule. The eaon onnued unl a onveene eon aheved. Ehe he um of quae of eo SSE o, he nom of he veo, ued... 5 nveaon of Eq. 9- how ha boh expeon nlude edual of he wae nflux em a he daa pon,.e. he dffeene beween obeved and alulaed value. Sne wae nflux value un no mllon of bael, he manude of SSE o, would be n he ode of. E8 fo a % elave dffeene of he edual. To oveome h poblem, he edual ae nomaled by dvdn eah em by he wae nflux a ha pon, We. elave dffeene of % n he edal wll eul n a um of quae of.e-4 mulpled by he numbe of daa pon. onveene eon of SSE.E-5 amoun o a elave eo of.% fo a un of daa pon. eul and uon The daa eneaed fo onan wae nflux ae ued o e he popoed mehod. ompue poam wen un he developed poedue. The mnmaon wea pefomed on he nomaled um of quae of edual. The expeon fo n Eq. wa dvded by We. The aumed value fo he aqufe paamee e, B, and wee 5,.8 e. Bbl/p, and.66 day - epevely. To e he pefomane of he alohm, wo value fo eah paamee wee ued, one oo mall and he ohe oo lae ompaed o he aual value. ll 8 ombnaon of he value wee ued a an pon. The eul of hee un ae hown n Table. een fom he eul ha all un onveed o value vey loe o he aual value le han.5% fo and.5% fo boh b and e. Table 4 how he eul fo one of he un. een ha he abolue peenae eo n he value of he wae nflux We ane beween a mnmum value of abou.% o a maxmum value of abou. %. Wh he value of SEE of.88 E-5 fo daa pon, he aveae abolue elave eo.%. The value of anmbly T and oavy S alulaed fom B and n Table 4 un Eq. and ae md.f/p and. x -5 f/p a ompaed o he aual value of md.f/p and. x -5 f/p epevely. The eo.7% n T and.5% n S. eul of able how ha he mehod onvee alway o he oe value of aqufe paamee eadle of he value a nal ue. o nal ue onveed o a dffeen oluon and o a unque oluon wa obaned. The nal ue pon wee hooen o ove a wde ane of paamee value and epeen he one of a eanula paallelepped aound he oe value. heefoe expeed ha f a faly eaonable emae of he aqufe paamee an be made fom. o 5 of he oe value, he poedue would onvee o he oe value. The aqufe onan B depend on he hne h, he pooy φ and he aqufe adu w. Boh h and φ an be eaonably emaed. value fo w.beween 5% and % of he u value an be emaed fom eoloal and eophyal daa wh eaonable eany. The me adumen fao depend on he pemeably, he voy µ, he pooy φ, and he ompebly C. ll hee popee an alo be emaed wh eaonable auay beween 5% and % of aual value. The ame an alo be aed fo he dmenonle aqufe adu e. f he nal ue wa made whn hee lm, he poedue hen expeed o onvee o he nehbohood of he ue value. Table : eul of eeon naly un un nal Gue Fnal eul o. e B C e B C SSE E E E E E E E E5

7 SPE Table 4: eul of eeon naly Value of Paamee eeon ual e B SSE.87667E E-9 Tme We We % b day Obeved Calulaed Eo Conluon The follown onluon an be made - mehod peened fo aqufe paamee emaon by he mehod of lea quae appled o uneady ae van Evednen and Hu oluon n Laplae doman. The paamee emaed ae he elave aqufe e, he aqufe onan B and he me adumen fao. The paamee B and an be ued o alulae he anmbly T and oavy S of he aqufe. - The Sehfe alohm fo numeal nveon of Laplae Tanfom wa ued o evaluae he wae nflux and oban analyal expeon fo he f and eond devave of he oluon wh epe o aqufe paamee. - The Levenbe-Maquad mehod wa ued n paamee emaon o avod dveene fo nal uee no loe o he oluon. The mehod aheved onveene o a unque oluon n he nehbohood of he eal paamee fo wdely epaaed an pon. o dveene o onveene o dffeen oluon wa obeved. 4- The peue hoy appoxmaed by a ee of lnea emen LP mehod ahe han he ale peue ep SP mehod. The Lp mehod wa foud o ve moe auae eul. 5- The developed poedue an be appled fo mulaneou emaon of onal hydoabon n plae OHP ol ab\nd/o a and aqufe paamee. omenlaue B aqufe onan, bbl/p m /Pa me adumen fao, day - - C oal fomaon ompebly, p - Pa - G nal a n plae, SCF m G P umulave a poduon, SCF m H Hean max l modfed Beel funon of f nd, ode l J l Beel funon of f nd, ode l l modfed Beel funon of eond nd, ode l abolue pemeably, md µm m lope of peue, p/day Pa/ nal ol n plae, STB m Sehfe alohm numbe p umulave ol poduon STB m oal numbe of daa pon P peue dop, p Pa q flow ae, bbl/d m / menonle wae nflux Laplae anfom of w eevo adu, f m e dmenonle aqufe adu a oulbly n ol, SCF/STB Laplae anfom paamee S oavy hφc, f.p - m Pa - Sw nal wae auaon, faon me, d dmenonle me T anmbly h/µ, md.f/p µm /Pa. We wae nflux, Bbl m Wp wae poduon, Bbl m Y l Beel funon of eond nd, ode l β O ol fomaon volume fao, e. Bbl/STB β a fomaon volume fao, e. F /SCF µ voy, p.pa. λ Levenbe paamee ϕ pooy, faon Subp a nal o ol w wae Supep Laplae anfom nowledemen The auho whe o anowlede he uppo povded by he Peoleum Enneen epamen of n Saud Unvey dun h udy. efeene. Havlena,.and.Odeh,.S.:"The Maeal Balane a an Equaon of Sah Lne," J. Pe. Teh. uu

8 8 SPE 848. Tehan,., H..: n naly of Volume Balane Equaon fo Calulaon of Ol n Plae and Wae nflux, J. Pe. Teh. Sep van Evednen,.F. and Hu,W.:"pplaon of he Laplae Tanfom o Flow Poblem n eevo," Tan. ME 94986, 5 4. ouhey, E.L., Zaala, B and hahah,.: " Genealed Mehod fo Man Opmal Emae of eevo-qufe Paamee," Tan. ME ,. 5. Waenbae,.. and amey, H.J., J.: "n nveaon of Wellboe Soae and Sn Effe n Uneady Lqud Flow:. Fne-ffene Teamen," SPEJ Sep Chen, T.L., Chen, Y.L. and Ln, Z.S.: "eemnaon of he Onal-Ga-n-Plae and qufe Popee n a Wae-ve eevo by Opmaon Tehnque," pape SPE 987 peened a SPE a Paf Ol & Ga Confeene, uala Lampu, Malaya, pl -, ln, M.., Bouhad,.J. and Cable, C.L.: " Polynomal ppoah o he van Evednen-Hu menonle Vaable fo Wae Enoahmen," SPEE Feb Vea, L. and Waenbae,..: "ew ppoah fo Smulaneou eemnaon of he OGPP and qufe Pefomane wh o Po nowlede of qufe Popee and Geomey," pape SPE 5978 peened a he SPE/CE Ga Tehnoloy Sympoum, Calay, lbea, Canada, pl Gada,.J., Waenbae,. and Saman,...: " ew Mehod of Mahn qufe Pefomane and eemnn Onal-Ga-n-Plae," SPEE uu Chaa,.T. and Malefam,H.: "The Emaon of qufe Popee fom eevo Pefomane n Wae-ve Feld." Pape SPE 97, unoled.. Sll, S..: "mpoved Maeal-Balane eeon naly fo Waedve Ol and Ga eevo," SPEE May Sehfe,, H.: "umeal nveon of Laplae Tanfom," Communaon of he CMJan. 97, o., 47. Vo, J.P. and Wan, B.: " Moe uae Wae nflux Fomula wh pplaon," Jou. Cnd. Pe. Teh., July-u Levenbe,.: " Mehod fo he Soluon of Cean onlnea Poblem n Lea Squae," ua. ppl. Mah.944, Maquad,.W.: "n lohm fo Lea Squae Emaon of onlnea Paamee," J. SM 96, 4. ppendx evaon of he Elemen of he Hean Max The um of quae of eo o be mnmed ln ln SSE We B m V..-.Whh may be wen a SEE B,, e. - whee he edual eo ven by ln ln We B m V - Fom Eq. -, he devave of he edual w,.. he paamee B, and e ae ven by ln ln m V.. -4 B e B m ln ln e V..-5 V ln { ln B m ln }.. -6 The eobd devave of he edual ae obaned by dffeenaon of Eq B ln ln m V.-8 B e e ln V m { }...-9 B ln V B m { }..- e e e B m ln ln B m e Fom Eq. and 6 Un V { 4 6 }.. - ln V - e e e e e. -

9 SPE ln whee e e e e.. -7 The paal devave of w... and e ae ven by e / 5/ e - / e e 5 - The paal devave of and wh epe o e and ae e e. - e e -4 e e e e e. -5. e e e e e The eond paal devave of and wh epe o e and ae e e e e e e e e e e e e e e e e e e e e.. -

10 SPE 848. e e e e e e e e e e e e e e Whee he devave of he Beel funon ae x x x x x x x x x x x x x x x x x x x x x x x x S Me Conveon Fao bbl x E m p x. * E Pa. f x.48 * E m f x * E m * Conveon fao exa.