A REVIEW OF CRITICAL CONING RATE CORRELATIONS AND IDENTIFYING THE MOST RELIABLE EQUATION

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1 A REVIEW OF CRITICAL CONING RATE CORRELATIONS AND IDENTIFYING THE MOST RELIABLE EQUATION A Dissetatin By Ali Kalili Submitted t te Scl f Petleum Engineeing f Univesity f Ne sut Wales In Patial fulfilment f te equiements f te degee f MASTER OF ENGINEERING July 005 Maj Subject: Petleum Engineeing Ali Kalili July 005

2 II ORIGINALITY STATEMENT I eeby declae tat tis submissin is my n k and t te best f my knledge it cntains n mateials peviusly publised itten by ante pesn, substantial pptins f mateial ic ave been accepted f te aad f any te degee diplma at UNSW any te educatinal institutin, except ee due acknledgement is made in te tesis. Any cntibutin made t te eseac by tes, it m I ave ked at UNSW elseee, is explicitly acknledged in te tesis. I als declae tat te intellectual cntent f tis tesis is te pduct f my n k, except t te extent tat assistance fm tes in te pject s design and cnceptin in style, pesentatin and linguistic expessin is acknledged. Signed Ali Kalili July 005

3 III ACKNOWLEDGEMENTS I uld like t acknledge my supevis, Pf. W. Val Pinczeski, f is guidance and suppt duing tis study. I als is t extend my special tanks t Petleum Univesity f Tecnlgy (PUT) ic pvided financial suppt f tis study. Ali Kalili July 005

4 IV ABSTRACT Te study f cning in il pductin is imptant because f uge ate pductin assciated it il pductin aund te ld eac yea. Estimatin f citical cning ate as been te subject f numeus studies and a numbe f celatins ave been epted. Tis study pesents a evie f te cuent available metds f estimating citical cning ate f bt vetical and izntal ells. Te vaius metds and celatins ae cmpaed and te assumptins n ic tey ae based evaluated. Flling cmpaisn made beteen te celatins, te mst eliable teies ae identified f bt vetical and izntal ells sepaately. Amng te celatins f vetical ells, tis study ecmmends t implicit metds pesented by Weatley and Aza Nejad et al. Tey detemined te il ptential distibutin influenced by ate cne it a emakable accuacy. F izntal ells, t metds, Jsi s equatin and Recem et al fmula, ae cnsideed t be te mst eliable. Jsi s equatin pvides le estimates tan Capen s celatin in ic te ate cne effect n il ptential as neglected. Te Recam et al fmula als gives a simila esult. On te le, te Recem et al metd is pefeed. Ali Kalili July 005

5 V TABLE OF CONTENTS ORIGINALITY STATEMENT...II ACKNOWLEDGEMENTS... III ABSTRACT... IV TABLE OF CONTENTS...V LIST OF TABLES... VII LIST OF FIGURES... VIII NOMENCLATURE...X Capte One: Intductin..... Wat is Cning?..... Pblems esulted fm cning Wate cning mecanisms Stable and unstable cne Evaluatin f cning penmena Fe cncepts in cning Citical cne ate Beaktug time Well Pefmance afte beaktug Optimizatin f cmpletin inteval Scpe f Tis Study...7 Capte T: Citical cning ate at vetical il ells8.. Citical cning ate at vetical il ells (single cning) Meye and Gade Metd Capen s Appac Te Abass and Bass Metd Te Gu-Lee Metd Te Hyland et al Metd Weatley s pcedue Te Aza Nejad-Ttike Pcedue... 3 Ali Kalili July 005

6 VI.. Simultaneus cning f gas and ate Te Meye and Gade metd Te Cieici-Ciucci Appac Cmpaisn Capte Tee: Cning at Hizntal il ells Cning beaviu at izntal ells Celatins used t calculate te citical cning ate in izntal il ells Capen s appac Te Gige Appac Jsi s Metd Te Yang-Wattenbage Celatin f Hizntal ells Cmpaisn Capte Fu: Summay Summay and cnclusin f celatins pesented f vetical ells80 4. Summay and cnclusin f celatins pesented f izntal ells Fute studies BIBLIOGRAPHY...8 REFERENCES APPENDICES...85 Appendix A Appendix B: Appendix C: Appendix D: Ali Kalili July 005

7 VII LIST OF TABLES Table. Example data used t plt Meye and Gade metd... 0 Table. Example data f Capen s appac... 3 Table.3 An example data f Abass-Bass metd... 6 Table.4 An example f esevi and fluid ppeties f Gu s metd.. Table.5 Data f Te Hyland metds... 4 Table.6 An example data f Weatley's metd Table.7 Resevi and fluids ppeties f te Cieici and Meye and Gade metd Table.8 Data f te Cieici metd Table.9 A Cmpaisn beteen diffeent celatins and Capen s metd Table 3. Typical data f capen s metd... 6 Table 3. An example data f Gige s tey (Bttm Wate dive mecanism) Table 3.3 An Example data f Jsi s celatin Table 3.4 An sample data f te Weiping-Wattenbage celatin Table 3.5 Te Citical il ates btained by diffeent appaces Ali Kalili July 005

8 VIII LIST OF FIGURES Figue. A Scematic f a esevi at static cnditins...3 Figue. A scematic da f ate and gas cning...3 Figue. Scematic illustatin f a esevi it Statinay cne...9 Figue. Te Meye-Gade equatin t detemine citical cning fl ate... 0 Figue.3 A scematic da f Capen s metd... Figue.4 te Cne eigt at diffeent Citical cning ate... 3 Figue.5 A mdificatin f Capen's metd by Jsi (99) Figue.6 Citical cning ate against dimensinless ell penetatin at diffeent distance fm tp f esevi, steady state... 6 Figue.7 Citical cning ate vesus dimensinless ell penetatin, unsteady-state... 7 Figue.8 L-pessue-gadient case, n unstable cne exists. (Qt<Qt<Qt3)... 9 Figue.9 Hig-pessue-gadient case, unstable cne exists (Qt3>Qt>Qt)... 9 Figue.0 RSC fl patten... Figue. A plt f citical cning ate vesus dimensinless ell penetatin... Figue. Dimensinless citical cning ate against factinal ell penetatin (Cited, Amad, 000)... 5 Figue.3 Cmpaisn beteen te Hyland metds... 6 Figue.4 Citical cning ate vesus ell penetatin f diffeent equatins... 3 Figue.5 Te Lcatin f elements accding t te Aza Nejad metd 33 Figue.6 A cmpaisn beteen te Aza Nejad-Ttike and te Weatley pcedue t calculate te value f citical ate Ali Kalili July 005

9 IX Figue.7 A cmpaisn beteen diffeent studies t detemine te citical cning ate Figue.8 A Scematic pfile f a simultaneus cning f gas and ate 40 Figue.9 A cmpaisn beteen te ate cning and simultaneus cning f gas and ate accding t te Meye and Gade Metd... 4 Figue.0 Diagammatic Repesentatin f a ate and gas cning system in a Hmgenus Fmatin Figue. Cmpaisn beteen te Cieici and Meye Metd at simultaneus cning Figue. Dimensinless functin f De=5 (Cited in Amad, 000) Figue.3 Dimensinless functin f De=0 (Cited in Amad, 000) Figue.4 Dimensinless functin f De=0 (Cited in Amad, 000) Figue.5 Dimensinless functin f De=30 (Cited in Amad, 000)... 5 Figue.6 Dimensinless functin f De=40 (Cited in Amad, 000)... 5 Figue.7 Dimensinless functin f De=60 (Cited in Amad, 000) Figue.8 Dimensinless functin f De=80 (Cited in Amad, 000) Figue.9 A scematic cmpaisn beteen celatins develped t calculate te citical il ate f vetical il ell Figue 3. A da f ate cesting bel a izntal ell Figue 3. Immbile ate cest bel a izntal ell Figue 3.3 A elatin beteen XA and Distance beteen cne apex and ell... 6 Figue 3.4 Nn-dimensinal WOC cuves Figue 3.5 Scematics f a izntal ell dainage vlume Figue 3.6 Te citical cning ate against te distance beteen izntal ell and WOC Figue 3.7 A vey simple sketc f y-z pfile f a izntal ell Figue 3.8 Te Yang-Wattenbage celatin Figue 3.9 Te Recem-Tuami Celatin Ali Kalili July 005

10 X NOMENCLATURE A=Well space (ft ) a = Nmalizing fact (=) a T = Tansfmatin fact B = Oil Fmatin Vlume Fact (bbl/stb) b pt = Dimensinless ptimum ell be lcatin D= Dainage idt alf distance beteen t izntal ell lines (ft) D b = Distance beteen WOC and izntal ell (ft) g = Gavity Cnstant, 9.8 m/s = Resevi tickness (ft) c = Cne Heigt (ft) p = Pefated inteval (ft) K = Bessel Functin f te de ze K= Resevi Pemeability (md) K = Resevi Pemeability elated t il (md) K = Resevi Pemeability elated t ate (md) K = Oil Relative Pemeability (Dimensinless) K = Wate Relative Pemeability (Dimensinless) L= Hizntal ell lengt (ft) L p = Lengt f pefatin (ft) M N p = Mbility ati = Cumulative il pductin (STB) Ali Kalili July 005

11 XI P = Oil Pessue (psi) P = Wate Pessue (psi) q q c q cd = Oil fl ate (STB/day) = Citical Cning ate (STB/day) = Dimensinless citical cning fl ate q c q cv = Citical Cning ate f izntal ell (STB/day) = Citical Cning ate f vetical ell (STB/day) e = Well adius (ft) = Effective ell adius (ft) e = Dainage adius (ft) De = Dimensinless dainage adius s = Skin fact S = Aveage il satuatin S c = Cnnate Wate Satuatin S = Residual il satuatin x a = Dainage idt (ft) X D = Dimensinless Dainage Widt Geek symbls ρ = Oil density (lb/ft 3 ) ρ = Wate Density (lb/ft 3 ) ρ = Gas Density (lb/ft 3 ) g µ = Oil viscsity (cp) µ = Gas viscsity (cp) g µ = Wate viscsity (cp) ϕ = Oil ptential (psi) Ali Kalili July 005

12 XII ϕ = Wate ptential (psi) ϕ De = Dimensinless ptential at ute bunday φ = Psity (factin) Ψ= Steam Line functin Ali Kalili July 005

13 Capte One: Intductin.. Wat is Cning? Petleum esevis usually cmpise tee segments, a gas cap, il zne and an aquife. As il pductin begins, due t pessue dadn aund te ell be, te fluid intefaces mve tad te pefated inteval causing te ate-il gas-il inteface t defm fm its initial sape t a cne sape. Tat is y tis penmenn is efeed t cning. If te ell is pduced at me tan a citical cning ate te unanted fluids eventually ill beak int te ell. Because f pducing a uge amunt f te esevi ate assciated it il evey yea, numeus studies ave been cnducted n ate and gas cning mecanism in te vetical and izntal il ells. Mst f studies ae cncened it pedictin f te citical cning ate... Pblems esulted fm cning Tee ae seveal pblems caused by ate gas cning. Oil esevis can seiusly be impacted by cning in tems f ell pductivity, depletin degee and te veall ecvey efficiency (Sbcinski, 965). Envinmental impact f uge vlumes f te esevi ate pduced t te suface is als a seius pblem, ic as diffeent ppeties fm suface ate even seaate, fls n te gund. As a esult, an extensive pat f te gund lcated aund te ell is damaged. Als, fling ate tug casing and suface facilities may esult in csin. Ali Kalili July 005

14 Meve, in tems f cst, tee ave t be sme exta suface facilities t sepaate te pduced ate fm il..3. Wate cning mecanisms Cnside Figue (.) t be a ell patially penetated in a esevi at static cnditins. In fact, tee ae tee essential fces playing key le in te cning mecanism. Tey ae capillay, gavity and viscus fces (Amad, 000). F simplicity, te pcess is assumed t be dminated by viscus fces and capillay fces ae teefe neglected. Befe pductin, te gavity fce, ic is a cnsequence f te density diffeence beteen te fluids, is dminant. Once a ell is alled t pduce il, te viscus fces ic esult fm pessue dadn incease. In de t cuntebalance te system, ate il cntact (WOC) defms and mves up until viscus fce is balanced by te gavitatinal fce at a cetain elevatin, tat is, at a cetain fl ate, tee is a pint at ic a balance can be acieved beteen te viscus fce and te gavity fce. If suc a balance is neve acieved te cne ill be dagged up until it ill beak int te ellbe (figue.)(ozkan and Ragavan, 990). Te sape and te natue f te cne depend n seveal facts suc as pductin ate, mbility ati, izntal and vetical pemeability, ell penetatin and viscus fces (Iniki, 00). Ali Kalili July 005

15 3 Figue. A Scematic f a esevi at static cnditins Figue. A scematic da f ate and gas cning.4. Stable and unstable cne Stable cne efes t a static cne fmed bel te pefatin f ic te viscus gadient is balanced by te gavitatinal fce esulting te defmatin f inteface. Tis is te nly pblem f te ell fling bel a citical cning ate. If te ell fls abve te citical cning ate, te viscus fce dminates and te cne is dagged int te ellbe. It is imptant t nte tat a stable cne can nly be maintained f a peid. Ali Kalili July 005

16 4 Because f te upad mvement f te fluids cntact duing pductin and pessue depletin, te il ptential distibutin canges aund te ellbe..5. Evaluatin f cning penmena.5.. Fe cncepts in cning A suvey f te liteatues ss tat a temendus amunt f eseac as been cnducted n cning. Tis anges fm expeimental cning studies t analytical and numeical simulatin studies aimed at undestanding and pedicting ate cning in vetical and izntal ells (Ku and DesBisay, 983). In de t evaluate ate and gas cning, tee ae tee essential cncepts, ic ae called citical cning ate, beaktug time and pst beaktug. Te citical cning ate efes t a maximum il fl ate fm a ell at ic te ell ill nt cne ate gas. Heve, due t ecnmic necessity, il cmpanies ften pduce at a ate f ige tan citical cning ate. Tis causes ate gas cning simultaneus cning f ate and gas. Teefe, if te il fl ate f a ell exceeds te citical cning ate calculated f tis ell te cne becmes unstable and ill beak int te ellbe afte a cetain time. Tis peid is called te beaktug time (Amed, 000)..5.. Citical cne ate Seveal studies ave been cnducted t estimate te citical cne ate. In geneal, te slutins pvided can be classified in tee categies; Ali Kalili July 005

17 5 (i) Analytical slutins, (ii) Expeimental studies and (iii) celatins btained using numeical esevi simulatin (Ku and DesBisay, 983) Beaktug time As nted ealie, in pactical cases, a stable cne exists nly f a limited time and nce te pductin ate exceeds te citical cning ate te cne mves tad te ell and subsequently beaks int te ellbe. At tis stage, aving knledge f beaktug time may elp t impve ell management and extend ell life itut pductin f ate fee gas (Wagenfe and Hatzignatiu, 996). Wen te unanted fluid beaks int a ellbe, te fluid distibutins and te elative pemeability cange. Teefe, by estimating te beaktug time ne can ptimize te pductin plan t maximize te delay f ate gas beaktug time. Tus, pedictin f te beaktug time is cucial f il ells subject t ate cning (Ozkan and Ragavan, 990). Te beaktug time as been te subject f vaius studies. F instance, Sbcinski and Cnelius (965) and Bunazel and Jeansn (97) ppsed empiical celatins f beaktug time pedictin. As ante example, Ozkan and Ragavan (990) develped an analytical metd t investigate te beavi f a ate/gas cne and als t pedict te beaktug time and cncluded tat a izntal ell may impve te beaktug time cmpaed t a vetical ell. Ali Kalili July 005

18 Well Pefmance afte beaktug Due t ecnmic necessity mst il ells fl at a ate ige tan te citical cning ate. Once ate beaks int a ellbe, te ell pefmance becmes imptant and meits caeful attentin. Tis pedictin migt elp a esevi enginee t plan te futue pductin t acieve an ptimum cumulative il pductin. Ante advantage f ate cut pedictin is tat te ell life abandnment time f a ell can be anticipated. A fe studies ave been cnducted n ate cut pefmance afte beaktug. F example, Bunazel and Jeansn (97) ppsed an empiical celatin t pedict te ate cut. Altenatively, te cmmecial numeical simulats can be used t evaluate te ate cut f bt vetical and izntal ells Optimizatin f cmpletin inteval Optimizatin f te lcatin f te pefated inteval f a ell subject t pssible cning as been a subject f investigatins. Taditinally, il cmpanies used t pefate ells at te cente f il zne t ptimize te distance beteen ell and fluids inteface (WOC, GOC). Heve, te gas and ate due t significant diffeence in density and mbility ati eac te ellbe at diffeent times. F tis easn, te pefated inteval suld be lcated itin te il zne s tat ee bt gas and ate beak int te ell simultaneusly and teefe a maximum beaktug time Ali Kalili July 005

19 7 and cumulative il pductin can be acieved (Wagenfe and Hatzignatiu, 996)..6 Scpe f Tis Study In tis study, seveal celatins develped f detemining te citical cning ate ill be discussed. Specifically, in capte t te appaces elated t citical cning ate in vetical ells ae cnsideed. It is assumed tat te esevi cntains nly ate and il. As a esult, nly tse celatins dealing it te ate cning ae cnsideed it te exemptin f t studies cnducted n simultaneus cning f ate and gas in vetical ells. In capte tee, te cuent available metds f izntal ells ae descibed. Like capte t, te system ic is cnsideed is ate-il system. At te end f captes t and tee, a cmpaisn is pesented beteen all celatins t identify te mst suitable metd t calculate te citical cning ate f eac f te vetical and izntal ells. Finally, te last capte pesents te cnclusins. Ali Kalili July 005

20 8 Capte T: Citical cning ate at vetical il ells.. Citical cning ate at vetical il ells (single cning)... Meye and Gade Metd Meye and Gade (954) develped an equatin t detemine te value f te citical cning fl ate f a static cne at te base f te pefatin. A ell it dept penetatin f D int a esevi it tickness f, ic is undelain by an aquife, is depicted in figue (.). Let H and H be te fl ptential f il and ate, espectively, defined as flls: H = z + P P') /( gρ ) (.) ( H = z + P P') /( gρ ) (.) ( Wee z is te cne eigt abve an abitay efeence level and P is efeence pessue, P and P epesent pessue at te il and ate pase espectively. Since te ate cne as assumed t be static, H is a cnstant. Neglecting capillay pessue e can ite, H = H ρ ρ z + ( ρ ρ ) ρ (.3) Dacy s la is applied as flls: q K = π gρ ( z) µ dh d (.4) H is substituted fm equatin (.), giving Ali Kalili July 005

21 9 q K = π g( ρ ρ )( z) µ dz d (.5) Aanging vaiables and integating, q e d = πg ( ρ K ρ ) µ 0 D ( z) dz (.6) Teefe, te citical cning ate in field unit is itten as q c = B ρ ρ ln( e / K ) µ ( D ) (.7) Wee: q c : STB/day, ρ, ρ : lb/ft 3, e,, : ft, K :md, µ :cp, B =bbl/stb Figue. Scematic illustatin f a esevi it Statinay cne Ali Kalili July 005

22 0 Bel, te citical cning ate against te factinal ell penetatin as been pltted by using data sn in table.. Table. Example data used t plt Meye and Gade metd Paamete H R e K ρ Μ Value Units ft Ft md g/cm 3 Cp ft 30 Citical fl ate(stb/day) Dimensinless dept penetatin( ell penetatin/esevi tickness) Figue. Te Meye-Gade equatin t detemine citical cning fl ate As expected, figue. ss tat te value f te citical cning ate inceases it deceasing factinal ell penetatin. Ali Kalili July 005

23 ... Capen s Appac Capen (986) develped a metd t detemine te Citical cning ate f gas cning (figue.3). Te metd is als diectly applicable t te case f ate cning. Figue.3 A scematic da f Capen s metd Te aut assumed tat te ell as a l penetatin s tat it culd be cnsideed as a pint suce f estimating te fl ptential ic cespnds t emispeical fl (see Appendix A.). In de t acieve a static equilibium, te fl ptential is equated t te gavity ptential (Eq.A.3). Te metd f images is applied t detemine te fl ptential diffeence beteen pints A and S (Figue.3). Te ell is cnsideed t be a pint suce lcated at te igin f a semi-infinite pus medium. A n fl plan is placed at z=0 and a n fl bunday at z=. Te equatin f calculating te gavity ptential is itten as, Ali Kalili July 005

24 Φ A Φ B = ρg Z ) ( s (.8) Tus, by equating te fl ptential diffeence t te gavity ptential diffeence (Equatin.8), te citical cning ate in field units is itten as, k q ) 4 c = ( ρ qc Bµ (.9) Wee q c is te value f te citical cning ate (STB/day) and q * is dimensinless fl ate (See Appendix A.). All vaiables ae in field units as flls: K and K v : izntal and vetical pemeability, espectively; md : Resevi tickness ; ft A : Dainage adius f steady-state(ee inteface elevatin is in te case pseud-steady-state A =0.607 e q c * : dimensinless fl ate B : il fmatin vlume fact; RB/STB In equatin.9, q* c is a functin f A k ( )( k v / ). Tis accunts f anistpy. As an example, te citical cning ate against diffeent cne eigts as been pltted in figue.4. Ali Kalili July 005

25 3 Table. Example data f Capen s appac Paamete H A K v K B ρ µ W Value Unit Ft ft Md md RB/STB g/cm 3 cp ft Citical il ate(stb/day) Dimensinless cne eigt Figue.4 te Cne eigt at diffeent Citical cning ate As sn, te cne eigt inceases it inceasing citical cning ate until te citical cning ate eaces t a peak, ic is smeat aund 33% f te esevi tickness. Afteads, te value f te citical cning ate declines because te gavity fces cannt pevail ve te viscus fces esulted fm ige fl ate. Cnsequently, t acieve a stable cne, te ell suld be pduced at a le ate. Ali Kalili July 005

26 4 Altenatively, Jsi (99) extended te Capen metd s tat it culd be applied f diffeent factinal ell penetatin (Cited in Amad, 000). k ( ) 6 p.943 q c = ρ (0.73+ ) (.0) µ B De As sn in figue.5, te Jsi mdificatin gives a maximal value f abut 68 STB/day at l ell penetatin. Tis can be cmpaed t te maximum citical cning ate btained fm te figue.4, ic is 54 STB/day Citical Oil Rate(STB/day) Factinal Well Penetatin Figue.5 A mdificatin f Capen's metd by Jsi (99)...3. Te Abass and Bass Metd Abass and Bass (988) deived t equatins t detemine te citical cning ate f bt te steady state and unsteady-state cnditins. He Ali Kalili July 005

27 5 cnsideed tat te fl as adial aund te ellbe. In additin, He pinted ut tat tee paametes cntlled te value f te citical cning ate.(i) Te Radius f te cne ( ). (ii) Te Well penetatin (z). (iii) Te cne eigt ( c ). Seveal cmpute uns ee made t find a elatinsip beteen and c (equatin.). c k k v (.) F steady state fl system te citical cning ate as detemined by te flling equatin in field units (see Appendix B.). q c = πk g ρx( N x) µ B ( / + ln( )) (.) And f Unsteady State fl cnditins (see Appendix B.) q c = µ B ( πk g ρx( N x) ln( ) 4 e + ) (.3) Wee g : Gavitatinal cnstant=9.83 m/sec x: Dimensinless ell penetatin N: distance beteen te tp f an il zne and ee a ell is cmpleted (ft) : Resevi tickness (ft) Ali Kalili July 005

28 6 As an example, te citical cning ate is pltted against ell penetatin f bt steady state and unsteady state cnditins in figues.6 and.7 Table.3 An example data f Abass-Bass metd Paamete H e K v k ρ Μ B Value Unit Ft Ft Md md g/cm 3 Cp RB/STB Citical il fl ate(stb/day) N/=0 N/=0. N/= Dimensinless Dept Penetatin Figue.6 Citical cning ate against dimensinless ell penetatin at diffeent distance fm tp f esevi, steady state Ali Kalili July 005

29 7 Citical il Rate(STB/day) N/=0 N/=0. N/= Dimensinless dept penetatin Figue.7 Citical cning ate vesus dimensinless ell penetatin, unsteady-state Nt supisingly, figues (.6) and (.7) eveal tat as te pefated inteval mves dn fm tp f te esevi te citical cning fl ate deceases s tat nce penetated inteval tuces te WOC, te citical cning ate becmes ze. Te citical cning ate diffeence beteen Steady-state and unsteady-state cnditins depends n te tem (- - )/4 e ic is clse t ze. Cmpaing figues (.7) and (.) eveals sme diffeences. F example, Abass and Bass cnsideed tat te pefated inteval culd be lcated at any distance fm tp f te esevi eeas Meye and Gade assumed tat te pefated inteval stated fm tp f te esevi. In additin, figue (.7) ss tat te maximum citical cning fl ate ccus at 50% ell penetatin; eve, it is acieved at t l penetatin f figue (.). Te maximum citical cning ates ae Ali Kalili July 005

30 8 STB/day and 5 STB/day f te Abass-Bass and te Meye-Gade metd, espectively...4. Te Gu-Lee Metd Gu and Lee (993) stated tat te existence f an unstable ate cne depended n te vetical pessue gadient beneat te ellbe. Wen te vetical pessue gadient is ige tan te ydstatic pessue gadient f ate, an unstable ate cne can fm. In additin, e pinted ut tat te citical cning ate ad t be defined as a ate at ic a stable cne se tad a ellbe and as te maximum ate-fee ate f te ell. Te analysis is limited t steady state cnditin. Tee is a similaity beteen is metd and te Abass-Bass metd. In bt metds te citical cning ate is ze en te factinal ell penetatin is ze and ne. Heve, Abass and Bass assumed fl system t be D adial fl ilst Byun cnsideed adial/speical/cmbined (RSC) 3D fl system. Te Gu and Lee vie f te ate cning mecanism He stated tat te upad dynamic fces esulting fm ellbe dadn pessue caused ate t ise t a eigt ee te viscus fce as balanced by te eigt f ate beneat tis pint. In de t explain te mecanism e classified cning penmena in t categies; (i) lpessue gadient case, ee tee as n unstable cne (fig.8),(ii) igpessue gadient, ee tee as an unstable cne (fig.9). Ali Kalili July 005

31 9 Figue.8 L-pessue-gadient case, n unstable cne exists. (Qt<Qt<Qt3) Figue.9 Hig-pessue-gadient case, unstable cne exists (Qt3>Qt>Qt) Figues.8 and.9 illustate t cases in ic te pessue distibutin alng te vetical diectin as been dan. Line A-B epesents te pessue distibutin in te il zne en te fl ate is ze (Statinay Oil Pessue) and Line B-C ss te pessue distibutin cuve in te ate zne. As can be seen in tese figues, en te fl ate inceases fm ze te il pessue distibutin is sifted tad left. Ali Kalili July 005

32 0 F te l-pessue-gadient case, te ck cnductivity, ic is defined as te pemeability f a esevi ck divided by fling fluid viscsity, is ig enug s tat te ell be pessue des nt fall bel te ydstatic ate pessue befe ate beaktug. Teefe, te il pessue distibutin cuve intesects te line B-C at nly ne pint. As a esult, te eigt f te intesectin is te eigt f te stable cne if te capillay pessue is assumed t be negligible. In cntast, f te igt-pessue gadient case te ck cnductivity is l ic esults in a ellbe pessue bel te ydstatic ate pessue. Te il pessue distibutin cuve can intesect te ate pessue distibutin line at t pints. Te le eigt is te stable cne eigt. As te il fl ate inceases te t pints mve tad eac te until tey meet at ne pint. Tis pint is igest pint at ic a cne can be stable. Matematical Deivatin In de t deive an equatin t appximate te citical cning ate, te fl system f cmpletin inteval as cnsideed adial abut te ell and a speical fl-patten patten dminated nn-penetated il-zne. In te ds, te adial speical cmbined (RSC) fl system as a cmbinatin f a unifm line-sink adial fl at uppe pat and a pint-sink semi-speical-fl field at le pat. Tus, te ttal fl ate as detemined by summing bt f t fl ates. Te maximum ate-fee il fl ate as appximated by te equatin (.4). Ali Kalili July 005

33 Figue.0 RSC fl patten(cited in Gu and Lee(993) q c k = µ v ρg ( e e ( x)) e ( k k v + k v + x( ) e ) e ln( ) (.4) Wee K v : vetical pemeability (md) K : izntal pemeability (md) g: gavitatinal cnstant(9.8m/sec ) e : dainage adius(ft) : esevi tickness(ft) ρ: density diffeence (g/cm 3 ) Ali Kalili July 005

34 Figue (.) depicts a typical cmpaisn beteen te Abass-Bass metd and te Gu-Lee metd. Table.4 An example f esevi and fluid ppeties f Gu s metd Resevi and fluid ppeties H e K v k ρ Μ Value Units Ft Ft Md Md g/cm 3 Cp ft Citical il Fl ate(stb/day) Abass & Bass Gu Dimesinless ell penetatin Figue. A plt f citical cning ate vesus dimensinless ell penetatin Te figue (.) ss a typical cmpaisn beteen t simila appaces. As sn, te value f te citical cning ate is ze f bt celatins at ze dimensinless ell penetatin. In egins ee te value f dimensinless penetatin is beteen ze and ne, bt cuves fll diffeent pats. Heve as tey appac unit dimensinless Ali Kalili July 005

35 3 penetatin, te value f te citical cning ate f bt te cuves is again ze. Meanile, tee is a significant diffeence in te value f maximum citical cning ate. F instance, in tis case te value f te maximum citical cning ate is 7 STB/day f te Abass-Bass equatin eeas Gu- Lee s celatin gives 53 STB/day. Tis diffeence migt be a cnsequence f te fact tat Abass and Bass cnsideed nly adial fl ile Gu and Lee cnsideed adial fl at pefated inteval and als semi-speical fl (RSC) beneat te ell. In te ds, in te Gu-Lee metd te ttal citical cning ate is sum f bt fl ates elated t adial fl and semispeical fl subsequently it gives ige esult tan te Abass-Bass equatin...5 Te Hyland et al Metd Hyland et al (989) pesented t metds t pedict te value f te citical cning ate. Te fist appac as based n te esults f a lage numbe f a tee-pase black il simulat uns. In te ds, sensitivity analysis uns ee made n tse paametes aving effect n te value f te citical cning ate s tat f eac set f paametes, te citical cning ate as detemined and ten tey used a egessin analysis t cme up it an equatin. Tis equatin can be applied f istpic esevis. q c ρ L (.5) 08B µ k ( ρ ) p = ( ( ) ) ln( e ) Ali Kalili July 005

36 4 Te secnd metd as a pcedue, ic as an extensin f te Muskat- Wyckff tey. Flling te Muskat and Wyckff tey, te cne influence n il ptential as neglected. As a esult, it is expected t pvide an ptimistic evaluatin f citical cning ate. One advantage f tis pcedue is tat it can be used t detemine te value f te citical cning ate f an anistpic esevi. Te flling pcedue is used t calculate te citical cning ate. -Calculate te dimensinless adius by using equatin (.6). = De e k v k (.6) -Detemine dimensinless citical cning ate f seveal factinal ell penetatins fm figue (.) f a dimensinless adius btained at step. 3- Plt dimensinless citical cning ate against te ell penetatin. 4- Estimate te dimensinless citical cning ate at given factinal ell penetatin. 5-Use equatin (.7) t find te citical cning ate. q c = (ρ ρ ) k (.7) B µ Table.5 Data f Te Hyland metds Resevi fluid ppeties & H e K v k ρ Μ Value 70 ft 000ft 0 md 00 md 8.7 lb/ft 3 cp.5 ft Ali Kalili July 005

37 5 Figue. Dimensinless citical cning ate against factinal ell penetatin (Cited, Amad, 000) Ali Kalili July 005

38 secnd metd fist metd Citical il ate(stb/day) Factinal ell penetatin Figue.3 Cmpaisn beteen te Hyland metds As depicted in figue (.3), te esult btained by te secnd celatin is te ige ate. Tis esult is expected because te cne sape effect n te il ptential distibutin as nt been taken int accunt in tei study...6 Weatley s pcedue Te pevius metds alled te citical cning ate t be calculated explicitly. We n cnside a numbe f pcedues en te cning ate is calculated implicitly. Weatley (985) pesented a ne tey based n ptential distibutin f t-pase fl t calculate te value f citical cning ate. Te esevi as cnsideed t be mgenus, bunded abve by a izntal impemeable baie and bel by WOC ic as dagged Ali Kalili July 005

39 7 upad. In additin, te il influx at te investigatin adius as assumed t be steady and adially symmetic. He expessed te fluid ptential in il pase as flls: Φ = q( A + aa + ba A4 ) /( ax ) (.8) Wee 3 b A (, z) = ln( ) g(, z y) + g(, z + ) (.9) y A (, z) = g(, z x) g(, z + ) (.0) x A (, z) = / f (, z x) + / f (, z + ) (.) 3 x A = A, z) + aa (, z) + ba (,) (.) 4 ( e e 3 e In ic f + (, z) = z (.3) g (, z) = ln( z + f (, z)) (.4) Te unknn paametes q, q DC, a, b and Y ae line suce stengt, dimensinless citical cning ate, elative line suce and pint suces stengt, espectively. Wen a, b, Y, q D ae detemined te line suce stengt can be calculated by: q = qd ρ g( ax b) (.5) Te WOC as cnsideed t be a steamline because f n fl acss te WOC, tus by applying (C-, see appendix C) te steam line culd be expessed as a functin f q, a and b. Ψ = q( B + ab + bb3) /( ax ) (.6) Wee b B (, z) = z + f (, z y) f (, z + ) (.7) y Ali Kalili July 005

40 8 B (, z) = f (, z + x) f (, z ) (.8) x And B (, z) = ( z x) / f (, z x) ( z + x) / f (, z + ) (.9) 3 x N, by cnsideing ze steamline at z=0 and n te ell axis f z>x, equatin.6 as eitten as Ψ = q (.30) Substituting.30 int te.6 ax b = B + ab + bb (.3) 3 A elatinsip beteen a and b in tems f y as btained by nting tat equatin.3 suld be satisfied at te pint ( e, ). Tus f lage value f e ax b = ( Y ) /( ) (.3) e e Equatin.3 implies tat tee is a stagnatin pint at sme pint z=z s n te ell axis ee φ / z = 0 Tus, fm equatin (.8) Y = z + z ( z x ) /( ax b bx /( z x ) (.33) s s s s In te case f nne-penetating ell, tat is, x=0, equatins (.3) and (.33) detemine b and Y it espect t z s f << x tis elatin is simplified t equatin (.34).( see figue C-) a ln( / x) b / = ln( x / Y ) (.34) In de t detemine a, b and Y, equatins (.3)-(.34) ee slved easily by iteatin metd stating it Y=. Ali Kalili July 005

41 9 N by applying cnditin (C-9) t substitute ptential functin, te equatin (.8) becmes z = + qd ( A + aa + ba ) (.35) 3 A4 Wee q D is dimensinless suce stengt. Equatin (.35) at cne eigt ee z=z c and =0 culd be itten q D = ( zc ) /( A4 A aa ba3 ) (.36) Wee A ln( Y z ) ln(4) (.37) A A = c + = ln(( z x) /( z )) (.38) c c + x = x /( z ) (.39) 3 c c x and A = / (.40) 4 ln( e ) + ( Y ax + b) e A gd matc f te WOC equatin and te steamline equatin as btained by equiing te apex f te cne (z c ) t cincide it te stagnatin pint (z s ). Equatin (.36) it z c =z s and it Y, a and ee calculated fm equatins (.3)-(.34) t give te dimensinless suce stengt q D in tems f te psitin f te cne apex. Weatley elated te il pductin ate t suce stengt tug equatin Q = 4π k ρgq / µ (.4) c O in il field units c D D Q = k ρ q / µ (.4) Q c :RB/day, :ft, k :md, ρ:gm/cm 3, µ :cp Weatley pvided te flling pcedue f calculating te value f te citical cning fl ate. ).Assign an initial value f z c =z s sligtly less tan. Ali Kalili July 005

42 30 ).Calculate Y, a and b iteatively fm equatins (.3)-(.34) stating it Y= 3). Detemine q D fm equatin (.35) 4).Reduce z c =z s and ecalculate a, b, Y and q D. 5).Repeat step (4) until a maximum in q D is btained 6).Heeinafte, educe z c and ecalculate q D fm equatin (.6) emain te value f z c, Y, a and b uncanged. 7).Repeat te abve step until a ne maximum in q D is btained. 8). te value f citical cning ate (Q c ) can be calculated by te maximum amunt f q D btained fm step (7). Since te WOC equatin (Eq..35) and bunding steamline (.3) ae equied t be identical, te value f te dimensinless dainage adius is t be beteen and 0 t satisfy tis cnditin. Teefe, tis pcedue gives me accuate esults f te values f te dainage adius beteen and 0. Due t cnsideing te cne sape effect n te il ptential distibutin, tis pcedue may give me accuate esults tan pevius celatins. A typical gap as been pltted accding t data given in table.6. Table.6 An example data f Weatley's metd Paamete H R e K ρ Μ Value Units Ft Ft md g/cm 3 Cp Ft Ali Kalili July 005

43 3 Te figue (.4) ss te beaviu f te citical cning ate value against te factinal ell penetatin by diffeent celatins. 60 Citical Oil fl ate (STB/day) Weatley Meye & Gade Abass & Bass Byun Gu Dimensinless ell penetatin Figue.4 Citical cning ate vesus ell penetatin f diffeent equatins As sn, te Weatley and te Meye-Gade metds fll a simila tend. Heve, as tey appac l ell penetatin te Weatley appac gives a ige value f te citical cning ate. In cntast, te Gu and Abass equatins pvide ze STB/day f citical cning ate at ze penetatin...7 Te Aza Nejad-Ttike Pcedue Aza Nejad and Ttike (995) develped an analytical metd t detemine te il ptential in te il zne, ic is a slutin f te Laplace equatin, in a ectilinea esevi f any cmpletin inteval. In tis metd, nce te il ptential distibutin in te esevi is calculated te Ali Kalili July 005

44 3 pductin ate and te cne eigt can be btained. In additin, te effect f te cne sape n te ptential distibutin as taken int accunt by applying a tansfmatin ule t tansfm te defmed dmain int a ectilinea dmain. Ptential Distibutin Unifm Flux alng te Wellbe: All te dimensins ee nmalized by b as flls: b =, = R / b, = R / b, = Z / b, = l b (.43) e e b p / Wee is te esevi tickness. Te fm f te Laplace equatin f ptential distibutin at steady state cnditin is ϕ ϕ ( ) + = 0 (.44) Wee ϕ = 0, z = 0, b / (.45) ϕ = cnst., = (.46) e Madelung (98) detemined te ptential in a ectilinea dmain itut cnsideing cnstant ptential at te ellbe i.e. ϕ p (,, p ) = 4q[ ( K 0 (πn)cs(πn)cs(πn p )) + ln( )] (.47) Wee p is te lcatin f a pint suce in Z axis and K 0 is Bessel functin f de ze. Te equatin f a line suce extending fm tp f a esevi it lengt f x is Ali Kalili July 005

45 33 ϕ L (,, x) = 4q[ ( K 0 (πn) cs(πn)sin(πnx) + x ln( ) (.48) π n Equatin (.48) guaantees a unifm flux alng te ell be, eve; it as a singula pint n te Z-axis at =0. It des nt satisfy te cnstant ptential cnditin at te ell be. Cnstant Ptential Alng te ellbe: A supepsitin metd as applied t btain a cnstant ptential alng te ellbe. In tis metd, six line suces and t pint suces ee used s tat te line suces ee patially penetating ell. In te case f unifm flux, evey t line suces fm a flux elements, teefe, tee ae five flux elements including t pint suces as flls: Figue.5 Te Lcatin f elements accding t te Aza Nejad metd -Flux Element (I) Tis element as te same lengt as te iginal ellbe, ic extends fm XS t XE. -Flux Element (II) Ali Kalili July 005

46 34 It is a line it pducing lengt f ne sixt f iginal ellbe lengt esticted beteen XS and XE.Tis element is ceated by supeimpsing t patially penetating ell extending fm tp f te esevi t XS and XE espectively. -Flux Element (III) Like te t te elements, it is a cmbinatin f t patially penetating ells stating fm XS and extending t XE. Flux Element (4 & 5) As sn in figue (.5), tese t flux elements ae t pint suces, ic ae lcated at te tp and te bttm f te fist line suce (sink). Xp=XS and Xp=XE Applying te supepsitin ule, ne can ite te ptential at any pint in te esevi as flls: ϕ(, ) = q ϕ ( xs, xe,, ) + q ϕ + q ϕ 5 (.49) p L ( xe,, ) L ( xs, xe,, ) + q ϕ ( xs, xe,, ) + q ϕ ( xs,, ) 3 L3 4 p In de t detemine te unknns a t a 5, ic ae elements stengt, te equatin (.49) as applied f five diffeent pints n te ellbe suface and tey ee fced t be an unique value ic as ell ptential. Ante equatin as needed t slve te system f equatins. Tis equatin as called te cnstain equatin meaning tat t make ptential independent f pductin ate all te ptentials ee calculated at unit ate. Ali Kalili July 005

47 35 5 i= q = (.50) i Once te ptential at any pint is btained, ne culd apply equatin (.5) t calculate te pductin ate f any abitay cne eigt and ell penetatin. q c = B µ [( 5 6πk q φ( x ρg,, )) 4( n n n= n= 5 n= q n x n 5 c q n x n )ln( )] (.5) Tansfmatin ule: Calculatin f pductin mainly depends n te accuacy f te ptentials appeaing at te denminat f te equatin (.5). In fact, cnsideing te WOC as a n fl bunday, ne as t deal it t pblems. Te fist pblem is tat te gemety f te WOC is unknn and e secnd is tat te bunday is iegula. Cnsideing an iegula gemety in Laplace diffusivity equatin is impssible. Cnsequently, in de t cnside te cne sape effect n te ptential distibutin, a tansfmatin ule as applied t tansfm te cne bunday t staigt line. In fact, it tansfms evey pint f te WOC t its cnjugate n te staigt line, tat is, evey vetical pint must be inceased by a cefficientα T. c α T = + (.5) (0.5 ) c Ali Kalili July 005

48 36 Wee c is te eigt f te cne at a paticula instant; teefe, te tansfmed pefated inteval may be calculated by using te flling equatin. L PT c = LP ( + ) (.53) 0.5 c N, by detemining te ne ell lengt in te ectilinea dmain te exact ptential at cne eigt ( c ) can be calculated. Aza Nejad and Ttike ppsed te flling pcedue t calculate te il fl ate.. Stat it small value f and apply tansfmatin ule.. Calculate ptential by equatin (.49) at given cne eigt. 3. Cmpute pductin ate by equatin (.5). 4. Incement by a small value. 5. Ceck te cne eigt it beneat te ell be and g t. 6. Repeat stages -5 until a maximum pductin ate, ic is citical cning fl ate, is btained. Ali Kalili July 005

49 37 Citical Oil Rate (STB/day) Weatley Aza Nejad Dimensinless ell penetatin Figue.6 A cmpaisn beteen te Aza Nejad-Ttike and te Weatley pcedue t calculate te value f citical cning ate Citical Oil fl ate (STB/day) Weatley Aza Nejad Meye & Gade Hyland (pcedue) Hyland (Simulatin) Dimensinless ell penetatin Figue.7 A cmpaisn beteen diffeent studies t detemine te citical cning ate Figue (.6) ss a cmpaisn beteen te Aza Nejad-Ttike and Weatley metds. Te cmpaisn ss tat bt metds beave Ali Kalili July 005

50 38 similaly. Heve, tey ae diffeent in sme cases. Aza Nejad and Ttike cnsideed te WOC t be a mving n-fl bunday and initial WOC emained at cnstant pessue eeas Weatley assumed tat te WOC culd be a steamline. In de t cnfigue te ptential functin, Weatley applied ne line suce and t pint suces ile Aza Nejad and Ttike applied tee line suces and t pint suces. As a esult, because f adding t exta line suces, te Aza Nejad-Ttike pcedue may be me pecise tan te Weatley metd. Te value f te citical cning ate against factinal ell penetatin as been pltted f seveal appaces f a specific example in figue (.7). Because tey display diffeent beaviu, te t celatins, te Abass- Bass and te Gu-Lee cuve, ave been emved. It is evident tat te Hyland et al pcedue cannt be eliable as it gives muc ige esults tan te te celatins cnsideed. Als, te Meye-Gade esult is t cnsevative. Heve, te citical cning ate btained fm te Weatley pcedue is vey clse t tat f Aza Nejad and Ttike s slutin because bt t metds ae based n deteminatin f te il ptential distibutin. Te value f te citical cning ate btained by te Hyland et al equatin (Simulatin) is sligtly ige tan tese t metds. Heve, as an altenative celatin, it is ecmmended because a simple calculat can be used t calculate te citical cning ate ate tan a lng pcedue... Simultaneus cning f gas and ate Ali Kalili July 005

51 39 Despite te fact tat te vast majity f esevis cntain all tee pases (Gas, Oil, and Wate) tgete, mst studies ave been cnducted n te ate cning gas cning. In simultaneus cning, cnsideatin is taken f bt te ate and te fee gas f te esevi. Te nly significant ay t evaluate te ate and gas cning is t apply esevi simulatin (Pinczeski, 003). Heve, seveal pcedues ave been ppsed t estimate te citical cning ate in te pesence f bt gas and ate cning.... Te Meye and Gade metd Flling te explanatin used f te ate cning in te Meye- Gade metd, n e cnside a esevi cmpising tee znes f gas, il and ate. Te pblem is t lcate te pefated inteval s tat te il pductin is maximal and te gas and ate pductin is minimal. Accding t assumptins nted ealie, te il ptential in te gas zne may be itten: ϕ = ϕ G ρ ρ G O ρg gz( ) ρ 0 (.54) S tat e ave ϕ (, z) = ϕ G ρ ρ G + g( D + p ρg )( ) ρ O (.55) Wee D z D + (.56) p Ali Kalili July 005

52 40 Wee p is te pefated eigt, is te ell adius and is te esevi tickness. Figue.8 A Scematic pfile f a simultaneus cning f gas and ate And il ptential in te ate zne can be itten: ρ ρ ϕ (, z) = ϕg g( D)( ) ρ ρ Wee (.57) D z D + (.58) p Meye and Gade assumed tat te il ptential at gas and ate zne as equal. As a esult, by equating equatin (.55) and (.57) te flling equatin as btained. ρ ρ (.59) G D = ( p ) ρ ρ G Similaly te il ptential at ell adius at diffeent cases migt be expessed as Ali Kalili July 005

53 Ali Kalili July = ), )( ( ), ( D z D D g z p ρ ρ ϕ ϕ ), ( ), )( ( z D gz D z z g W W p G ρ ρ ρ ρ ϕ ρ ρ ρ ρ ρ ρ ϕ (.60) Als by cmbining te Dacy equatin and te Hubbet ptential functin, te Oil fl ate as expessed in te fm f integal as flls: = k ρ π dz z q 0 ), ), ( ( ) / ln( ϕ ϕ µ (.6) Wee φ is te ptential value at te dainage adius, Teefe, by substitutin te expessins in equatin (.59) and (.60) in (.6) te value f te citical cning fl ate can be detemined. Eventually, te maximum citical cning fl ate can be detemined if pefated inteval p appaces ze. Tus, Meye and Gade deived te flling equatin in field units: ) ln( ) )( ( ) )( (( max e g g g g g c B K µ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ + = q (.6)

54 4 30 Citical Oil fl ate(stb/day) Simultaneus cning ate cning Dimensinless ell penetatin Figue.9 A cmpaisn beteen te ate cning and simultaneus cning f gas and ate accding t te Meye and Gade Metd. Te figue (.9) illustates a cmpaisn beteen te ate cning and te simultaneus cning f gas and ate. As sn, te value f te citical cning ate btained by tis Appac f simultaneus cning f ate and gas is le tan tat f te nly ate cning case. As can be seen at equatin (.6), te nly key paamete is te density diffeence beteen te tee pases. Als, te value f te maximum citical cning fl ate is independent f te lengt f te pefated inteval. Te independence f te citical cning ate n te lengt f te pefated inteval esults fm te assumptin tat te maximum ate ccus en te ate and gas cmes fist meet. Ali Kalili July 005

55 43... Te Cieici-Ciucci Appac Cieici-Ciucci (964) used a ptentimetic mdel tecnique t study te ate and gas cning in ic a systematic study as cnducted by means f an electical analgy tecnique. In tis study te main assumptins ae:. Hmgenus esevi (eite istpic anistpic).. Te aquife as assumed t be s limited tat it did nt cntibute t te enegy f te esevi.3. Te gas cap expends at a vey l ate s tat te ptential gadient in te gas cap is negligible. His esults ee pesented in te fm f a set f cuves t be used in te flling t cases: a- Detemining te value f te citical cning ate at given esevis and fluid ppeties, as ell as te lengt and psitin f te pefated inteval. b- Optimizing te psitin and lengt f te pefated inteval at ic te ell is pduced at citical cning ate, at given esevis and fluid caacteistics. He came up it te equatins (.63) and (.64) t detemine te value f te citical cning ate. Te citical cning ate is smalle ate btained fm te flling equatins. q = ( ρ ρ ) kψ ( B µ De, ε, δ ) (.63) q g ( ρ ) 4 ρ g = kψ g ( De, ε, δ g ) B µ (.64) Ali Kalili July 005

56 44 Wee De as defined as = De e k (.65) v k And δ, δ g and Є ave been defined in figue (.0) Figue.0 Diagammatic Repesentatin f a ate and gas cning system in a Hmgenus Fmatin. F any il pductin ate geate tan q q g, il-ate inteface mves up te gas-il inteface mves dn until it eventually eaces te ell and te unanted fluid beaks int te ellbe. Wit tis in mind, te flling equatins ae t be satisfied in de t acieve a maximum il ate itut ate and gas. qc q (.66) q c q g As can be seen in equatins (.63) and (.64), te citical cning ate is elated t te fluid caacteistics tug a dimensinless functin, ψ Ali Kalili July 005

57 45 ( De, Є, δ). Te functin ψ as detemined by applying a ptentimetic analyse utilizing te analgy existing beteen te steady state fl in pus media and te electical cuent fl in cnducts. Te esults ee pesented by figues (.) tug (.8). Tese esults ae valid nly itin te flling anges. 5 De 80 0 ε δ 0.9 (.67) It is imptant t nte tat te maximum citical cning ate can be detemined at te flling cnditin. q = q = q (.68) g H t slve te pblem? a- Well aleady pefated: In tis case Є and δ g ae given. Besides, fluids and te esevi caacteistics suc as De,, k R, K v, ρ and ρ ae knn. Once te value f ψ is ead ff by using te apppiate gap amng te figues (.) tug (.8), ne can detemine te value f te citical cning ate by equatins (.63) and (.64). Te btained value can be cecked by applying equatin (.67). b- Well as nt been pefated In tis case, te value f e,, k R, k v and te fluids ppeties ae assumed t be knn. Te pblem can be slved by applying figues (.) tug (.8) and using te flling pcedue. A value f ε is speculated Ali Kalili July 005

58 46 and ten te cespnding value f δ g and ψ ae ead ff fm suitable ( ρ g / ρ ) cuve. Te accuacy f te values depends n te inteplatin metd applied. Tus, te pefated inteval is lcated by kning te distance f te pefated inteval fm gas cap ic is calculated by means f te equatin given in figue (.0). Finally, aving values f δ g and ψ, ne can detemine te citical cning ate f tis penetatin. Table.7 Resevi and fluids ppeties f te Cieici and Meye and Gade metd Paamete H e k v k ρ ρ g Μ Value Unit Ft Ft md md g/cm 3 g/cm 3 Cp Ft Table.8 Data f te Cieici metd ε δgas Ali Kalili July 005

59 ceiici-ciucci Meye & Gade Citical Oil Rate(STB/day) Dimensinless ell penetatin Figue. Cmpaisn beteen te Cieici and Meye Metd at simultaneus cning Te figue. ss a cmpaisn beteen te Cieici and te Meye- Gade metd f simultaneus cning f ate and gas. As sn, te Cieici metd gives a ige esult tan te Meye-Gade equatin. As nted ealie, Meye and Gade assumed tat te ate as at est in te cne. Als te esevi pemeability as cnsideed unifm tugut te esevi. Te Cieici metd may be citicized n te basis tat te cne sape effect as nt taken int accunt. Ante disadvantage f tis metd is tat te esevi ppeties must satisfy te cnditins f equatin (.67). F instance, as sn, te factinal ell penetatin must be beteen ze and Ali Kalili July 005

60 48 Figue. Dimensinless functin f De=5 (Cited in Amad, 000) Ali Kalili July 005

61 49 Figue.3 Dimensinless functin f De=0 (Cited in Amad, 000) Ali Kalili July 005

62 50 Figue.4 Dimensinless functin f De=0 (Cited in Amad, 000) Ali Kalili July 005

63 5 Figue.5 Dimensinless functin f De=30 (Cited in Amad, 000) Ali Kalili July 005

64 5 Figue.6 Dimensinless functin f De=40 (Cited in Amad, 000) Ali Kalili July 005

65 53 Figue.7 Dimensinless functin f De=60 (Cited in Amad, 000) Ali Kalili July 005

66 54 Figue.8 Dimensinless functin f De=80 (Cited in Amad, 000).3. Cmpaisn Figue.9 ss a cmpaisn beteen all te celatins cnsideed in tis study. Depending n te assumptins and basic lgic cnsideed in develping te celatins, diffeent esults ae btained. Te flling cnclusins can be dan fm figue (.9). Te cuves ae categised int t gups. Te fist gup cnsists f t cuves, te Abass-Bass Cuve and Gu-Lee cuve. As can be seen, tee is a significant diffeence beteen tese t cuves and te te Ali Kalili July 005

67 55 celatins. Paticulaly, te value f te citical cning ate f te cuves inceases damatically as te factinal ell penetatin inceases till it eaces a maximum citical cning ate. It may be explained by stating tat te limited ellbe penetatin mves fm ze ell penetatin t mst il dminant zne. Tis pseak value ccus at factinal ell penetatin f 50% and 33% f te Gu-Lee and te Abass cuves, espectively. Aftead, it stats t decline until it eaces ze at fully ell penetatin. Even tug te beaviu is qualitatively smalle, te Gu-Lee ates ae a gd deal ige tan te Abass-Bass ates. Te secnd gup cnsists f all cuves except te Abass-Bass and te Gu-Lee Cuves. In tis gup, geneally, te cuves begin it ze ate f full penetatin and ten as te factinal ell penetatin appaces ze te value f te citical cning ate inceases. F instance, te utcme pvided by te Hyland et al pcedue ises damatically as te cuve appaces l penetatin. Te easn is tat tis estimatin flls te Muskat-Wyckff tey in ic te cne sape effect n il ptential distibutin as nt cnsideed (Hyland et al, 989). In cntast, te Meye-Gade fmula pvides te lest esult f te factinal ell penetatin f less tan 4%. Teefe, te Hyland et al and te Meye cuves can be defined as te uppe limit and te le limit espectively. Evidently, te esults f te Weatley, Aza Nejad-Ttike and Hyland et al s equatins ae sligtly diffeent. T be me pecise, te Weatley and Aza Nejad-Ttike pcedues ae almst te same. One f te main Ali Kalili July 005

68 56 advantages f tese studies is tat unlike mst f te fme celatins te cne sape effect n il ptential distibutin as cnsideed. Apat fm te equatin develped by Jsi f Capen s tey, it is desiable t make a cmpaisn beteen Capen s iginal metd and te te metds. Since Capen develped e tey f nly l penetatin case, e esult ill be cmpaed against te te metds, pvided tat te ell is pefated at te tp f te il zne. Capen s equatin pvides ige esults cmpaed t te te metds. F instance, te Capen metd gives te citical value f 6% ige tan te mst eliable equatins suc as te Weatley and Aza Nejad Ttike pcedues. Table.9 A Cmpaisn beteen diffeent celatins and Capen s metd Citical ate(stb/day) Celatin Meye-Gade Gu-Lee Abass-Bass Hyland et al() Hyland et al() Weatley Aza Nejad-Ttike Cieici- Ciucci(simultaneus) Capen Ali Kalili July 005

69 57 Citical Oil fl ate (STB/day) Capen Hyland (pcedue) Weatley Gu Aza Nejad Hyland (Simulatin) Meye & Gade Abass & Bass Dimensinless factinal ell penetatin Figue.9 A scematic cmpaisn beteen celatins develped t calculate te citical cning ate f vetical il ell Ali Kalili July 005

70 3 Capte Tee: Cning at Hizntal il ells 3.. Cning beaviu at izntal ells Unlike te vetical ells ee te upad mvement f te ate ceates a cne sape, te ising ate at izntal ells fms a cest ic is called te ate cest (Fig 3.). Tee ae sme advantages in dilling izntal ells ve te cnventinal vetical ells. Fistly, te pessue dadn is cncentated in te vicinity f te vetical ells eeas in te case f a izntal ell it is distibuted ve te dainage vlume f te izntal ell. Teefe, te pessue dadn f a izntal ell may be muc smalle tan tat f a vetical ell (Pinczeski, 003). Secndly, te value f te citical cning ate f a izntal ell is me tan t tee times tat f te citical cning ate estimated f a vetical ell at te same cne eigt (Kace and Gige, 986). Finally, in tems f seep efficiency, a izntal ell is me effective because te il vlume invaded by te ate t ceate a cest bel a izntal ell is cnsideably geate tan te il vlume sept t fm a cne bel a vetical ell (Kace and Gige, 986). Cnsequently, a ige ultimate il ecvey is expected fm a izntal ell ate tan a vetical ell. Ali Kalili July 005

71 58 Figue 3. A da f ate cesting bel a izntal ell 3.. Celatins used t calculate te citical cning ate in izntal il ells 3... Capen s appac Capen (986) develped an analytical celatin ic as te fist study dealing it te citical cning ate in izntal ells. Te aut pvided a simple estimatin f te citical cning ate at steady state cnditin f bt istpic and anistpic fmatins. F simplicity, te ell as assumed t be nea te tp f te esevi t decease te cance f te ate cning. Ceatin f a stable cest depends n a balance existing beteen te viscus fces and te gavity fces (pint A n te inteface is aay fm te ellbe and pint S is n te apex f te cest, figue (3.)). An Ali Kalili July 005