( /100 pts) Total. Name: Key (TAB/DI ) Section: One staple or binder clip here.

Transcription

1 One ale o binde cli hee. Name: Key (TAB/I.. Peon on my lef i: Peon on my igh i: xam Sing Thuday Mach (:-: in RICH, and Oienaion: You ae icly fobidden o collaboae by any mean collaboaion/cheaing F coue gade. The exam i oen book/oen noe. Rule: (no exceion Sign he Hono Code Saemen NOW! Pu you name on evey age NOW!! You mu only ie on he fon of any given age! o no ue en! You mu ue you on aigh-edge(, iangle(, encil, eae, calculao, ec. You mu u you name and numbe he hee (age you ine ino he exam. You ae emied o go o he eoom, ake a beak, have food o dink HOWVR, you mu do hi alone uden obeved alking ill be aumed o be collaboaing, and hi ill be een a an ac of cholaic dihoney, and all hoe involved ill be given an F gade fo he exam. Scholaic dihoney don' even hink abou i you do no an o kno ha ide of me. Invenoy of Poblem: ( /. Coelaion of Peohyical aa k veu φ Coelaion ( / ( / ( / ( / ( / ( / ( / ( / ( / Toal. Skin Faco eivaion. Mulile-Rae adon Te Analyi. Peue Buildu Te Analyi. Veificaion of Conan Rae Soluion fo he iffuiviy uaion. Hyebolic Cumulaive Poducion-Rae Relaion. Radial Flo Reevoi Peue iibuion. Peue iibuion in a Muliell Syem. Analyi of Mulioin Ga Well Te aa Suggeed Wok Saegy: The oblem ih he lee oin ae "eaie" you may an o a ih hee. Poblem i "aighfoad". Poblem i he mo ediou (in em of comuaion. Poblem and ae no comlicaed bu do euie conideable ok. The ay i i: Give you be effo no exceion. Neane coun moe han you hink. Kee you ok comlee and NAT. Aggie Code of Hono: An Aggie doe no lie, chea, o eal o oleae hoe ho do. Reuied Academic Inegiy Saemen: (Texa A&M Univeiy Policy on Academic Inegiy "On my hono, a an Aggie, I have neihe given no eceived unauhoized aid on hi academic ok." (you ignaue

2 xam /Poblem (Coelaion of Peohyical aa Sing Thuday Mach (:-: in RICH, and (. Coelaion of Peohyical aa k veu φ Coelaion Name: Key (TAB/I.. Given: In hi oblem you ae given he meaued emeabiliy and ooiy coe daa aken fom a andone eevoi. Samle (φ Pooiy (facion (k Pemeabiliy (md Reuied: Fi he model belo o he dominan daa end on you log-log k veu φ daa lo given on he nex age. k αφ β... ( α x (md β. (dimenionle Fi he model belo o he dominan daa end on you emi-log k veu φ daa lo given on he nex age. k α ex( β φ... ( α x - (md β (dimenionle Commen on you eul hich model efom bee fo hi cae? Why?

3 xam /Poblem (Coelaion of Peohyical aa Sing Thuday Mach (:-: in RICH, and. Coelaion of Peohyical aa k veu φ Coelaion (Coninued Name: Key (TAB/I..

4 xam /Poblem (Coelaion of Peohyical aa Sing Thuday Mach (:-: in RICH, and. Coelaion of Peohyical aa k veu φ Coelaion (Coninued Name: Key (TAB/I..

5 xam /Poblem (Skin Faco eivaion Sing Thuday Mach (:-: in RICH, and (. Skin Faco eivaion Given: You ae given acy' La in diffeenial fom fo a finie (eady-ae adial eevoi yem: (acy uni πh d k( (liuid cae B and μ ae conan Bμ d k( indicae a adial diibuion of emeabiliy. Secific o hi cae, e have: fo < < : k( α β (aleed emeabiliy fo < < e : k( k (unaleed emeabiliy Name: Key (TAB/I.. Reuied: (You mu ho all ok fo cedi a. eive an exeion fo eevoi eue ( fo he enie eevoi. b. eive an exeion fo he kin faco a a efomance/efficiency index fo hi adial eevoi yem. c. xlain he behavio of he β-aamee fo: β < (damage o imulaion? β > (damage o imulaion? Reul: a. Reevoi eue exeion: b. Skin faco exeion: Bμ ln πh αβ β β k k ln αβ β β c. β-aamee < : Simulaion cae β-aamee > : amage cae

6 Name: Key (TAB/I.. xam /Poblem (Skin Faco eivaion Sing Thuday Mach (:-: in RICH, and. Skin Faco eivaion (Coninued Pa a. Reevoi eue exeion Saing: πh d k(... ( Bμ d Reaanging: πh d d... ( k( Bμ Recalling: k( α β ; fo < < k( k; fo < < e : Inegaing., e have: πh d d... ( k( Bμ Uing he oeie of inegaion, e ill ake he inegal in he inne zone ( << and in he oue zone ( <: πh d d k( k Bμ Taking he inegal of he fi em: d... ( ( β d d... ( β ( α α Comuing he inegal e have: β ( β d α... ( α β Comleing:... ( αβ β αβ β β The inegal of. i given a: πh ln (... ( αβ β k β Bμ Finally e obain he exeion fo he eevoi eue a: Bμ ln (... ( πh αβ k β β Reaanging, e have: Bμ ln... ( πh αβ β β k

7 Name: Key (TAB/I.. xam /Poblem (Skin Faco eivaion Sing Thuday Mach (:-: in RICH,, and. Skin Faco eivaion (Coninued Pa b. Skin faco exeion β β αβ π μ h B... ( β β αβ π μ h B... ( We noe ha i he ellboe floing eue ha exi becaue of damage o imulaion and ould be he eue ha e ould acually meaue in he ell. In ode o elae he damage/imulaion cae (. ih he undamaged/unimulaed cae e ecall. and inegae beeen << : d B h d k * μ π... ( Comleing he inegaion: Bμ πh k ( ln *... ( We define * a he "baeline" (i.e., no damage/imulaion cae ellboe eue. πkh Bμ ln *... ( We define he "kin zone eue do" a: ( * Δ... ( Subiuing. and ino give: Δ β β αβ π μ h B πkh Bμ ln... ( Facoing Δ k πkh Bμ ln β β αβ... ( A a efomance/efficiency index, e define he "kin faco" a: Δ k Bμ πkh ln β β αβ... (

8 (. Mulile-Rae adon Te Analyi xam /Poblem (Mulile-Rae adon Te Analyi Sing Thuday Mach (:-: in RICH, and Name: Key (TAB/I.. Given: You ae given a vaiable-ae eue dadon daa e fo a oducing oil ell. The "ime-eue-ae" daa ae abulaed belo. Poin, N Poducing Time, (hou Poducion Rae, (STB/ Floing Peue, f (ia Nomalized Peue, i - f / (i/stb/ Sueoiion Time, u Oienaion: The geneal euaion fo anien eue behavio caued by a vaiable floae i given belo: N i f ( m' b'... ( N N Whee Bμ m'. (loe... ( kh k b' m' log.. φμc (inece... ( Theefoe, mulile-ae e daa hould aea a a aigh line (fo he infinie-acing adial flo egime hen loed a: (Nomalized eue v. Sueoiion ime: i f ( N v. N... ( N I i imoan o undeand ha he ae coeonding o each eue oin i N he la ae ha can affec he eue. Noe ha a,. Fo examle, fo N; he coeonding ueoiion ime i calculaed a: N... (

9 . Mulile-Rae adon Te Analyi (Coninued Reuied: (You mu ho all ok fo cedi xam /Poblem (Mulile-Rae adon Te Analyi Sing Thuday Mach (:-: in RICH, and Name: Key (TAB/I... You ae o comue he nomalized eue and he ueoiion ime coeonding o each oin, and hen you ae o comlee he able given on he eviou age and lo you eul on he lo given in he nex age.. You ae o eimae he emeabiliy and he kin faco uing he loe and he inece of he aigh line ha you obain fom he lo.. The euied eevoi oeie ae given belo.. Fo efeence, he lo of he floing eue ( f veu he logaihm of ime ( i ovided belo. Reevoi oeie: Oil oeie: φ.. f i ia h f B. RB/STB μ. c c x - ia - f v Plo

10 . Mulile-Rae adon Te Analyi (Coninued xam /Poblem (Mulile-Rae adon Te Analyi Sing Thuday Mach (:-: in RICH, and Woking Plo Name: Key (TAB/I..

11 Name: Key (TAB/I.. xam /Poblem (Mulile-Rae adon Te Analyi Sing Thuday Mach (:-: in RICH, and. Mulile-Rae adon Te Analyi (Coninued Soluion:.. The comuaion of he nomalized eue i aighfoad. We ill comue he ueoiion ime coeonding o each eue oin. Fo N, N N. Fo N, N N. Fo N, N N. Fo N, N N. Fo N, N N.

12 Name: Key (TAB/I.. xam /Poblem (Mulile-Rae adon Te Analyi Sing Thuday Mach (:-: in RICH, and. Mulile-Rae adon Te Analyi (Coninued Fo N, N N. Fo N, N N. Fo N, N N. Fo N, N N.

13 xam /Poblem (Mulile-Rae adon Te Analyi Sing Thuday Mach (:-: in RICH, and. Mulile-Rae adon Te Analyi (Coninued Bμ (. RB/STB(. c k.. md m' h (. i/stb/ cycle( f b'. log. m' φμ k c (. ( md log.. (. (.(. c( i (. f Name: Key (TAB/I..

14 xam /Poblem (Peue Buildu Te Analyi Sing Thuday Mach (:-: in RICH, and Name: Key (TAB/I.. (. Peue Buildu Te Analyi Inoducion: RA THIS FIRST! Thi oblem coni of a eue buildu e efomed on an oil ell afe a mulile-ae e euence. Thee ae no "ick" involved hi hould be a aighfoad analyi/ineeaion. Be ue o efom all analye and label each even and co-check/double-check you ok heneve oible. Given: Thee aached daa ee aken fom a eue buildu e efomed on an oil eevoi he eevoi i aumed o be homogeneou. Reevoi oeie: φ.. f c.x - ia - i ia h f A ace (uae eevoi Oil oeie: B. RB/STB μ. c Poducion aamee: STB/ h f (Δ. ia Reuied: Peue Buildu Analyi you ae o eimae he folloing aamee: ( Pa a Middle Time Analyi (Pemeabiliy and Skin Faco Semilog (Hone Analyi: (IARF Infinie-Acing Radial Flo Fomaion emeabiliy, k. md Skin faco,. dimenionle ( Pa c Lae Time Analyi (Aveage Reevoi Peue Caeian Analyi: (Lae Time aa "Muka-A-Smih" Mehod Aveage eevoi eue, (if alicable. ia

15 . Peue Buildu Te Analyi (Coninued aa Funcion: Peue Buildu Cae xam /Poblem (Peue Buildu Te Analyi Sing Thuday Mach (:-: in RICH, and Δ, h, ia Δ, i Name: Key (TAB/I..

16 xam /Poblem (Peue Buildu Te Analyi Sing Thuday Mach (:-: in RICH, and. Peue Buildu Te Analyi (Coninued Pa a Middle Time Analyi (Pemeabiliy and Skin Faco Semilog (Hone Analyi: (IARF Infinie-Acing Radial Flo Fomaion emeabiliy, k. md Skin faco,. dimenionle Name: Key (TAB/I.. Pemeabiliy, k: Bμ ( STB/ (. RB/STB (. c k. (.. md mlh (. i/cycle ( f Skin Faco, : (,h f Δ k. log log. m l φμc ( ia (. ia ( h. log (. i/cycle ( h. (. md log. (.(. c (.x i (. f

17 . Peue Buildu Te Analyi (Coninued Pa b Lae Time Analyi (Aveage Reevoi Peue xam /Poblem (Peue Buildu Te Analyi Sing Thuday Mach (:-: in RICH, and Caeian Analyi: (Lae Time aa "Muka-A-Smih" Mehod Aveage eevoi eue, (if alicable. ia Name: Key (TAB/I.. Muka-A-Smih Mehod: The "Muka-A-Smih" mehod i a igoou elaion ued o eimae he aveage eevoi eue duing lae-ime eue buildu (if he bounday effec ae eablihed. The fundamenal elaion given by Muka i: a ex( bδ Lae, A and Smih efomulaed he "Muka elaion" ino a loing funcion a follo: d b Δ In he cae of he "Muka-A-Smih" mehod, i loed veu d /dδ o eimae he aveage eevoi eue a he inece of he aigh line end a d /dδ. Commen:

18 Name: Key (TAB/I.. xam /Poblem (Veificaion of Conan Rae Soluion fo he iffuiviy uaion Sing Thuday Mach (:-: in RICH,, and (. Veificaion of Conan Rae Soluion fo he iffuiviy uaion Given: In hi oblem you ae o veify he exonenial inegal ( (x (i.e., he full oluion by ubiuion ino he dimenionle diffuiviy euaion. imenionle iffeenial uaion: (Homogeneou Radial Flo Syem... ( xonenial Inegal Soluion (Full Radial Flo Soluion: e e (, ex... ( e Reuied: (You mu ho all ok fo cedi You ae o "validae" he oluion (. by ubiuion i.e., you ae o ubiue. ino. (he diffuiviy ideniy and eolve he calculu and algeba ino he loe oible fom. You ae o hen commen a o he "validiy" of.. If he oluion i exac, hen he lef and igh hand-ide ill cancel uon ubiuion. Oheie, he ooed oluion i eihe an aoximaion, o i no a oluion of he diffeenial euaion. Hin: Chain Rule: fo a geneal funcion, f(z d dz d f ( z f ( z... ( dx dx dz eivaive of he xonenial Inegal Funcion: (z d ex( z ( z... ( dz z eivaive of he xonenial Funcion: ex(z d d ex( z ex( z and ex( z ex( z... ( dz dz Genealized eivaive of he Naual Logaihm Funcion: ln(z d ln ( z... ( dz z Commen: Lage e and lage aoximaion

19 Name: Key (TAB/I.. xam /Poblem (Veificaion of Conan Rae Soluion fo he iffuiviy uaion Sing Thuday Mach (:-: in RICH,, and. Veificaion of Conan Rae Soluion fo he iffuiviy uaion (Coninued Taking he deivaive of. ih eec o dimenionle ime and adiu and ubiuing: ] [ ex ] [ ex ] [ ex e e e e e e e e e... ( Uon imlificaion e have: e e e ex... ( Mulilying hough by give e e e ex... ( We ill ake he deivaive em by em: ex... ( ex ex... ( ex... ( e e ex... (

20 Name: Key (TAB/I.. xam /Poblem (Veificaion of Conan Rae Soluion fo he iffuiviy uaion Sing Thuday Mach (:-: in RICH,, and. Veificaion of Conan Rae Soluion fo he iffuiviy uaion (Coninued e e ex ex ex e... ( e e ex ex ex... ( e e ex ex ex ex e e Finally e have: ex e... ( e Ou eul imlie ha he oluion (. i an aoximaion and i only valid fo lage e and lage value.

21 xam /Poblem (Hyebolic Cumulaive Poducion-Rae Relaion Sing Thuday Mach (:-: in RICH,, and Name: Key (TAB/I.. (. Hyebolic Cumulaive Poducion-Rae Relaion Given: You ae given he "hyebolic" ae-ime elaion deived emiically fom obeved ae-ime efomance daa by eveal auho and ummaized by A in he. Hyebolic Rae Relaion: (ASSUM valid fo eudoeady-ae ga flo gi g (... ( / b [ bi ] Reuied: a. You ae o deive he cumulaive oducion elaion fo he hyebolic ae-ime elaion (i.e., uing.. The hyebolic cumulaive oducion-ime elaion (i.e., he eul fo hi cae i given a: gi ( / b G ( g ( d [ ( bi ]... ( ( b i Obviouly, you mu SHOW ALL WORK fo cedi on hi oblem. b. You ae o develo he hyebolic ae-cumulaive oducion elaion given belo hoing all ok in full deail: G ( b gi g ( gi hee G... ( G ( b i Hin: A convenien "e-caing" of. i: C g ( C B i gi [ A ] B [ A ] (hee A b ; B / b; C Pa a.: (Hyebolic cumulaive oducion-ime elaion Taking he deivaive of. : g ( gi... ( [ ] / b b i Re-caing he oblem: (Ab i ; B/b; C gi g C ( C[ A ] B... ( [ ] B A The definiion of he cumulaive ga oducion (G ( i: G ( g ( d... ( Subiuing. ino., e have: G ( C [ A ] B d... ( efining a vaiable of ubiuion: z [ A ] z( dz A z( A d... (

22 xam /Poblem (Hyebolic Cumulaive Poducion-Rae Relaion Sing Thuday Mach (:-: in RICH,, and. Hyebolic Cumulaive Poducion-Rae Relaion (Coninued Subiuing. ino. yield: Name: Key (TAB/I.. A C G ( z B dz... ( A Solving he inegaion in. give u: C A ( B G ( z... ( A ( B xanding., e have: C G ( [( (B A ]... ( A ( B Subiuing Ab i ; B/b; and C gi ino. yield: gi G ( [( (/ b bi ] bi ( / b Reducing give u: gi G ( [( (/ b bi ] i ( b And finally, e have: gi G ( [ ( ( / b bi ]... ( ( b i Pa b.: (Hyebolic ae-cumulaive oducion elaion Saing ih he hyebolic cumulaive oducion-ime elaion (i.e.,., e have: gi G ( [ ( ( / b bi ]... ( ( b i efining: gi G... ( ( b i Subiuing. ino., e have: G ( [ ( ( / b G bi ]... ( Solving fo he G (/G aio, e obain: G ( (/ b ( bi... ( G Seaaing he ( b ( / b i em, e have: (/ b / b ( b i ( b i ( b i... ( Recalling he hyebolic ae definiion (i.e.,. : g ( gi / b gi ( bi... ( [ ] / b b Reaanging. yield: i b g ( ( bi /... ( gi

23 Name: Key (TAB/I.. xam /Poblem (Hyebolic Cumulaive Poducion-Rae Relaion Sing Thuday Mach (:-: in RICH,, and. Hyebolic Cumulaive Poducion-Rae Relaion (Coninued Solving. fo (b i, e have: b gi g i b ( (... ( Subiuion of. and ino., e obain: gi g b gi g b i b ( ( ( / ( O: b gi g b i b / ( ( (... ( Subiuing of. ino. give u: b gi g G G ( (... ( Reaanging: G G b gi g ( (... ( Solving fo g (/ gi, e have: b gi g G G ( (... ( Solving fo g ( i ou final eul: i gi b gi g b G G G ( hee ( (... (

24 xam /Poblem (Radial Flo Reevoi Peue iibuion Sing Thuday Mach (:-: in RICH, and (. Radial Flo Reevoi Peue iibuion Given: Tanien Radial Flo Soluion: (ime and diance-vaian oluion Full Tanien Flo Soluion: (xonenial Inegal B c (, i. μ kh φμ k "Log Aoximaion" Tanien Flo Soluion: Bμ k (, i. ln kh φμc Seady-Sae Radial Flo Soluion: (ime invaian oluion Name: Key (TAB/I..... (... ( Bμ ( e i. ln kh... ( Peudoeady-Sae Radial Flo Soluion: (ime and diance-vaian oluion, noe ha Aπ e Bμ B e ( ( i. ln. kh... ( ( φhac e Full Radial Flo Soluion: (ime and diance-vaian oluion in dimenionle fom (, e e e ex ex... ( e e Whee he definiion fo he dimenionle vaiable ae: kh ( i...(...(. Bμ e e... ( k. φμc... ( Reuied: (You mu ho all ok fo cedi. eive he "adiu of inveigaion" conce uing he "log aoximaion" oluion.. eive he ime,, o eady-ae fo a bounded cicula eevoi uing he "adiu of inveigaion" fomula. (Hin: Thi eul ill be a geneal exeion in em of φ, μ, k, c, and e.. eive he ime,, o eudoeady-ae fo a bounded cicula eevoi. (Hin: Thi eul ill be a geneal exeion in em of φ, μ, k, c, and e.. You ae o calculae he eue ofile a h uing. and lo hee eue on he aached lo. The eevoi oeie ae given on he nex age. (f (, h (ia......

25 xam /Poblem (Radial Flo Reevoi Peue iibuion Sing Thuday Mach (:-: in RICH, and. Radial Flo Reevoi Peue iibuion (Coninued Hin: Name: Key (TAB/I... The definiion of eudoeady-ae flo condiion fo a ell ceneed in a ymmeically-haed eevoi i given by: df df... ( d anien d. Fo x<., ue he logaihmic aoximaion fo he exonenial inegal oluion: ( x ln ( e γ.... ( e γ x. Fo x>., (x Reevoi oeie: φ.. f i ia k md h f e f Oil oeie: B. RB/STB μ. c c.x - ia - Poducion aamee: STB/ Woking Plo

26 xam /Poblem (Radial Flo Reevoi Peue iibuion Sing Thuday Mach (:-: in RICH, and. Radial Flo Reevoi Peue iibuion (Coninued -i(-x Table fom Lee Tex ( [noe ha (x-i(-x] Name: Key (TAB/I..

27 xam /Poblem (Radial Flo Reevoi Peue iibuion Sing Thuday Mach (:-: in RICH, and. Radial Flo Reevoi Peue iibuion (Coninued Soluion:. eive he "adiu of inveigaion" conce uing he "log aoximaion" oluion. inv i he adial diance comued hen i e o i uing he log aoximaion oluion. Bμ. ln kh ln k φμc xoneniaing k φμc k φμc Name: Key (TAB/I..... (... (... ( o k... ( φμc o k inv... ( φμc. eive he ime,, o eady-ae fo a bounded cicula eevoi uing he "adiu of inveigaion" fomula. Solving fo he ime a hich inv e give he ime o "eady-ae" k e... ( φμc k e... ( φμc φμc e... ( k. eive he ime,, o eudoeady-ae fo a bounded cicula eevoi. Taking he ime deivaive of. df d Bμ d k Bμ d [ i ]. ln. [ln[ ]]... ( d d kh d kh d anien φμc o d f Bμ.... ( d kh anien Taking he ime deivaive of. d f d Bμ d ( B d [ ]. ln e i. [ ] d d kh d... ( ( hac d e φ

28 xam /Poblem (Radial Flo Reevoi Peue iibuion Sing Thuday Mach (:-: in RICH, and. Radial Flo Reevoi Peue iibuion (Coninued O: df B. d φhac Subiuing. and ino., e obain Name: Key (TAB/I..... ( Bμ B..... ( kh φhac Solving fo ( give. φμc A... (. k O ince Aπ e c. φμ e π... (. k finally φμc e... ( k. You ae o calculae he folloing eue a he ecibed ime uing. and lo hee eue on he aached lo. Luming em in he dimenionle aamee e have a( i... ( kh ( md( f a.... (. Bμ. ( STB/(. RB/STB(. c. ( i... ( hee... ( ( f e e... ( (.f b... ( k ( md b ( φμc (.(. c(.x i (.f.... ( Ineing,,, and ino. e have ( (. ( i (. (.... ( (. ( ( ( ex ex ( (. ( (.

29 xam /Poblem (Radial Flo Reevoi Peue iibuion Sing Thuday Mach (:-: in RICH, and. Radial Flo Reevoi Peue iibuion (Coninued Simlifying. e obain he geneal elaion fo hi oblem a. ( i. Name: Key (TAB/I.. ex.x... ( ex Fo hou, e have. ( i. ex.... ( x ex ( ( ( (. ( i [ ] [ ]. ex[ ]. x ex[ ] A ex[-], e can neglec he la o em in he euaion given above. Fo. f [ (. ] [ (. ]. ( i i ln e γ. ia Fo f, (x>., (x. ( i i. ia [.] /....(a... (b...(c [ ( ] o. ( i ln γ...(a e [.] /. Fo f, (x>., (x :. ( i i... (b...(c [ ( ] [.] /....(a... (b. ia...(c

30 xam /Poblem (Radial Flo Reevoi Peue iibuion Sing Thuday Mach (:-: in RICH, and. Radial Flo Reevoi Peue iibuion (Coninued Fo f, (x>., (x :. ( i i. ia [ ( ] [.] /. Fo f, (x>., (x :. ( i Name: Key (TAB/I.....(a... (b...(c [ ( ]...(a i... (b ia...(c (f (, h (ia......

31 xam /Poblem (Peue iibuion in a Muliell Syem Sing Thuday Mach (:-: in RICH, and (. Peue iibuion in a Muliell Syem Given: You ae given he folloing configuaion of ell. (noe: OA OC f, OB O f Name: Key (TAB/I.. Reuied: (You mu ho all ok fo cedi a. The eue a he obevaion ell fo h, h, h, h. b. The ime a hich he eue do in he obevaion ell i. i. Reevoi oeie: φ.. f i ia k md h f Oil oeie: B o. RB/STB μ o. c c x - ia - Poducion aamee: Well A: A STB/ (oducion A oa f Well B: B STB/ (oducion B ob f Well C: C STB/ (inecion C oc f Well : STB/ (inecion o f Hin:. You hould aume infinie-acing homogeneou eevoi behavio (i.e., he adial flo oluion.. You mu ue he exonenial inegal fom of he oluion (i.e., he (x fomulaion do NOT ue he logaihmic aoximaion.. Fo a b: You ill obain he oluion u o one decimal oin by ial and eo. Ty ime value beeen and hou. Reul: a. The eue a he obevaion ell fo h:. ia The eue a he obevaion ell fo h:. ia The eue a he obevaion ell fo h:. ia The eue a he obevaion ell fo h:. ia b. The ime a hich he eue do in he obevaion ell i. i:. h

32 xam /Poblem (Peue iibuion in a Muliell Syem Sing Thuday Mach (:-: in RICH, and. Peue iibuion in a Muliell Syem (Coninued -i(-x Table fom Lee Tex ( [noe ha (x-i(-x] Name: Key (TAB/I..

33 xam /Poblem (Peue iibuion in a Muliell Syem Sing Thuday Mach (:-: in RICH, and. Peue iibuion in a Muliell Syem (Coninued a. The eue a he oduce ell fo h. To begin, e define he oal eue change a he um of he eue change in all of he ell: Δ ΔoA ΔoB ΔoC Δo Subiuing he aoiae eue change model (i.e., he (x oluion, e have: Bμ φμc B c Δ oa μ φμ ob i o. A. B kh k kh k B μ φμc B c oc μ φμ. o C. kh k kh k Collecing conan em, e obain: Δ c oa c ob c oc c o A α B α C α α Whee: Bμ (. RB/STB (. c c...x kh ( md ( f φμc - (. (. c (.x i α x k ( md Subiuing he value of c and α, e obain: Δ (.x ( STB/ (.x ( STB/ ( f (x ( f (x ( f (.x ( STB/ (x x x ( f (. ( STB/ ( Which educe o he folloing geneal eul (in ime fo hi cae: Δ.... Δ.. Comuing he eue do a he obevaion ell a h: Δ.. ( h [.]. [.] ( h Δ. Uing he -i(-x able (ecall ha (x-i(-x, e have: Δ (. (.. (.. i Theefoe, he eue a he obevaion ell fo h i o.. ia Name: Key (TAB/I..

34 xam /Poblem (Peue iibuion in a Muliell Syem Sing Thuday Mach (:-: in RICH, and. Peue iibuion in a Muliell Syem (Coninued Comuing he eue do a he obevaion ell a h: Δ.. ( h ( h [.]. [.] Δ. Uing he -i(-x able (ecall ha (x-i(-x, e have: Δ (. (.. (.. i Theefoe, he eue a he obevaion ell fo h i: o.. ia Similaly, he eue a he obevaion ell fo h h ae o.. ia ( h o.. ia ( h Name: Key (TAB/I.. b. The ime a hich he eue do in he obevaion ell i. i. Thi i faily aighfoad, e ue he eul fom a a, and "evee calculae" he ime fo a given eue change. In hi cae, he eue change i. i. Ou eviou eul i given by: Δ...b. xac Soluion Fo hou e have: (eul of a a. Δ... i Fo hou e have: Δ... i Fo hou e have: Δ... i Fo. hou e have: Δ... i.. Fo. hou e have: Δ... i.. Fo. hou e have: Δ... i..

35 xam /Poblem (Peue iibuion in a Muliell Syem Sing Thuday Mach (:-: in RICH, and. Peue iibuion in a Muliell Syem (Coninued.b. Aoximae Soluion (- oin Seing he eue change o. i, e have: Δ... i Uing he "log aoximaion" of he xonenial Inegal oluion: ln ln. e γ e γ Poeie of he logaihm can be ued o imlify he above euaion: ln. e γ e γ Fuhe imlificaion yield: e. e γ e γ Solving fo, e obain e. e γ e γ (hee γ.. h (diffeen han he eviou eul, obviouly no coec ule' conan Name: Key (TAB/I..

36 xam /Poblem (Analyi of Mulioin Ga Well Te aa Sing Thuday Mach (:-: in RICH, and Name: Key (TAB/I.. (. Analyi of Mulioin Ga Well Te aa Given: In hi oblem you ae given he ell efomance daa (aveage eevoi eue, floing boomhole eue, and ga floae fo a oducing ga ell. The obecive i o eablih a "deliveabiliy" elaion and an inflo efomance elaion (o IPR fo hee daa, and comae he eul ediced fom hee elaion in hi cae, he maximum heoeical ga floae ( g, max. Mulioin Well Te aa: Gazom Ikuk- Ga Well (Ruia Poin (ia f (ia g (MSCF Reuied: (You mu ho all ok fo cedi a. eliveabiliy Analyi: Uing he elaion belo (i.e.,. and he lo ovided on he nex age, you ae o eimae α, β, and g, max hee g, max i an exaolaed value eimaed hen f. ia (i.e., amoheic eue. Fo hi cae you can aume f ia fo imliciy. ( β ( α β f f g g... ( α Whee α and β ae he inece and loe of he ( f veu g lo, eecively. α (inece:. ia β (loe:. dimenionle g, max :, MSCF b. Modified IPR Analyi: Uing he elaion belo (i.e.,. and he lo ovided on he nex age, you ae o eimae ν g and g, max hee g, max i an exaolaed value eimaed hen f. ia (i.e., amoheic eue. Fo hi cae you can aume f ia fo imliciy. f a g bg... ( Whee a and b ae he inece and loe fom a lo of a (inece:.- /MSCF b (loe:.- (/MSCF f veu g lo, eecively. g g, max :, MSCF (olved a a uadaic oo of.

37 xam /Poblem (Analyi of Mulioin Ga Well Te aa Sing Thuday Mach (:-: in RICH, and. Analyi of Mulioin Ga Well Te aa (Coninued a. eliveabiliy Analyi: Name: Key (TAB/I.. b. Modified IPR Analyi: