RATE TRANSIENT ANALYSIS OF GAS/WATER TWO-PHASE RESERVOIRS: A DENSITY-BASED APPROACH

Transcription

1 Th Pnnsylvania tat Univsity Th Gaduat chool John and Willi Lon Family Datmnt of Eny and Minal Eninin RATE TRANIENT ANALYI OF GA/WATER TWO-PHAE REERVOIR: A DENITY-BAED APPROACH A Thsis in Eny and Minal Eninin by Yao Fn 016 Yao Fn ubmittd in Patial Fulfillmnt of th Ruimnts fo th D of Mast of cinc Auust 016

2 Th thsis of Yao Fn as vid and aovd* by th folloin: Luis Ayala Pofsso of Ptolum and Natual Gas Eninin Associat Datmnt Had fo Gaduat Education Thsis Adviso Hamid Emami-Mybodi Assistant Pofsso of Ptolum and Natual Gas Eninin Eun Moan Assistant Pofsso of Ptolum and Natual Gas Eninin *inatus a on fil in th Gaduat chool ii

3 ABTRACT As natual as bcoms an imotant ny souc, a numb of studis hav attmtd to solv as oblms numically o analytically. Fom th intoduction of al as sudo-ssu (Al- Hussainy t al. 1966) to th dimnsionlss ty cuv oosd by Ftovich (1980), continuous oss is mad alon ith mo ustions aisd. In 014, Zhan and Ayala oosd a dnsitybasd scald xonntial modl fo dy as oducd und constant bottomhol ssu duin bounday dominatd flo, and imovd it fo vaiabl-ssu/vaiabl-at cass and as sv diction. This study xtndd thi modl fom dy as svois to as and at tohas svois, involvin constant bottomhol ssu scnaios, vaiabl bottomhol ssu cass and sv diction. Fluid dnsitis and th dnsity fom of matial balanc uation a usd in this mthod. Factos such as satuation and lativ mability a add as th y of this oblm, hil fluid miscibility, caillay ssu and at influx a not considd. Modl outcoms a comad ith CMG numical simulations, and snsitivity analyss of initial at satuation, iducibl at satuation and daina adius a don to tst th modl's adatation and limitation. Th modl is ovn to b fasibl but limitd to svois ith lo at satuation to avoid th imact causd by th simlification of diffusivity uations. Th tansfomation of th modl invsly dicts oiinal as in lac and oiinal at in lac ith svoi and fluid otis ovidd. iii

4 Tabl of Contnts LIT OF FIGURE... vi LIT OF TABLE... ix NOMENCLATURE... x ACKNOWLEDGEMENT... xiv INTRODUCTION... 1 Chat 1 Constant Bottomhol Pssu Dclin Analysis of Gas/Wat To Phas Rsvois Usin a Dnsity-Basd Aoach Chat ummay Bacound Rscald Exonntial Modl fo Gas/Wat To-Phas ystm in BDF Poducin Und Constant BHP Cas tudis nsitivity Analysis Concludin Rmas... 7 Chat Vaiabl Bottomhol Pssu Dclin Analysis of Gas/Wat To Phas Rsvois Usin a Dnsity-Basd Aoach Chat ummay Bacound Rscald Exonntial Modl fo Gas/Wat To Phas ystm in BDF Poducin Und Vaiabl BHP Cas tudis nsitivity Analysis Concludin Rmas Chat 3 Poduction Data Analysis of Gas and Wat To-Phas Rsvois Usin a Dnsity-Basd Aoach Chat ummay Bacound Rscald Exonntial Modl fo Constant BHP Gas/Wat To Phas ystm in BDF Cas tudis Concludin Rmas... 6 CONCLUION REFERENCE iv

5 FUTURE WORK Andix A Divation of Gas Flo Euation Andix B Divation of Wat Flo Euation Andix C Pocdu of Gas and Wat Rat Estimations in th Extndd To-Phas Modl 75 Andix D imlification of Diffusivity Euations v

6 LIT OF FIGURE Fiu 1-1 Comaison of as at dictions btn numical simulation and as/at tohas scald xonntial modl fo Cas tudy 1-A Fiu 1- Comaison of at at dictions btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-A Fiu 1-3 Comaison of as at atio dictions btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-A Fiu 1-4 Gas at dviation fo Cas tudy 1-A Fiu 1-5 Wat at dviation fo Cas tudy 1-A Fiu 1-6 Gas at atio dviation fo Cas tudy 1-A Fiu 1-7 Comaison of as at dictions btn numical simulation and as/at tohas scald xonntial modl fo Cas tudy 1-B Fiu 1-8 Comaison of at at dictions btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-B Fiu 1-9 Initial at satuation snsitivity analysis fo as at (constant BHP)... 0 Fiu 1-10 Initial at satuation snsitivity analysis fo at at (constant BHP)... 1 Fiu 1-11 Pssu distibution on last day of oduction fo i=0.9 (constant BHP)... Fiu 1-1 Gas at dviation fo i snsitivity analysis (constant BHP)... 3 Fiu 1-13 Wat at dviation fo i snsitivity analysis (constant BHP)... 3 Fiu 1-14 Iducibl at satuaion snsitivity analysis fo as at (constant BHP)... 4 Fiu 1-15 Iducibl at satuaion snsitivity analysis fo at at (constant BHP)... 5 Fiu 1-16 Daina adius snsitivity analysis fo as at (constant BHP)... 6 Fiu 1-17 Daina adius snsitivity analysis fo at at (constant BHP)... 6 vi

7 Fiu -1 Comaison of as ats btn numical simulation sults and to- has scald xonntial modl fo vaiabl BHP cas study Fiu - Comaison of at ats btn numical simulation sults and to- has scald xonntial modl fo vaiabl BHP cas study Fiu -3 Comaison of as-at atio btn numical simulation sults and to- has scald xonntial modl fo vaiabl BHP cas study Fiu -4 Gas-at atio dviation fo vaiabl BHP cas study Fiu -5 Initial at satuation snsitivity analysis fo as at (vaiabl BHP) Fiu -6 Initial at satuation snsitivity analysis fo at at (vaiabl BHP)... 4 Fiu -7 Pssu distibution on last day of oduction fo i=0.9 (vaiabl BHP) Fiu -8 Iducibl at satuaion snsitivity analysis fo as at (vaiabl BHP) Fiu -9 Iducibl at satuaion snsitivity analysis fo at at (vaiabl BHP) Fiu -10 Daina adius snsitivity analysis fo as at (vaiabl BHP) Fiu -11 Daina adius snsitivity analysis fo at at (vaiabl BHP) Fiu 3-1 Poduction histoy fo Cas tudy 3-A Fiu 3- Fist itation of as has in Cas tudy 3-A Fiu 3-3 Fist itation of at has in Cas tudy 3-A Fiu 3-4 Final itation of as has in Cas tudy 3-A Fiu 3-5 Final itation of at has in Cas tudy 3-A Fiu 3-6 Poduction histoy and ssu ofil fo Cas tudy 3-B Fiu 3-7 Fist itation of as has in Cas tudy 3-B Fiu 3-8 Fist itation of at has in Cas tudy 3-B Fiu 3-9 Final itation of as has in Cas tudy 3-B vii

8 Fiu 3-10 Final itation of at has in Cas tudy 3-B viii

9 LIT OF TABLE Tabl 1-1 Hyothtical Rsvoi Infomation fo Cas tudy 1-B Tabl -1 Pssu ofil fo vaiabl BHP cas study Tabl 3-1 Bottomhol ssu ofil of Cas tudy 3-B ix

10 NOMENCLATURE A daina aa, ft 3 b, dimnsionlss sudo-stady comonnt D ss c as comssibility, si -1 c i as comssibility, si -1 c t total comssibility, si -1 c ti initial total comssibility, si -1 c at comssibility, si -1 D i initial dclin cofficint fo as has, days -1 D i initial dclin cofficint fo at has, days -1 G cumulativ as oduction, scf h svoi nt ay, ft absolut mability, md as lativ mability as lativ mability at ava svoi condition at lativ mability at lativ mability at ava svoi condition m al as sudo-ssu, si /c M as mobility at ava svoi condition, c -1 x

11 M at mobility at ava svoi condition, c -1 OGIP oiinal as in lac, scf OWIP oiinal at in lac, TB i initial svoi ssu, sia nomalizd sudo-ssu, sia f bottomhol ssu (ll floin ssu), sia ava svoi ssu, sia sc as flo at at standad condition, scf/d i initial as dclin at, scf/d sc at flo at at standad condition, TB/d i initial at dclin at, scf/d Q n nomalizd cumulativ oduction, scf daina adius, ft as dnsity dadon atio at dnsity dadon atio llbo adius, ft as satuation as satuation at ava svoi condition i initial as satuation xi

12 at satuation at satuation at ava svoi condition i initial at satuation i iducibl at satuation m mobil at satuation t a ~ t a adjust tim, days al as sudo-tim, days(si/c) t amb matial balanc tim, days t n nomalizd tim, days W cumulativ at oduction, TB z z i as comssibility facto initial as comssibility facto dltion-divn dimnsionlss tim dislacmnt facto fo dltion-divn dimnsionlss tim dislacmnt facto fo as scific avity M cofficint fo as mobility M cofficint fo at mobility dimnsionlss dadon colatin aamt xii

13 ava viscosity-comssibility dimnsionlss atio as viscosity, c as viscosity at ava svoi condition, c i initial as viscosity, c at viscosity, c oosity as dnsity, lb/ft 3 as dnsity at ava svoi condition, lb/ft 3 i initial as dnsity, lb/ft 3 sc as dnsity at standad condition, lb/ft 3 f as dnsity at llbo condition, lb/ft 3 at dnsity, lb/ft 3 at dnsity at ava svoi condition, lb/ft 3 i initial at dnsity, lb/ft 3 sc at dnsity at standad condition, lb/ft 3 f at dnsity at llbo condition, lb/ft 3 xiii

14 ACKNOWLEDGEMENT Fist of all, I ould li to xss my dst aciation to my adviso, D. Luis F. Ayala H., fo his at hl and atint uidanc thouhout my nti sach. With his ich xinc and continuous suot, I as abl to lan ho to conduct sach and succssfully finish my study. I am atful as ll to my committ mmbs, D. Hamid Emami-Mybodi and D. Eun Moan, fo thi usful commnts and sustions. In addition, I ould li to than Miao Zhan and Xion Li fo thi ofssional advic on my sach and thsis itin, and also than my family fo thi mntal suot hn I as und stss. xiv

15 INTRODUCTION Natual as has bcom an ny souc sinc th tntith cntuy, and th study of as svois is main continuous oss du to th incasin dmand and oduction of natual as. As aly as 1945, As oosd th tys of at dclin cuv: xonntial dclin, hamonic dclin and hybolic dclin, hich laid a foundation fo many futu sachs on hydocabon. Gas, as a comssibl fluid, mas as oblms mo difficult to solv bcaus of nonlinaity. In 1966, Al-Hussainy t al. achivd a at bathouh by atially linaizin as oblm usin th al as sudo-ssu. Haft, many simila tansfomations - i.. al-as sudo-tim (Aaal 1979), nomalizd sudo-tim (Faim and Wattnba 1987), matial balanc sudo-tim (Palacio and Blasinam 1993)- oosd to futh analyz as oblms. Th intoduction of dimnsionlss vaiabls has contibutd to ty cuv studis. Ftovich (1980) usd dimnsionlss flo at and dimnsionlss tim in his ty cuv analysis to simlify th calculations. Cat (1985) dfind th dimnsionlss dadon colatin aamt λ to dscib th influnc of as viscosity and as comssibility. Palacio and Blasinam (1993) dfind th matial balanc sudo-tim to modl at-tim lation and xlicitly dict oiinal as in lac. This modl as lat modifid usin nomalizd flo at by Matta and Andson (003). Datin fom vious sach basd on sudo-vaiabls, Ayala and Zhan (013; 014) oosd a dy as modl adin as dnsity to valuat as ll fomanc duin boundaydominatd flo. Th scald xonntial modl is a tim-at modl ioously divd fom th dnsity foms of th diffusivity uation and matial balanc uation. Gas comssibility is tan into account via th al as sudo-ssu and th ava dimnsionlss viscositycomssibility atio. Zhan and Ayala (014) lat oosd a tansfomation of th modl that 1

16 invsly dicts oiinal as in lac. This snt study is conductd on th basis of th scald xonntial modl to valuat as and at to-has svois.

17 Chat 1 Constant Bottomhol Pssu Dclin Analysis of Gas/Wat To Phas Rsvois Usin a Dnsity-Basd Aoach 1.1 Chat ummay Of th many ays to analyz oduction data, Zhan and Ayala (014) oosd a dnsity-basd scal xonntial modl fo dy as at tansint analysis. This chat adjusts thi modl by tain satuation and lativ mability into account so that as and at to-has svois oducd und constant bottomhol ssu (BHP) duin bounday-dominatd flo (BDF) can b valuatd as ll. Th xtndd aoach modls as and at ats fo closd as/at tohas svois. To cas studis a ovidd to vify th modl, and snsitivity analyss on initial at satuation i, iducibl at satuation i and daina adius a ivn to tst th modl s adatation. Th modl os fo vaious conditions but hih at satuation svois h th simlification of diffusivity uations can caus dviation. 1. Bacound Rat tansint analysis, on of th most commonly usd tchnius to valuat as svs, has bn studid by many authos in th ast dcads. Comssibl as oblms a difficult to solv bcaus of its non-linaitis. Al-Hussainy t al. (1966) intoducd th al as sudo-ssu, dfind as: m 0 d z Euation 1-1 3

18 hich is idly usd in many alications to linaiz th diffusivity uations. And a simila ~ tansfomation, th al as sudo-tim t a, is intoducd to futh linaiz th oblm by Aaal in Oth osss a mad in studyin as svois. Ftovich (1980) oosd to simlify calculations by alyin dimnsionlss vaiabls, and built his ty cuvs fo as and oil lls oducin at constant bottomhol ssu. Cat (1985) dvlod ty cuvs fo boundaydominatd flo by intoducin th dimnsionlss dadon colatin aamt: ic i m m i z i f z f Euation 1- hich tas th as viscosity-comssibility tm into considation as a function of bottomhol ssu. In th 1990s, Aaal s dfinition fo sudo-tim as found inalicabl fo boundaydominatd flo. Lat in 1988, Blasinam and L dfind matial balanc sudo-tim as: t amb ic ti t 0 c t dt Euation 1-3 basd on th nomalizd sudo-tim tn intoducd by Faim and Wattnba (1987). Palacio and Blasinam (1993) divd th as ll dlivability uation involvin th folloin function of nomalizd sudo-ssu: iz i i 0 d z Euation 1-4 4

19 and futh studid th lationshi btn ssu-do nomalizd at and sc i f tamb. In 014, Zhan and Ayala ovidd th folloin scald xonntial modl basd on as dnsity fo as lls oducin at constant BHP duin BDF: sc x D t i i Euation 1-5 h is th ava viscosity-comssibility dimnsionlss atio imovd on th basis of th λ aamt viously dfind by Cat (1985): m ici m f f c i i c, f Euation 1-6 ith as its tim-avad volution: 1 t dt t 0 Euation 1-7 Th scald xonntial modl ovids a dnsity-basd aoach and avoids th calculation of sudo-tim. Tan th associatd at in as svois, th as and at to-has scald xonntial modl in this chat xtnds Zhan and Ayala s modl fom dy as svois to as svois ith mobil at, h th to fluids a immiscibl and at is assumd to b slihtly comssibl (ith constant viscosity and comssibility). 5

20 1.3 Rscald Exonntial Modl fo Gas/Wat To-Phas ystm in BDF Poducin Und Constant BHP Th dnsity-basd as dlivability uation (Andix A and Andix B) divd by Ayala and Zhan (013) can b ittn as: sc b D, ss h c sc i i f Euation 1-8 and fo th at has, it can b ittn as: sc b D, ss h c sc f Euation 1-9 fo a as/at to-has svois. Th lationshi btn ths to hass is dtmind by th lativ mability, hich is a function of satuation. Hnc solvin a dnsity-basd tohas oblm uis th connction btn satuation and dnsity. Th oducts of ava fluid dnsitis and satuations a ssu-latd. By alyin th oduct ul and th dfinition of as and at comssibility, on can oos th folloin lationshis btn dnsity and satuation: d d d d d d d c d Euation

21 7 and d d c d d d d d d Euation 1-11 In as/at flo, has satuations a futh latd thouh th xssion +=1, hich mans d/d=-d/d. Pssus of as and at a ual hn caillay ssu is not considd. Thus th lation btn as and at satuation can b xssd as: d d c d d c d d c d d c d d Euation 1-1 Th diffusivity uations of as and at a t Euation 1-13 and t Euation 1-14 h all fluid otis a tim/location-dndnt. To simlify th diffusivity uations, th dnsity and mobility chan alon location is nlctd (Andix D). It ill not caus o hn it is small comad to th st of th uations, but th ill b dviation in som cass. Fo

22 homonous, isotoic and incomssibl svois, th simlifid diffusivity uations bcom t Euation 1-15 and t Euation 1-16 Dividin Euation 1-15 by Euation 1-16, th dnsity basd fom that fo ava satuations could b ittn as: d d M M Euation 1-17 h M / Euation 1-18 and M / Euation 1-19 a th mobility of as and at. 8

23 Th at of ava satuation chan und ssu dltion is solvd by combinin Euation 1-1 and Euation 1-17: d d c M M M c M Euation 1-0 Th dnsity-basd matial balanc uations fo as and at can b xssd as i i G 1 OGIP Euation 1-1 and i i W 1 OWIP Euation 1- Th tan matial balanc uation fo at is divd fom WIP=OWIP-W. Th oducts of ava fluid dnsitis and satuations chan ith tim. Tain th tim divativ of Euation 1-1 and Euation 1-, th uations can b xssd usin as and at flo ats: d dt d G sc sc ii 1 dt OGIP Ah Euation 1-3 and d dt d W sc ii 1 dt OWIP Ah sc Euation 1-4 9

24 By alyin th oduct ul, th dfinition of fluid comssibility and Euation 1-0, th tim divativs of ava dnsity-satuation can also b ittn as d dt M M M c c c d dt Euation 1-5 and d dt M M M c c c d dt Euation 1-6 ubstitut Euation 1-3 and Euation 1-4 into Euation 1-5 and Euation 1-6, th divativ of ava fluid dnsity ov tim can b ittn as d dt M sc Ah i sc Euation 1-7 and d dt M sc Ah i sc Euation 1-8 h M c i c c M M M Euation

25 M ic c c M M M Euation 1-30 Aain, by substitutin th dlivability uations Euation 1-8 and Euation 1-9 into Euation 1-7 and Euation 1-8, th divativs bcom: d dt A i icibd, ss f Euation 1-31 and d dt A i icibd, ss f Euation 1-3 By intation btn th initial condition and ava svoi dnsity, th dnsity dclin can b xssd in th xonntial fom: i f f x D t i Euation 1-33 and i f f x D t i Euation 1-34 h 11

26 D i A i c i b i D, ss Euation 1-35 D i A i c b D, ss Euation t t 0 M dt Euation t t 0 M dt Euation 1-38 inc th intation is a ocss und constant ll floin ssu, Euation 1-33 and Euation 1-34 o only fo svois oducin at constant BHP. ubstitution of Euation 1-33 and Euation 1-34 into dlivability uations Euation 1-8 and Euation 1-9 lads to th scald xonntial modl: sc x D t i i Euation 1-39 And, fo at: sc x D i i t Euation 1-40 h 1

27 i sc h c i i i b D, ss Euation 1-41 i h sc c i b D, ss Euation 1-4 i i f Euation 1-43 i i f Euation 1-44 Comad to th sinl as has modl by Zhan and Ayala, this to-has modl tas satuation and lativ mability into account. Wat influx ffcts a nlctd in th modl. 1.4 Cas tudis Cas tudy 1-A. Zhan & Ayala Cas This cas study is natd basd on th scnaio ith initial svoi ssu of 5,000 sia in Zhan & Ayala (014), containin th svois of diffnt siz (=175, 350 and 700 ft). Wat has ith 1 c viscosity, 3.5x10-6 si -1 comssibility and lb/ft 3 standad condition dnsity is addd to th systm. Iducibl at satuation (i) of 0.15 and initial at satuation (i) of 0.3 a usd. Th lativ mability-satuation lation follos Coy modl fo as and at to-has systm: 13

28 1 1 1 i 1 i i Euation 1-45 i 1 i 4 Euation 1-46 Gas viscosity and as comssibility a calculatd usin mthods dvlod by L t al. (1966) and Abou-Kassm t al. (1990), sctivly. Fiu and Fiu dmonstat that as and at flo ats dictd by th scald xonntial modl fo as and at to-has systm a consistnt ith numical simulation solutions duin BDF. Dviation is shon in Fiu 1-4 and Fiu 1-5. Fiu 1-3 and Fiu 1-6 futh nsud th modl s validity via as at atio (GWR) and its dviation. 14

29 Fiu 1-1 Comaison of as at dictions btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-A Fiu 1- Comaison of at at dictions btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-A 15

30 Fiu 1-3 Comaison of as at atio btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-A Fiu 1-4 Gas at dviation fo Cas tudy 1-A 16

31 Fiu 1-5 Wat at dviation fo Cas tudy 1-A Fiu 1-6 Gas at atio dviation fo Cas tudy 1-A 17

32 Cas tudy 1-B. A Hyothtical Cas This is a hyothtical svoi cas study ith data natd basd on atial infomation of Ti Rid Gas Fild in Montana (Cox, 1978). Rsvoi and fluid otis utilizd in this study a listd in Tabl 1-1. Tabl 1-1 Hyothtical Rsvoi Infomation fo Cas tudy 1-B Initial svoi ssu i, sia 1300 Rsvoi tmatu T, F 140 Thicnss h, ft 35 Poosity φ 0.15 Pmability, md 15 Gas scific avity ϒ 0.65 Wat comssibility c, si -1 3.x10-6 Wat viscosity μ, c 1.0 Wat dnsity at standad condition ρ sc, lb/ft Initial at satuation i 0.38 Iducibl at satuation i 0.15 Rsvoi adius, ft 1300 Initial as in lac OGIP, BCF 1.50 Initial at in lac OWIP, MMTB 1.86 Wllbo adius, ft Bottomhol ssu f, sia

33 Fiu 1-7 Comaison of as at dictions btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-B Fiu 1-8 Comaison of at at dictions btn numical simulation and as/at to-has scald xonntial modl fo Cas tudy 1-B 19

34 Th as and at oduction ats a dislayd in Fiu 1-7 and Fiu 1-8, sctivly. This hyothtical svoi has much lo initial svoi ssu and hih mability than Cas tudy 1-A. Th as and at to-has scald xonntial modl aain shos ood match ith numical simulation sults. 1.5 nsitivity Analysis Initial Wat atuation i In a as and at to has svoi, as at ill do consuntially as initial at satuation incass. To invstiat th imact of initial at satuation on th scald xonntial modl fo to-has svois, th scnaio ith 175 ft daina adius in Cas tudy 1-A is xtndd to vaious initial at satuations. Gas and at ats ith i fom 0.3 to 0.9 a shon in Fiu 1-9 and Fiu It is obsvd that modl sults do not match numical solutions as i os, scially hn i is hih than 0.6. Fiu 1-9 Initial at satuation snsitivity analysis fo as at (constant BHP) 0

35 Fiu 1-10 Initial at satuation snsitivity analysis fo at at (constant BHP) On ossibl ason is bcaus of bounday-dominatd flo. Ta th situation ith initial at satuation 0.9 as an xaml. It is ossibl that th oduction ocss has not achd boundaydominatd flo yt. To invstiat hth th svoi is in bounday-dominatd-flo iod o not, th svoi bloc ssu at th xtnal bounday on th last day of oduction nds to b xamind. In CMG numical simulation, th cylindical svoi is dividd into 100 blocs in -diction. Bloc ssus on th last day hn initial at satuation is 0.9 a shon in Fiu 1-11, and th xtnal bounday is at sia, much lo than initial ssu. Thfo, th svoi is alady und bounday-dominatd flo. 1

36 Fiu 1-11 Pssu distibution on last day of oduction fo i=0.9 (constant BHP) inc th oduction ocss has achd bounday-dominatd-flo iod, th only xlanation fo th mismatch of as ats in Fiu 1-9 is th nlct of th divativ of fluid mobility ith sct to location in Euation 1-15 and Euation Gas and at at dviations a shon in Fiu 1-1 and Fiu It is cla that flo ats a lss accuat at hih i.

37 Fiu 1-1 Gas at dviation fo i snsitivity analysis (constant BHP) Fiu 1-13 Wat at dviation fo i snsitivity analysis (constant BHP) 3

38 1.5. Iducibl Wat atuation i To xlo th imact of iducibl at satuation on th modl, Cas tudy 1-B tstd ith i fom 0.05 to 0.35 (i has to b lo than i). Fiu 1-14 and Fiu 1-15 do not sho aant inaccuacy in flo ats. But as xlaind in initial at satuation snsitivity analysis, th modl may not o oly ith la amount of at. Iducibl at is immobil, thus it is th initial mobil at satuation m (m=i-i) that matts. To futh confim this conclusion, th modl has un sval cass ith constant initial mobil at satuation. Fiu 1-14 Iducibl at satuaion snsitivity analysis fo as at (constant BHP) 4

39 Fiu 1-15 Iducibl at satuaion snsitivity analysis fo at at (constant BHP) Daina Radius In od to nsu th modl s adatability on diffnt svoi sizs, Cas tudy 1-A is tstd aain ith daina adius of 50, 100, 00, 400, 700 and 1000 ft. Fiu 1-16 and Fiu 1-17 dislays as and at ats sctivly. No obvious inconsistncy is shon in th lots. Thfo, th modl is aoiat fo any svoi siz. 5

40 Fiu 1-16 Daina adius snsitivity analysis fo as at (constant BHP) Fiu 1-17 Daina adius snsitivity analysis fo at at (constant BHP) 6

41 1.6 Concludin Rmas Th as and at to-has scald xonntial modl succssfully dicts at tansint bhavios fo constant BHP as lls in as svois ith immiscibl associatd at duin BDF, h satuations and lativ mability a ssu-dndnt vaiabls. Cas studis and snsitivity analyss vify that this modl is suitabl fo divs svoi ssus and daina sizs, but has limitations on initial at satuation and iducibl at satuation. It ovids inaccuat sults fo svois ith hih at satuation du to th simlification of diffusivity uations. 7

42 Chat Vaiabl Bottomhol Pssu Dclin Analysis of Gas/Wat To Phas Rsvois Usin a Dnsity-Basd Aoach.1 Chat ummay To accod ith actic, at tansint analyss a imovd fom lls oducin at constant bottomhol ssu (BHP) to vaiabl bottomhol ssu. In 013 and 014, Zhan and Ayala succssfully xtndd thi dnsity-basd scald xonntial modl fo dy as svois und bounday-dominatd flo (BDF) fom constant BHP to vaiabl BHP condition. Tan satuation and lativ-mability, this chat adds at has into thi svoi systm. An imovd modl is dvlod to stimat both as and at ats fo to-has svois ithout at influx. A hyothtical cas is studid and futh xtndd to invstiat th imact of initial at satuation i, iducibl at satuation i and daina adius. Cas studis and snsitivity analyss on ths th otis tstify that th scald xonntial modl is succssfully adjustd to as and at to-has svois oducin at vaiabl BHP, but fails to dict oduction ats hn at satuation is hih, h th nlctd divativ of fluid mobility ith sct to location nds to com into considation.. Bacound Rat tansint analysis fo as lls oducin at constant bottomhol ssu (BHP) duin bounday-dominatd flo (BDF) is commonly sn. But lls a actually oducd und vaiabl BHP and vaiabl flo at condition in th fild. In 1988, Blasinam and L built an dvlod a fo vaiabl BHP scnaios usin adjustd tim 8

43 t a c i ti t 0 1 c t dt Euation -1 Palacio and Blasinam (1993) oosd a staiht-lin analysis fo as oduction duin BDF involvin nomalizd sudo-ssu and matial balanc sudo-tim tamb: iz i i 0 d z Euation - t amb ic ti t 0 c t dt Euation -3 In 013, Ayala and Zhan oosd a dnsity-basd aoach to dscib as dclin fo vaiabl BHP lls duin BDF by intoducin an ava viscosity-comssibility dimnsionlss atio m ici m f f c i i c, f Euation -4 m(ρ), usually ittn as m(), is th al as sudo-ssu dfind by Al-Hussainy t al. (1966): m 0 d z Euation -5 9

44 Th al as sudo-ssu is commonly usd in as svoi studis to atially linaiz th oblm. In this a, th dnsity-basd dclin modl is adatd to analyz as svois ith associatd at..3 Rscald Exonntial Modl fo Gas/Wat To Phas ystm in BDF Poducin Und Vaiabl BHP Adatd fom Ayala and Zhan s (013) dnsity-basd as dlivability uation (divd in Andix A and Andix B), dnsity-basd dlivability uations fo as and at to has svoi a sc b D, ss h c sc i i f Euation -6 and sc b D, ss h c sc f Euation -7 h m c i i f m f c i i c, f Euation -8 30

45 31 is th ava viscosity-comssibility dimnsionlss atio. Alyin Dacy s La to dnsity-basd diffusivity uation, th diffusivity uation fo as and at hass can b ittn as t Euation -9 and t Euation -10 and futh simlifid to t Euation -11 and t Euation -1 To lat OGIP and OWIP ith th flo ats, th dnsity fom tan matial balanc uations fo as and at lay imotant ols. Fo as has, th matial balanc uations can b ittn as

46 i i G 1 OGIP Euation -13 and fo at has, i i W 1 OWIP Euation -14 Th diffusivity uation and matial balanc uation toth solvs th lation btn satuation and ssu, hich is th y in this modl. By usin as and at mobility M=/μ and M=/μ, th divativ of as satuation ov ssu can b xssd as d d c M M M c M Euation -15 Poduction at vsus tim uis a lationshi ith tim instad of ssu. Euation -15 toth ith th oduct ul and th dfinition of fluid comssibility ovid on connction ith tim: d dt M M M c c c d dt Euation -16 and 3

47 d dt M M M c c c d dt Euation -17 Th tim divativ of dnsity and satuation s oduct ovids anoth lationshi ith tim: d d d d d d d c d Euation -18 and d d d d d d d c d Euation -19 By substitutin Euation -18 and Euation -19 into Euation -16 and Euation -17, th dnsity chan ov tim can b found: d dt M sc Ah i sc Euation -0 and d dt M sc Ah i sc Euation -1 h 33

48 M c i c c M M M Euation - and M ic c c M M M Euation -3 Rcall th as and at at uations at th binnin of this chat (Euation -6 and Euation -7), th tim divativs of as and at dnsity can b xssd as d dt A i icibd, ss f Euation -4 and d dt A i icibd, ss f Euation -5 Th vious scald xonntial modl is limitd to lls oducin at constant BHP. But in ality, most lls oduc und vaiabl BHP and flo at. If Euation -18 and Euation - 19 a intatd btn tims t=tj and t=t, j 1 f d t D t j i d t Euation -6 34

49 and j 1 f d t D t j i d t Euation -7 Th xonntial fom duin a constant BHP iod ill b,, j f, f, j x D i t t j Euation -8 and,, j f, f, j x D i t t j Euation -9 ubstitut Euation - and Euation -3 into dnsity-basd as and at dlivability uations, th dnsity-basd flo at uations a sc t i x Di Mdt t j Euation -30 and sc t i x Di Mdt t j Euation -31 h 35

50 i sc h c i b i i D, ss Euation -3 i h c b sc i D, ss Euation -33 i i f Euation -34 i i f Euation -35 Th lativ-mability lation is natd basd on Coy s as and at to-has modl: i 1 i i Euation i i 4 Euation -37 Gas viscosity follos L t al. (1966), 36

51 K x X Y Euation -38 h K MW T 09 19MW T 1.5 Euation X MW T Euation -40 Y.4 0. X Euation -41 hil comssibility facto and as comssibility a dtmind by Abou-Kassm t al. (1990)'s mthod..4 Cas tudis Cas tudy - Ayala and Zhan (013) This is a hyothtical cas study fo lls oducin und vaiabl BHP in as and at tohas svois. Rsvoi and fluid otis a basd on Y and Ayala s (013) cas study. Th ll floin ssu ofil usd is listd in Tabl -1, and at ith viscosity of 1 c and comssibility of 3.5x10-6 si -1 is usd. Th initial at satuation and iducibl at satuation a st to b 0.50 and 0.15, sctivly. 37

52 Tabl -1 Pssu ofil fo vaiabl BHP Cas tudy t (days) P f (sia) Fiu -1 Comaison of as ats btn numical simulation sults and to-has scald xonntial modl fo vaiabl BHP cas study 38

53 Fiu - Comaison of at ats btn numical simulation sults and to-has scald xonntial modl fo vaiabl BHP cas study Fiu -1 and Fiu - coma as and at ats btn numical simulation and th tohas scald xonntial modl. Th modl ovids at match on th vaiabl BHP cas. Th bottomhol ssu dcass vy ya to stimulat th oduction ats, and th modl is accuat in ach iod. Gas at atio in Fiu -3 also indicats that th modl is mostly accuat, and Fiu -4 shos that th atio of GWR fom numical simulation to GWR fom th modl fluctuats aound 1. 39

54 Fiu -3 Comaison of as at atio btn numical simulation sults and to-has scald xonntial modl fo vaiabl BHP cas study Fiu -4 Gas at atio dviation fo vaiabl BHP cas study 40

55 .5 nsitivity Analysis.5.1 Initial Wat atuation i Wat amount in a as and at svoi can atly affct as oduction. Th lo th initial at satuation, th hih th as oduction at. Thfo, it is ncssay to tst th modl on diffnt initial at satuation. Fiu -5 Initial at satuation snsitivity analysis fo as at (vaiabl BHP) 41

56 Fiu -6 Initial at satuation snsitivity analysis fo at at (vaiabl BHP) Th vious cas study ith initial satuations of 0.3 to 0.9 a dmonstatd in Fiu -5 and Fiu -6. Aantly, th modl fails to dict as ats cisly ith initial at satuation of 0.9. This may b causd by bounday dominatd flo o th simlification of diffusivity uations (Andix D). If th oduction is und aly iod, th modl cannot succssfully dict flo ats bcaus it is dvlod und bounday dominatd flo. Whth th svoi has achd bounday dominatd flo o not can b dtmind by th ssu of th svoi xtnal bounday. Fiu -7 is th ssu distibution of th scnaio ith initial at satuation 0.9. Th svoi is dividd into 100 bloc, and th xtnal bloc ssu is, sia at th last day of oduction. Comad to th initial svoi ssu 5,000 sia, th svoi is und bounday dominatd flo. Thfo, th influnc causd by th uation simlification is th only asonabl xlanation. 4

57 Fiu -7 Pssu distibution on last day of oduction fo i=0.9 (vaiabl BHP).5. Iducibl Wat atuation i Th cas study is xtndd to scnaios ith iducibl at satuation fom 0.05 to 0.35 (initial at satuation is 0.5). Iducibl at cannot b oducd thus OGIP and OWIP ill not b influncd, but flo ats vais und diffnt i. Fiu -8 shos that ith hih iducibl at satuation, as at is hih at aly sta but lo at lat iod. 43

58 Fiu -8 Iducibl at satuation snsitivity analysis fo as at (vaiabl BHP) Fiu -9 Iducibl at satuation snsitivity analysis fo at at (vaiabl BHP) 44

59 In Fiu -9, numical solutions of at flo at baly chan hn iducibl at satuation is 0.05, 0.10 o 0.15, but th modl ovids a adually incasin tnd. Und a constant initial at satuation, lo iducibl at satuation lads to hih initial mobil at satuation, hich is dfind as m i i Euation -36 Bcaus of th simlification of diffusivity uations, th stimation of flo ats ill b influncd if th svoi has la amount of mobil at..5.3 Daina Radius Th imact of daina adius, o svoi siz, is invstiatd by tstin svois ith 50, 100, 00, 400 and 700 ft. All svoi and fluid otis bsid of daina adius main th sam ith th vious cas study. Gas and at ats fo lls oducin at vaiabl bottomhol ssu a ivn in Fiu -10 and Fiu -11. Th fct match btn CMG numical simulation and th to-has scald xonntial modl illustats that this modl is fasibl adlss of th svoi siz. 45

60 Fiu -10 Daina adius snsitivity analysis fo as at (vaiabl BHP) Fiu -11 Daina adius snsitivity analysis fo at at (vaiabl BHP) 46

61 .6 Concludin Rmas Th as and at to has scald xonntial modl can dict th as and at ats fo as and at to-has svois und BDF ith lls oducin at vaiabl BHP. Wat satuation and lativ mability a functions of ssu and tim thus influnc flo ats. nsitivity analysis shos that daina siz dos not affct th accuacy of this modl hil at satuation dos. A svoi ith hih at satuation is not suitabl fo th modl bcaus th fluid mobility divativs nlctd in th diffusivity uations affct th sults. 47

62 Chat 3 Poduction Data Analysis of Gas and Wat To-Phas Rsvois Usin a Dnsity-Basd Aoach 3.1 Chat ummay This chat intoducs a dnsity-basd aoach analyzin oduction data of as and at tohas svois. Basd on th dnsity fom of tan matial balanc uations and an xtnsion of Ayala and Zhan s (013) scald xonntial modl fo dy as, a n st of uations a ivn to dict oiinal as in lac (OGIP) and oiinal at in lac (OWIP). With oduction histoy and bottomhol ssu (BHP) ofil, OGIP and OWIP can b stimatd usin a staiht lin mthod on condition that all svoi and fluid otis a ovidd cisly. To cas studis a ovidd fo constant and vaiabl BHP scnaios, sctivly. 3. Bacound Oiinal as in lac (OGIP) is on of th most imotant lmnts hn valuatin a as fild. It can b volumtically stimatd o, mo ioously, focastd via matial balanc uation. Blasinam and L (1988) dictd OGIP fo vaiabl-at, ost-tansint flo usin adjustd ssu a and adjustd tim ta. Lat in 1993, thy intoducd th hamonic dclin fo as that dscibs vaiabl at and vaiabl ssu conditions on th basis of at-nomalizd ssu and matial balanc sudo-tim tamb: i f do sc 48

63 i sc f b a, ss OGIP c b i a, ss t amb Euation 3-1 h and tamb a dfind as iz i i 0 d z Euation 3- and t amb ic ti t 0 c t dt Euation 3-3 Euation 3-1 can b ittn in a staiht-lin analysis fom: i sc f 1 OGIPc i t amb b a, ss Euation 3-4 h OGIP can b comutd fom th slo of th staiht-lin lot. Lat in 003, th uation as ittn in th floin matial balanc fom of sc/[m(i)-m(f)] vsus nomalizd cumulativ oduction Qn by Matta and Andson: 49

64 m m i sc f 1 Q OGIP b * n b * Euation 3-5 h m() is th al as sudo-ssu, a commonly usd tm in as oblms, dfind by Al-Hussainy t al. in 1966: m m d z Euation 3-6 Rcntly, Zhan and Ayala (014) intoducd a dnsity-basd aoach to dict OGIP fo dy as svois und bounday dominatd flo (BDF): 1 G 1 OGIP sc sc i Euation 3-7 Th staiht-lin analysis of G vsus lads to a slo of OGIP ciocal, h sc sc and snts ava viscosity-comssibility dimnsionlss atio and dnsity dadon atio. This a ovids a mthod volvd fom Zhan and Ayala s aoach to stimat OGIP and oiinal at in lac (OWIP) fo as and at to-has svois. Associatd at ithout at influx is tan into considation. Dissolvd as in at o dissolvd at in as is nlctd in th systm. 50

65 3.3 Rscald Exonntial Modl fo Constant BHP Gas/Wat To Phas ystm in BDF tatin ith matial balanc consvation: GIP OGIP G Euation 3-8 and WIP OWIP W Euation 3-9 th dnsity-basd tan matial balanc can b xssd as i i G 1 OGIP Euation 3-10 and i 1 i W OWIP Euation 3-11 Fo as and at to-has svois, dnsity fom flo at uations (Andix A and Andix B) a 51

66 sc b D, ss h c sc i i f Euation 3-1 and sc b D, ss h c sc f Euation 3-13 h is th ava viscosity-comssibility dimnsionlss atio: m c i i f m f c i i c, f Euation 3-14 With th intoduction of dimnsionlss initial flo ats i and atios and, Euation 3-1 and Euation 3-13 can b ittn as i, and th dnsity dadon sc i 1 i Euation 3-15 and 5

67 sc i i 1 Euation 3-16 h i sc h c i i i b D, ss Euation 3-17 i h sc c i b D, ss Euation 3-18 i i f Euation 3-19 i i f Euation 3-0 ubstitutin th dnsity-basd matial balanc uations (Euation 3-10 and Euation 3-11) into Euation 3-15 and Euation 3-16, th uations can b aand to 53

68 i 1 1 G i 1 sc OGIP sc 1 i Euation 3-1 and i 1 W i 1 sc OWIP sc 1 1 i Euation 3- Euation 3-1 and Euation 3- dmonstat that OGIP and OWIP can b calculatd via staihtlin analysis. Plottin i 1 vsus sc 1 i G sc in a Catsian coodinat systm, th slo is th ciocal of OGIP. Th sam ocss os fo OWIP diction. Itations a ndd hn calculatin OGIP and OWIP. atuation, lativ mability and a tim-dndnt, but daily svoi and fluid otis a not ovidd oiinally. All ths otis should b udatd duin itation. Th ocdu of calculatin OGIP and OWIP a as folloin: 1. Plot i 1 vsus i sc condition, fo th fist stimation of OGIP; G sc, h all otis a calculatd at initial svoi. Guss an OWIP stimation to calculat fluid otis (,, tc.) and udat ith OGIP fom st 1; 3. Calculat a n OWIP usin fluid otis fom st and Euation 3-; 54

69 i 1 4. R-lot 1 and sc i G sc fo a n OGIP valu; 5. Rat sts to 4 until OGIP convs. Iducibl at satuation in this modl is uid as an inut, hil initial at satuation should b udatd ith OGIP and OWIP in th itation ocss usin th folloin uation: i OWIP OWIP OGIP * Euation 3-3 h OGIP* is th uivalnt OGIP in TB unit. 3.4 Cas tudis Cas tudy 3-A. Constant BHP Cas Th OGIP diction modl is fist tstd on a constant BHP cas ith oduction data shon in Fiu 3-1. Th ll is assumd to b oducin at 100 sia. Fiu 3- and Fiu 3-3 a th fist i 1 i G and last itation lots of 1 vsus, and Fiu 3-4 and Fiu sc sc 3-5 a th last itation lots fo as and at hass. Numical simulation ovids OGIP and OWIP of MMCF and 78,50 TB hil th OGIP and OWIP dictd by this modl is MMCF and 8,803 TB. 55

70 Fiu 3-1 Poduction histoy fo Cas tudy 3-A Fiu 3- Fist itation of as has in Cas tudy 3-A 56

71 Fiu 3-3 Fist itation of at has in Cas tudy 3-A Fiu 3-4 Final itation of as has in Cas tudy 3-A 57

72 Fiu 3-5 Final itation of at has in Cas tudy 3-A Cas tudy 3-B. Vaiabl BHP Cas In ality, most as lls a oducd und vaiabl BHP and flo at. Fiu 3-6 is th oduction histoy of a hyothtical vaiabl BHP and flo at cas hich has an OGIP of BCF and OWIP of 76,981 TB. Th ssu ofil is listd in Tabl 3-1. Fiu 3-7 and Fiu 3-8 dislay th fist itations of th to has dnsity aoach hil Fiu 3-9 and Fiu 3-10 sho th last itations, h OGIP and OWIP conv at BCF and 75,779 TB. 58

73 Fiu 3-6 Poduction histoy and ssu ofil fo Cas tudy 3-B Tabl 3-1 Bottomhol ssu ofil of Cas tudy 3-B t (days) P f (sia)

74 Fiu 3-7 Fist itation of as has in Cas tudy 3-B Fiu 3-8 Fist itation of at has in Cas tudy 3-B 60

75 Fiu 3-9 Final itation of as has in Cas tudy 3-B Fiu 3-10 Final itation of at has in Cas tudy 3-B 61

76 3.5 Concludin Rmas This chat intoducs a dnsity-basd aoach analyzin oduction data of as and at tohas svois. Basd on th dnsity fom of tan matial balanc uations and an xtnsion of Ayala and Zhan s (013) scald xonntial modl fo dy as, a n st of uations a ivn to dict oiinal as in lac (OGIP) and oiinal at in lac (OWIP). With oduction histoy and bottomhol ssu (BHP) ofil, OGIP and OWIP can b stimatd usin a staiht lin mthod. To cas studis a ovidd fo constant and vaiabl BHP scnaios, sctivly. 6

77 CONCLUION Th dnsity-basd scald xonntial modl oosd by Ayala and Zhan (013; 014) stimats as ats of dy as svois duin bounday dominatd flo (BDF), and alis to lls oducin und both constant and vaiabl bottomhol ssu (BHP). Th xtnsion in this study can dict both as and at ats in th abov to situations. Additional studis on initial at satuation and iducibl at satuation illustat that th amount of mobil at in a as svoi can affct th accuacy of th to-has scald xonntial modl. Hih at satuation causs dviation du to th divativ of mobility ith sct to loaction, hich is nlctd in th modl. Daina aa dos not influnc modl accuacy. In oth ods, th modl is suitabl fo any svoi siz. Zhan and Ayala (014) modifid thi scald xonntial modl to dict OGIP ith as oduction and ssu histoy. Th xtndd to-has modl invsly dicts OGIP and OWIP ith as oduction, at oduction, and ssu ofil. Th disadvanta of this mthodoloy is th uimnt of accuat svoi and fluid otis. In actic, lac of data ill caus la dviation in sv diction. 63

78 REFERENCE Abou-Kassm, J., L. Matta, and P. Danchu Comut Calculations Of Comssibility Of Natual Gas. Jounal of Canadian Ptolum Tchnoloy 9 (5). doi:10.118/ Aaal, Ram G Ral Gas Psudo-Tim - A N Function Fo Pssu Buildu Analysis Of MHF Gas Wlls. PE Annual Tchnical Confnc and Exhibition. doi:10.118/879-m. Al-Hussainy, R., H.J. Ramy J., P.B. Cafod, H.J. Ramy, and P.B. Cafod Th Flo of Ral Gass Thouh Poous Mdia. Jounal of Ptolum Tchnoloy 18 (05): doi:10.118/143-a-pa. As, J. J Analysis of Dclin Cuvs. Tansactions of th AIME 160 (1): doi:10.118/9458-g. Ayala H., Luis F., and Miao Zhan Rscald Exonntial and Dnsity-Basd Dclin Modls: Extnsion to Vaiabl- Rat/ssu-Dadon Conditions. Jounal of Canadian Ptolum Tchnoloy 5 (6): doi:10.118/1683-pa. Blasinam, T.A., and W.J. L Th Vaiabl-Rat Rsvoi Limits Tstin of Gas Wlls. PE Gas Tchnoloy ymosium. doi:10.118/17708-m. Cat, Robt Ty Cuvs fo Finit Radial and Lina Gas-Flo ystms: Constant- Tminal-Pssu Cas. ocity of Ptolum Enins Jounal. doi:10.118/1917-pa. Cox, Dav O. "Rsvoi limit tstin usin oduction data." Th Lo Analyst 19, no. 0 (1978). Faim, M.L Gas Rsvoi Dclin-Cuv Analysis Usin Ty Cuvs With Ral Gas 64

79 Psudossu and Nomalizd Tim. PE Fomation Evaluation (04): doi:10.118/1438-pa. J, M, and M.J. Ftovich Dclin Cuv Analysis Usin Ty Cuvs. Jounal of Ptolum Tchnoloy 3 (6): doi:10.118/469-pa. L, Al, Mh Gonzalz, and B Eain Th Viscosity of Natual Gass. Jounal of Ptolum Tchnoloy 18 (8): doi:htt://dx.doi.o/10.118/1340-pa. Matta, L, and D M Andson A ystmatic and Comhnsiv Mthodoloy fo Advancd Analysis of Poduction Data. Pocdins - PE Annual Tchnical Confnc and Exhibition, 373. doi:10.118/8447-m. Palacio, J. C., and T. A. Blasinam. "Dclin cuv analysis usin ty cuvs analysis of as ll oduction data." a PE 5909 (1993): Y, P, and L Ayala taiht-lin Analysis of Flo Rat vs. Cumulativ Poduction Data fo th Exlicit Dtmination of Gas Rsvs. Jounal of Canadian Ptolum Tchnoloy, no. in ss. Zhan, Miao, and Luis Ayala Gas-Rat Focastin in Bounday-Dominatd Flo: Constant-Bottomhol-Pssu Dclin Analysis by Us of Rscald Exonntial Modls. PE Jounal 19 (03): doi:10.118/16817-pa. Zhan, Miao, and Luis F Ayala H Gas-Poduction-Data Analysis of Vaiabl- ystms : A Dnsity-Basd Aoach, no. Novmb. 65

80 FUTURE WORK 1. Th simlification of diffusivity uations can b futh tstd to find out ho xactly it ill influnc th modl s accuacy by matchin Euation 1-0 ith CMG numical simulations.. Th to-has modl can b modifid fo oil/at to-has svois. 66

81 Andix A Divation of Gas Flo Euation Th nalizd dnsity-basd diffusivity uations fo as has is v t Euation A-1 By intatin th lft hand sid of Euation A-1, V v dv v nd scsc Euation A- By intatin th iht hand sid of Euation A-1, V t dv d dt V dv d dt V Euation A-3 Combin Euation A- and Euation A-3, d dt V Euation A-4 In cylindical coodinat systm ith incomssibl oc, d dt scsc Ah sc sc h Euation A-5 Alyin Dacy s La, as la and th dfinition of al as sudo-ssu to Euation A-1, th uation can b ittn as 67

82 m t Euation A-6 h RT MW Euation A-7 Rlativ mability bhavio in th svoi is dtmind by satuation; hov, th satuation distibution of th svoi is nith non no can b calculatd thouh oth vaiabls. Thfo, th as lativ mability in Euation A-6 is simlifid to b tim-dndnt only. This simlification may caus dviation in futh calculation of as and at flo ats. Intatin th lft hand sid of Euation A-6, hav V m dv h 0 0 m m dddz h Euation A-8 h th bounday condition is 0 Euation A-9 Intatin th iht hand sid of Euation A-6, hav 68

83 69 dt d h t dv t V Euation A-10 Combinin Euation A-8 and Euation A-10, hav dt d m Euation A-11 Intatin Euation A-11 ivs th xssion of as sudo-ssu: sc sc f h m m ln ` Euation A-1 Th ava as sudo-ssu of th svoi is h V d m dz dd m V dv m V m Euation A-13 hn >>. ubstitutin Euation A-1 into Euation A-13, and hav 4 3 ln sc sc f h m m Euation A-14

84 Euation A-14 can also b ittn as sc sc h c b i i D, ss f Euation A-15 h b D, ss ln 3 4 Euation A-16 m c m i i f f c i i c, f Euation A-17 70

85 Andix B Divation of Wat Flo Euation Th nalizd dnsity-basd diffusivity uations fo at has is v t Euation B-1 By intatin th lft hand sid of Euation B-1, V v dv v nd scsc Euation B- By intatin th iht hand sid of Euation B-1, V d d dv dv V t dt V dt Euation B-3 Combin Euation B- and Euation B-3, d dt V Euation B-4 In cylindical coodinat systm ith incomssibl oc, d dt sc Ah sc h sc sc Euation B-5 Euation B-1 can b ittn as 71

86 t Euation B-6 Alyin at comssibility ( d d c c ) to Euation B-6, hav t Euation B-7 h th at lativ mability is tatd as a location-indndnt vaiabl. This assumtion ill caus inaccuacy to som xtnt. But lativ mability is a function of satuation only, and satuation at any location in th svoi is an indndnt vaiabl. In oth ods, th satuation distibution of a svoi cannot b dtmind by oth vaiabls, such as ssu and tim. Wat viscosity and at comssibility a constant. Intatin both th lft hand sid and iht hand sid of Euation B-7 ov th svoi volum, hav d c h t h dt Euation B-8 ith bounday condition 0 Euation B-9 7

87 73 Thn th -divativ of at dnsity bcoms dt d c Euation B-10 By intatin Euation B-10, hav th xssion fo at dnsity: sc sc f h c ln ` Euation B-11 To dscib at flo at ith ava at dnsity, Euation B-11 nds to b intatd ov th nti svoi volum. Th ava at dnsity can b ittn as h V d dz dd V dv V Euation B-1 ubstitut Euation B-11 into Euation B-1 and hav 4 3 ln sc sc f h c Euation B-13 h >>. Raan th uation and hav th flo uation fo at:

88 74 f ss D sc sc b c h, Euation B-14 h 4 3 ln, ss D b Euation B-15

89 Andix C Pocdu of Gas and Wat Rat Estimations in th Extndd To-Phas Modl To bin ith, and lativ mabilitis should b calculatd at initial svoi condition: 1 m c m i i i i f f Euation C-1 1 i 1 i 1 1 i 1 i i Euation C- 1 i i 1 i 4 Euation C-3 Thn as and at ats at th fist tim st can b dtmind: 1 h 1 1 sc bd, ssscici i f Euation C-4 and 1 h 1 sc bd, ssscc i f Euation C-5 Th cumulativ oductions a calculatd fo th involvmnt of matial balanc: 75

90 G sc t Euation C-6 W sc t Euation C-7 Fo any lat tim st, itation is uid hn analyzin th cunt svoi condition. Usin th ssu fom th vious tim st as an initial uss ( [i] = <1> =i), n dnsitis and satuations at tim st a calculatd as [ i1] i x c i [ i] Euation C-8 [ i1] i i [ i1] 1 W 1 OWIP Euation C-9 [ i1] 1 [ i1] Euation C-10 1 [ i1] i G i 1 [ i1] OGIP Euation C-11 [ i 1] [ i1] M ZRT Euation C-1 76

91 inc caillay ssu is not considd, at th ocss fom Euation C-8 to Euation C-1 until th ava svoi ssu convs. Dnsitis and satuations und this ssu a usd as otis at tim st. and lativ mabilitis at tim st a calculatd: i 1 i i Euation C-13 i 1 i 4 Euation C-14 m c m i i i f f Euation C-15 Gas and at ats at tim st is thn calculatd: 1 h sc bd, ss sc ici f Euation C-16 and h sc bd, ssscc f Euation C-17 77

92 78 Cumulativ as and at oductions a 1 t G G sc Euation C-18 1 t W W sc Euation C-19 Rat th abov ocdu fo futh tim sts.

93 79 Andix D imlification of Diffusivity Euations Fo as has in as/at to has svois, th diffusivity uation is t Euation D-1 and fo at has: t Euation D- Th xansion of Euation D-1 and Euation D- is t M M Euation D-3 and t M M Euation D-4 In th modl, th fist tms in Euation D-3 and Euation D-4 hich contain th divativ of fluid mobility a nlctd. Whn thy a not small nouh to b nlctd, th modl fails to dict accuat sults.