SPE p p. Abstract

Transcription

1 SPE Inflw Pefmance Relatinship Euatin f Slutin Gas-Dive Resevis unde Gavity Segegatin Rbet Padilla-Sixt, SPE, Cnsultant, and Rdlf Camach-Velázuez, SPE, Pemex E&P Cpyight 14, Sciety f Petleum Enginees This pape was pepaed f pesentatin at the SPE Latin Ameican and Caibbean Petleum Engineeing Cnfeenceheld in Maacaib, Venezuela, 1 3May 14. This pape was selected f pesentatin by an SPE pgam cmmittee fllwing eview f infmatin cntained in an abstact submitted by the auth(s). Cntents f the pape have nt been eviewed by the Sciety f Petleum Enginees and ae subject t cectin by the auth(s). The mateial des nt necessaily eflect any psitin f the Sciety f Petleum Enginees, its ffices, membes. Electnic epductin, distibutin, stage f any pat f this pape withut the witten cnsent f the Sciety f Petleum Enginees is phi bited. Pemissin t epduce in pint is esticted t an abstact f nt me than 3 wds; illustatins may nt be cpied. The abstact must cntain cnspicuus acknwledgment f SPE cpyight. Abstact Based n synthetic esults btained by Padilla and Camach f the fluid flw f gas and il in esevis pducing by slutin gas-dive (SGD) and gavity dainage (GS), in this pape the behavi f the il mbility functin in patially and fully penetated wells is studied; als its influence n the develpment f a pactical IPR euatin is analyzed. It is cncluded that the Fetkvich s staight line appximatin with an dinate t the igin eual t ze is valid nly when the pductin cmes fm the middle f the pductin thickness with and withut gavitatinal effects. Als, changes f the gas satuatin with depth due t the influence f gavity dainage and als the measuement psitin causes that the dinate f the mbility functin, (p)=ap+b, be diffeent fm ze; hweve, this linea appximatin is easnable at lng pducing times duing the bunday dminated flw peid aund the wellbe. F patially penetated wells this cnsideatin is nt entiely apppiate. Cnsideing the abve esults, a pductivity cefficient (ϑ) defined as the ati f il t the ttal pductivity index is defined. A uadatic IPR euatin valid f any psitin f the pductin inteval is deived. The uadatic flw efficiency definitin given by Camach and Raghavan is cnsideed. A table f nmalized efeence euatins f diffeent pductivity cefficient values is pesented, whee the classic euatins f Vgel, Fetkvich and cnstant pductivity index ae paticula cases. Finally nmalized cubic celatins f ttally and patially penetated wells als ae btained. The euatins wee pved with infmatin f eal field cases f satuated natually factued esevis (NFR s) f Mexic. Intductin A basic calculatin f the IPR behavi cnsides that the ate is diectly pptinal t the diffeence f pessue, ΔP P -P. The cnstant f pptinality is called pductivity index (J), such that, =J ΔP, and its nmalized fm is given by / max =1-(P /P ), whee max is the maximum il flw ate ( AOF) and the flw ate cespnding t P f a specific aveage pessue (P ). As a ule, when P <P b, J will decease as the ate inceases, k g will incease educing the il mbility (k /μ B ), geneating a nn-linea behavi ( IPR). F il adial flw in steady state, we have the fllwing euatin 1, Pe Pe k( Sg ) ( p) dp dp, (1) ( p) B ( p) P P whee x kh/ln(.47 e / w ). F esevis pducing by SGD, Vgel based n synthetic infmatin and field data deived a celatin which cnside a cicula hmgeneus and istpic clsed esevi with a well in the cente and FE=1, defining the fllwing nmalized expessin,,max p p 1.., () p p Standing 3 based n the wk f Vgel, include the effect f damage ( P s ) using the definitin f flw efficiency f slightly cmpessibility liuid flw, FE=P -P /(P -P )=P -P - Ps/(P -P ), which is inadeuate f esevis pducing by SGD 4, thus f FE>1 will pduce incnsistent esults. An il flw euatin analgus t that established f gas wells was pesented by Fetkvich 5, which cnsides the mbility as a linea functin f pessue, (p)=ap, and given by;

2 SPE n p 1., (3), max p whee n is the expnent f tubulence which is in the ange f n 1. Euatins () and (3) ae used t descibe the IPR behavi f SGD systems, and have been used extensively in the petleum industy f seveal yeas. Jnes et al. 6 ppsed a gaph f Δp/ and Δp / vs. flw ate f liuid and gas flw, espectively; these gaphs can be used as a diagnstic tl t detect the pesence f nn-lamina flw. Camach et al. 7 using the uadatic flw euatin f Fchheime 8 pesented a wide study f inetial effects n the IPR cuves in systems pducing by SGD unde nn-dacy flw in the esevi. Klins-Majche 9 develped an expessin including the bubble pint pessue value (P b ) inside the expnent f the IPR euatin. Wiggins et al. 1 deive analytical IPR euatins f the phases f il, gas and wate cnsideing the behavi f mbility. Cheng 11 using esults f simulatin and fllwing the methdlgy f Vgel define a set f IPR euatins f diffeent degees f deviatin f the well ( t 9 ), whee f als epduce the Vgel s E. Bendakhlia-Aziz 1 pesents a celatin f hizntal wells as a functin f a fact V (cefficient) and n (expnent). F NFR s, Qasem 13 deives the expessin given by / max =1-(P /P ) B, whee the expnent B is a functin f the paametes f the factued medium, ω and λ. Regading the lineaity f the mbility with pessue, Kelka-Cx, 14 cnside (P)=k /μ B =ap+b, the value f mbility f P= is b=(k /μ B )P a = (P a ) and slpe a. Using the functin X f Whitsn, X= (P a )/ (P ) establishes, (P)=X (P )+(P/P ) (P )(1-X), deiving the expessin, / max =1-(X)/(1+X)(P /P )+(1-X)/(1+X)(P /P ), such that f X=1/9 we get Vgel s euatin, while f X= the Fetkvich s euatin. In all wks dne n IPR cuves, the cmbined effects f SGD and GS has nt been cnsideed; Padilla and Camach 15 cnsideed these tw effects establishing the fllwing expessin, n( t) ( p ) 1. (4), max ( p ) Φ(p ) -Φ(p ) euies the knwledge f the vetical psitin whee the aveage esevi pessue is lcated. Unde SGD with and withut GS this psitin is clse t h/, whee Φ is the ptential functin f Hubbet. It is imptant t pint ut that the pesent study is a cntinuatin f Refs.15 and 16. Numeical Mdel In this study it was used the same base f infmatin f the pecedent jbs 4,7,15-18,. The synthetic esults wee btained using a -z finite diffeence black-il simulat. Fully (FPW) and patially penetated wells (PPW) lcated at the cente f a cnstant thickness hmgeneus esevi with a clsed ute bunday ae analyzed. The skin zne is mdeled with the Hawkins`s fmula. 19 Inetial effects and capillay pessue fces ae negligible. The initial pessue thughut the esevi is abve the P b. The cases I, II and III f PPW cespnd t the psitin f the pductin inteval f thickness h w, at the bttm, middle and tp, espectively. F me details see Ref.. Backgund Figues 1 (a)-(d) shw the IPR cuves as is suggested by Vgel. The symbls and lines cespnd t esults f simulatin with and withut GS effects, espectively. The chaacteistic uadatic fm in the IPR cuves is bseved, whee the depth f measuement affects the well pductivity. At the tp f the esevi (Fig. 1a) at lw ates the infmatin with GS falls belw that withut these effects, the ppsite happens when the psitin is lcated at the bttm (Fig. 1c). F these tw cases, at / max =, the value f p /p ae diffeent fm unity as a esult f the gas satuatin vaiatins, which affect the il mbility (λ ) and theefe the pductivity. The data pints in Fig. 1(b) wee ecded at the middle f the pducing inteval at aveage cnditins, theefe Es.() and (3) shuld be useful, hweve bth euatins d nt cnside gavitatinal effects. Fig. 1(d), f s= cnsides tw exteme depletin levels, at the bttm f the esevi, the lines cespnd t the euatins f Vgel and Fetkvich (n=1). Vgel yields easnable esults at high ates, while at lw ates unde lw depletin; the euatin f Fetkvich shws a bette fit. Thus, the use f these expessins is esticted t cnditins whee viscus fces ae dminant. Figue (a) f the psitin at the bttm f the esevi shws the IPR cuves in tems f p, the symbls cespnd t the espnse including GS effects and the dashed lines t the cases withut GS. The diffeences ae inceasing with depletin and at lw ates, and it is me pnunced at the tp and bttm psitins f measuement. The expnent n nt nly depends n pductin time, als depends n the measuement psitin (z). The influence f the flw ate n n is geate when gavitatinal effects ae cnsideed. It is clea that the data pints d nt yield a staight line. In Fig. (b) the IPR cuves in tems f (p ) - (p ) is shwed, we can bseve that the data pints ae aligned with expnents clse t unity; hweve, the cmputatin f the ptential functin euies the knwledge f the vetical psitin whee the aveage esevi pessue is lcated. F PPW, Figues 3(a) and 3(b) shw the IPR behavi f cases I and II, whee the pductive inteval h w ae lcated at the bttm and the middle f the well, espectively. In bth cases a uadatic behavi f all levels f depletin als is bseved. The best pductin cnditins ae defined at late times in the case I, but at ealie times the middle psitin is bette. Results in tems f p f the case II ae pesented in Fig. 3(c), a me linea behavi than thse bseved in a FPW ae defined. Again, the use f Vgel s and Fetkvich s expessins ae esticted t cnditins whee viscus fces ae dminant, see Fig. 3(d).

3 SPE Results Behavi f the Oil Mbility Functin Figue 4(a) f FPW shws the behavi f k / B at diffeent pducing times and measuement psitins. The staight line appximatin (dashed lines), as it is suggested by Fetkvich ( (p) =ap) is easnable at lng times; hweve, at cetain depths the dinate t the igin (b) it is nt necessaily eual t ze, then a bette linea appximatin is given by, ( p) ap b( z), (5) At the tp (z ) in a gas zne, the functin (p) =. This is als valid f PPW, but fa fm the well (p) behavi is nn linea. The cnsideatin f Fetkvich is easnable when the data pints ae measued at an aveage depth f the pducing inteval. The value f intesectin b, will be dependent f the measuement psitin (z), pducing time and theefe f gas satuatin vaiatins in the neighbhd f the pducing inteval. F PPW, Fig. 4(b) pesents the behavi f (p) vs. p at diffeent dimensinless times and tw psitins f the pen inteval, at the bttm (case I, Z Di =.99) and at the middle (case II, Z Di =9), espectively. F PPW the hypthesis that the mbility is a linea functin f pessue, as a pactical appximatin is easnable nly at lng pductin times duing the bunday dminated flw peid and clse t the wellbe, see fig. 5 (case II) and fig. 6 (case I). The mbility functin is nt stictly linea, a bette appximatin is t use E.(5), whee b=f(z,t), and in this sense the cnsideatin f Fetkvich f b=, epesents a paticula case. Als Figue 5, f case II shws that fa away fm the well the mbility functin fllws a nn-linea behavi, which is accentuated at sht and intemediate times. Likewise, in Figue 6 f the case I,in a gaph f vs., the mbility shws a stng decline clse t the wellbe but fllwing a linea behavi, and being cnstant fa away fm the well (p>p b ). The lineaity f the functin depends n the magnitude f the gavitatinal effects; this is elated t pductin ate, depth f the pducting inteval and pducing time. With high ates the mbility pfiles ae me linea, when the viscus fces ae highe than the gavitatinal effects. With lw ates in the il zne in bth PPW and FPW have been bseved that small vaiatins in esevi pessue pduce stng vaiatins in the mbility, which geneate pfiles with lage slpes, see Figue 7, which leads t negative values f b. Figue 8 shws a schematizatin f the linea behavi f (p) with pessue (left gaph) and the behavi f S g vs.depth (ight gaph). The tends wee defined unde diffeent pfiles f S g bseved unde diffeent values f pemeability, ates, type f fluids, skin fact, etc. With egad t the behavi f S g, at sht times at any depth will take values slightly abve the S gc, but at lng times the satuatin behavi will depend n the depth and theefe f pessue. Twads the tp f the fmatin the S g inceases apidly t its maximum value, affecting the il mbility (k ) and theefe the pductivity index. Pducing intevals lcated at intemediate psitins and t the bttm f the esevi at ealy and intemediate times will allw the develpment f a gas zne and anthe f il, this als depends n the vetical pemeability (k v ) which will affect in an imptant pptin the behavi f the il mbility. Pfiles f S g (z) vs. depth at diffeent times in PPW ae shwn in Figue 9, duing the unsteady state and the bunday dminated flw peids. In the case III a gas cap it is nt develped due t the geate pptin f the gas segegated and fee gas is pduced, then (p). In the cases I and II the gas evlves twads the tp f the fmatin fming a gas cap. An incease f S g clse t the wellbe will educe the k and theefe the il mbility, see Figues1 and 11. The il mbility at lng times can take values clse t ze due t the pesence f a high gas satuatin in the neighbhd f the pducing inteval. The apid evlutin f the gas t the tp f the fmatin and the fmatin f a gas cap ae caused by favable effects f gavity dainage. Cnsideing the abve mentined, it is imptant t nte that bth the magnitude f the slpe (a) and the dinate t the igin (b) f the linea appximatin (E.5) will depend n the depth and time. Figue 1 f case I, smalle vaiatins in the il mbility behavi ae bseved in the esevi than thse f case II (see als Fig.11). In tems f il pductivity at lng times the case I shws the best cnditins, f case II the best cnditins ae at sht and intemediate times. F case III the pesence f gas pduces negative esults due t the cmbined effects f SGD and GS. Geneal IPR Euatin f SGD and GS Unde the hypthesis f lineaity f the functin f mbility with pessue, the fllwing appach f k / B is established, P P P P k( p, Sg ) ( p ) dp dp ( ap b) dp, (6) ( p) B ( p) P P Cnsideing the esult eached in the pesent investigatin that the dinate t the igin may be diffeent fm ze (b ); duing the clsed bunday dminated flw peid, the fllwing euatin is defined, P P b P P a, (7) whee is a cnstant that depends n the petphysical ppeties and the esevi gemety. If the mbility is defined by the E.(6) as a linea functin f pessue, slving f the slpe, the next elatinship is btained,

4 4 SPE a ( p ) b( z) P, (8) Using E.(8) in (7), we btain the fllwing expessin f the il ate pductin, ( p ) b( z) P P P P, (9) P ( p ) In the last expessin f b= the Fetkvich s euatin is btained. E.(9) cnsides the effects f changes f gas satuatin with depth. Als based n E.(9) the fllwing tw atis f pductivity index ae defined: and J ( p ), (1) P J * b( z) J, (11) ( p ) Replacing the elatinships (1) and (11) in E.(9), we each the next euatin, P P J * P P J, (1) E.(1) is a functin f tw pductivity indeces, J and J *. It is imptant t bseve that, if a> and b=, this is J * =, the euatin is educed t the Fetkvich s euatin f n=1 ( (p)= ap). Nmalizing E.(1) by max, we btainthe fllwing geneal euatin f IPR, max P P 1 1, (13) P P valid f values f 1.The vaiable is a fitting paamete and it is defined as the "multiphase pductivity cefficient", given by the fllwing elatinship, * J J, (14) J J J * t * E.(13) can be used t fit the well inflw pefmance which euies at least thee data pints f P and ecded at the esevi pessue (P ). Nte that nly f the specific case = it is pssible t attach an expnent n f the analysis. It is imptant t nte that E.(13) epesents diffeent IPR behavis pesented peviusly in the liteatue, i.e. J * = (a>, b=) Fetkvich s E. with n=1 J * = 1/9 J (a>, b>) Vgel s E..5 J * = J (a=b>) Cnstant pductivity E. The value f 1/9J f the euatin f Vgel agees with the value f the paamete X detemined by Kelka and Cx. 17 The euatin f Fetkvich is btained with X= ( (P a )/ (P )=, with (P ) ); while f X=1, (P a )= (P ) cnstant pductivity. The paamete X is a functin f mbility, while in the pesent wk it is used the ati f pductivity indices defined thugh the paamete which can be estimated based n the behavi f the IPR cuves. Thee is a cmbinatin f values f J and J * that define a same value f the multiphase pductivity cefficient. E. (9) cnsideing that a= (p )/P may be ewited as, a The fllwing special cases ccu: b( z) P P P P, P (15) Case 1.Oil zne cnstant il mbility E.(15) f a= and b> and using Es.(1) and (11), becmes t the fllwing euatin,

5 SPE P P J, (16) * In its nmalized fm is expessed as, max P P 1, (17) P P This euatin epesents a paticula slutin f the E.(13) f he shape f the IPR cuves ae cncave dwn and ae elated t favable pductin cnditins simila t thse bseved in hetegeneus systems f high cnductivity. By the hand, if b(z)= fm E.(15), als the Fetkvich s E. is btained. Case.Cnstant pductivity index E.(15) f a> and b> (whee a=b= ), becmes the euatin f cnstant pductivity index (J), J P P, (18) * whee J=α. It s vey well-knwn nmalized fm is given by, / max =1-P /P, pvided that J =J (see E. 14). This leads t a staight line at 45 with a slpe eual t the value f the dinate t the igin. Als it is a special case f E.(13) f Case 3. Gas zne ze il mbility In the gas zne, a=b=, whee J * = J =, theefe =. Case 4.Oil mbility with high slpes in PPW and FPW This is bseved in PPW and FPW when a> and b<, fm E.(7) we btain t the fllwing IPR euatin, P 1 1 P 1 P 1 P max, (19) This euatin has a limited ange f applicability f values f vey clse t ze. Outside this limited set f values this euatin pduces incnsistent esults. F = als Fetkvich s euatin (f n=1) is btained. Table 1 pesents a summay f the peviusly deived IPR euatins. Hweve, thugh the use f E.(13) it is pssible t establish the apppiate IPR. It is imptant t nte that as 1 the cnditins f pductivity tend t incease. As it is knwn, the specific case f cnstant pductivity ( =.5) tends t veestimate the il pductin ptential. Evaluatin f PductivityIndex An estimatin f J may be btained thugh a lg-lg gaph f p vs., while is btained fm the fit f E.(13). With J and and using the Figue 13 it is pssible t estimate an appximate value f J *. Flw Efficiency (FE) Flw efficiency is defined by the ati inflw pefmance elatinship with damage and the ideal IPR, FE f (, p, () f (, p, p ), p ) This allws us t knw the maximum il ate f FE 1, i 1 FE 1 FE f (, p, p ), (1) FE max max If f(ϑ,p,p )<f(ϑ,p,p ) i, we have a damaged well (s>), and if f(ϑ,p, p )>f(ϑ,p, p ) i, a stimulated well (s<). Thus, Figue 14 pesents the behavi f / max vs. P /P f and which epduces the tend f the euatins f Fetkvich and Vgel, espectively. Standing s 3 euatin f FE definitin f slightly cmpessible liuid flw and the espnse using the uadatic FE definitin as is ppsed by Camach and Raghavan 4 ae shwn. Based n E.(13), Figue15 shws the Fetkvich and Vgel s efeence cuves f diffeent values f FE. Refeence Cuves f 1 Figue 16 shws the behavi f the efeence cuves based n E.(13). The IPR cuves ae shwn f FE=1 (well withut damage) f diffeent values f the cefficient. The gaphs cespnding t FE 1 can be geneated easily. In the same way, in Figue 17 the efeence cuves f diffeent flw efficiency (wells damaged, withut damage and stimulated) and =1 ae shwn. The use f E.(13) will be exemplified with tw eal cases f fields pducing by slutin gas dive in NFR. Cubic IPR Celatins f PPW and FPW In Figs. 1 t 3 as aleady was mentined, we can bseve the classical uadatic shape f the IPR cuves; hweve unde cetain cnditins it is nmal t bseve values f p /p diffeent fm unity (at / max =), as a esult f the stng vaiatins f gas satuatin. An imptant esult is that the IPR cuves in tems f (p) shws a ttal aligned behavi, which is valid f FPW and PPW. Figue 18

6 6 SPE pesents infmatin f fully penetated wells in tems f ptential, cnsideing values egisteed at seveal psitins espect t a efeence level at the middle f the pductin inteval (z=h/). These esults indicate the pssibility t btain a celatin whee gavitatinal effects ae taken int accunt but in tems f ptential instead f pessue. Likewise Figue 19 shws the same type f infmatin but f patially penetated wells. In these figues the effect f depth is included in the ptential functin, the data pints ae aligned t a uniue aveage tend. A simila pcedue t thse used by Vgel was applied t define the IPR celatins valid unde the cmbined effects f SGD and GS. F PPW and FPW the synthetic data (symbls) ae adjusted with a high celatin cefficient using nn-linea egessin t a cubic plynmial functin (slid line), which is the best fit. It is cnsideed a cnfidence inteval f the 95%. In bth cases the suae f the esidue was.998, which is cnsideed a vey gd fit. Thus, f fully and patially penetated wells, the next celatins was btained, p * p *.38 p* 3, () max Whee p*=φ(p )/Φ(P ) f fully penetated wells, and max p*.34 p*.1 * p, (3) f patially penetated wells. Figue shws efeence cuves f diffeent values f FE, using Es.() and (3). F pupses f cmpaisn the cuve f Fetkvich f n=1 and FE=1 is als shwn. The pcess t evaluate the ptential based n a efeence level is nt tivial, it is necessay t have infmatin abut fluids satuatin and pessue at diffeent depths. Hweve, if the depth f measuement is clse t half level f the fmatin, these celatins can be used diectly in tems f pessue. The ptential functin f Hubbet is given by =ʃdp/γ(p)-z. In the pesent wk p ef cespnd t the vlumetic aveage pessue lcated t a psitin f h/ f the fmatin thickness. The tem f γ(p)is the specific weight (psi/ft), whee γ(p)=ρ g/g c, g is the gavitatinal acceleatin (ft/s ), g c the cnvesin cnstant (lb m -ft/(lb f -s )) and ρ the density (lb m /ft 3 ). Als z is a distance (ft) with espect t a efeence psitin. Field Cases Tw field cases ae cnsideed, test A cespnds t a patially penetated well and test B t a fully penetated well, bth pducing by slutin gas-dive in NFR. The wells ae cmpleted nea the middle f the pductin inteval and the data cnside the existence f inetial effects in the esevi. Test A The esults btained f,max ae f the same de f magnitude t thse btained using the euatin f Fetkvich (J =.8, n=55) with,max =14,939 STB/D, the shape f the IPR cuve and fit is shwed in Figue 1. Results f the well test analysis define a wellbe slightly damaged. The esults with E.(13) yields a FE=.95 which is in ageement t the damage cnditins (n=5 and ) and,max =14,67 STB/D. A cleaning wuld impve slightly the pductin, as can be seen with the esults with FE=1,,max =14,87 STB/D. E. (3) was als used t fit the data pints f the test btaining,max EF=1 =14,644 STB/D. The maximum il ate is simila t that btained with E.(13). Test B Figue shws the data pints and cuve fitted. The esults btained with E.(13) (with n=.7 and cespnd t,max FE=1 =99,56 STB/D, while using Fetkvich s expessin,max FE=1 = 9,4 STB/D with a pductivity index f J =.5 and n=8. Fm E. (),max FE=1 =99,6 STB/D. The maximum il ate is simila t that btained with E.(13). The esults f bth examples ae cnsistent with thse f Fetkvich, because the wells ae cmpleted nea t middle f the pducing inteval. As aleady mentined, the euatin f Fetkvich is valid nly at aveage cnditins when the pducing inteval is at the middle f the pductive thickness, with withut gavitatinal effects. Cnclusins The fllwing cnclusins may be eached based n the esults pesented in this pape: 1. At lng pducing times duing the bunday dminated flw peid and clse t the wellbe, the assumptin f cnsideing the il mbility functin, (p)=k / B, as a linea functin f pessue is appximately satisfied f ttally and patially penetated wells; but the dinate t the igin it is nt always necessaily clse t ze.. The staight line appximatin f the mbility functin as is suggested by Fetkvich ( p)=ap) is easnable at aveage cnditins, it can be assciated t depths clse t the middle f the pducing thickness (h/), whee viscus fces ae dminant in the esevi. 3. F bth PPW and FPW pducing at lw ates thee ae behavis f the mbility functin, with stng psitive slpes and negative values f the dinate t the igin (b). 4. Unde the appximatin f (p)=ap+b(z), a uadatic geneal euatin valid f any measuement psitin is deived, whee Vgel, Fetkvich and cnstant pductivity euatins ae paticula cases. The value f 1/9J f Vgel euatin is in ageement with the X paamete f Whitsn btained by Kelka and Cx. 5. A set f nmalized IPR cuves (efeence cuves) f damaged, undamaged and stimulated wells ae pesented. 6. Tw cubic celatins f IPR f FPW and PPW ae pvided, wee als a set f efeence cuves ae established.

7 SPE The IPR euatins wee tested with tw eal field cases in NFR, and cmpaed with the cespnding esults using Fetkvich s euatin. Nmenclatue AOF = abslute pen flw, STB/D b = (p) at p=, md/cp B f =il gas FVF: RB/STB, RB/scf c t = system cmpessibility, psi -1 g = gavitatinal acceleatin, ft/s g c = unit system cnvesin cnstant, lb m ft/(lb f - s ) GS = gavity segegatin h = fmatinthickness, ft h w =thickness f pductive inteval, ft h Di = dimensinless pductin thickness (h w /h) J = multiphase pductivity index unde SGD, STB/(D-(psi ) ) J * = multiphase pductivity index at P=, STB/(D-(psi ) ) J t = J + J *= Ttal multiphase pductivity index. STB/(D-(psi ) ) k = abslute pemeability, md k f = elative pemeability (f=il gas) n = expnent f tubulence in Fetkvich s E. NGS = nn gavity segegatin p = wellbe flwing pessue, psi p = aveage esevi pessue, psi = il pductin ate, STB/D,max = maximum il ate (AOFP), STB/D e = extenal dainage adius, ft ed = dimensinless dainage adius ( ed =/ w ) s = skin zne adius, ft w = wellbe adius, ft R = pducing GOR, scf/stb s = skin fact S f = satuatin f the phase f (f=il gas) SGD = slutin gas dive t = time, days t D = dimensinless time (βkt/ϕμc t w ) X =F(p a )/F(p): Whitsn s paamete f mbility z = distance espect t a efeence psitin, ft Z Di = dimensinless psitin f pductin inteval (z i /h) Geek Symbls: f = specific weight ( f = f g/g c = f /144, f=il gas), psi/ft p = pessue dp, psi ceficient f multiphase pductivity (p) = mbility functin, md /cp f = il gasviscsity, cp f = il gas density, lb m /cu-ft = Hubbet s ptential functin, psi/ (lb m /cu-ft) Acknwledgments The suppt f IMP, Univesity f Mexic (UNAM) and Pemex E&P is acknwledged. Refeences 1. Evinge, H.H., and Muskat, M.: Calculatin f theetical pductivity fact, Tans. AIME 146: , Vgel, J. V.: Inflw pefmance elatinships f slutin gas dive wells, JPT (Jan. 1968) 83-9; Tans. AIME, Standing, M.B.: Cncening the calculatin f inflw pefmance f wells pducing fm slutin gas esevis, JPT (Sept. 1971) Camach, V. R., and Raghavan, R.: Inflw pefmance elatinships f slutin-gas-dive esevis, JPT (May. 1989) Fetkvich, M.J.: The ischnal testing f il wells,pape SPE 459 pesented at the 1973 Annual Technical Cnfeence and Exhibitin, Las Vegas, Sept. 3-Oct Jnes, L.G., Blunt, E.M., and Glaze, O.H.: Use f sht-tem multiple ate flw tests t pedict pefmance f wells having tubulence, Pape SPE 6133 pesented at the 1976 Annual Fall Technical Cnfeence and Exhibitin, New Oleans, Oct Camach,V. R., Padilla, S.R., and Vásuez, C. M.: Inflw pefmance elatinships with inetial effects in the esevi, Pape SPE 5481 pesented at the 1993 Pductin Opeatins Sympsium, Oklahma City, Mach Fchheime, P.H.: WassebewegungDuchBden, ZeitschiftveIngenieu (191) 45, 1731.

8 8 SPE Klins, M.A., Majche, M.W.: Inflw pefmance elatinship f damaged impved wells pducing unde slutin-gas dive,jpt 44(1): , Wiggins, M.L., Russell, J.E., Jennings, J.W.: Analytical inflw pefmance elatinships f thee-phase flw in bunded esevis, Pape SPE 455 pesented at the 199 West. Reg. Meet., Bakesfield, CA. 11. Cheng, A.M.: Inflw pefmance elatinships f slutin-gas-dive slanted/hizntal wells, Pape SPE 7 pesented at the199 Annual. Tech. Cnf. Exhib., New Oleans, LA. 1. Bendakhlia, H., Aziz, K.: Inflw pefmance elatinship f slutin-gas-dive hizntal wells, SPE 1983 pesented at the1989 Annual Tech. Cnf. Exhib., San Antni, TX. 13. Qasem, F.H.: Pefmance and ecvey pedictin in hetegeneus natually factued esevis unde the slutin gas dive pcess. PhD Dissetatin, Univ. f Suthen Califnia, Kelka, B.G. and Cx, R.: Unified elatinship t pedict futue ip cuves f slutin gas-dive esevis, Pape SPE 1439 pesented at the 1985 Annual Technical Cnfeence and Exhibitin, Las Vegas, Sept Padilla, S.R., Camach, V.R., Castejón, A.R., and Samanieg, V.F.: Inflw pefmance elatinships unde gavity segegatin f slutin gas-dive esevis, Junal f Enegy Resuces Technlgy f ASME, V. 131/331-1, Septembe Padilla, S.R., and Camach, V. R.: Resevi pefmance unde slutin gas-dive and gavity dainage, Pape SPE 9186 pesented at the SPE Intenatinal Petleum Cnfeence in Méxic, Nvembe 7-9, Camach, V. R.: Well Pefmance Unde Slutin Gas Dive, PhD. Dissetatin, U. f Tulsa, Tulsa, Ok., Camach, V. R., and Raghavan, R.: Sme theetical esults useful in analyzing well pefmance unde slutin-gas dive, SPEFE (June 1991) Hawkins, M.F.J.: A Nte n the Skin Effect, Tans. AIME (1956) 7, Padilla, S.R., Camach, V.R., Castejón, A.R., and Samanieg, V.F.: Inflw pefmance elatinships unde gavity segegatin f slutin gas-dive esevis, Pape ETCE--411, pesented at the ETCE and OMAE Enegy f the New Millenium, New Oleans, Luisiana, Febuay 14-17,. (a) (b) (c) (d) Figue 1. IPR as suggested by Vgel:FPW, s=-; a) Z Di=.6; b) Z Di=9; c) Z Di=.99; d) Cmpaisn f Vgel and Fetkvich ses, s=. (afte Padilla et al., ) (a) (b) Figue. FPW: a) Lg-lg IPR cuves with and withut GS, using p,z Di=.99, s=-; b) Lg-Lg IPR cuves in tems f (P ) - (p ) at diffeent depletin levels, Z Di=.99, s=. (afte Padilla et al. 9)

9 SPE (a) (b) (c) (d) Figue 3. PPW: a) case I; b) Case II; c) Lg-Lg IPR with and withut GS using p at Z Di=.5; d) Cmpaisn f Vgel and Fetkvich s Es. s=. (a) (b) Figue 4. Cmpaisn f mbility functin pfiles as a functin f time and measuement psitins: a) FPW, b) PPW. Figue 5. Pfiles f il mbility vs. pessue f PPW, case II at diffeent pducing times, =55 STB/D.

10 Oil mbility (md/cp) Depth, ft gas zne gas-il tansitin il zne 1 SPE Figue 6. Pfiles f il mbility vs. distance f PPW, case I at diffeent pducing times, =6 STB/D Figue 7. Pfiles f il mbility vs. pessue f PPW, case I at diffeent pducing times. z=h h>z>h/ z=h/ c/g- b Pessue, lb/pg z= c/g- S g S gmax Figue 8. Schematizatin f vs. pessue and S g vs. z f diffeent measuement psitins int the pducing intevals.

11 SPE hw Case I Case II Case III Depth, ft hw gas zne gas zne hw S g S g S g Figue 9. Pfiles f S gvs. depth f PPW, Case I, II and III. Figue1. Pfiles f S gvs. distance f PPW, case II at diffeent pducing times. Figue11. Pfiles f il mbility vs. distance f PPW, case II at diffeent pducing times.

12 1 SPE Figue 1. Pfiles f il mbility vs. distance f PPW, case I at diffeent pducing times. Table 1 Refeence Euatins J * Euatin Ref. E. Depth Fetkvich E. Z=h/.1 J /9 Vgel E.. J /4.3 J /.33 J /1.5.5 J /1 Cnstant Pd. E. h>z>h/ 1.5 J.7.33 J 4 J.9 9 J 1 J = Z=h

13 /max SPE J J* x x E x1-3 1x1-4 1x1-5 1E-9. 1E Figue13. Refeence gaph f J * using values f J and the cefficient f multiphase pductivity (. /max FE = 1. FE = Fetkvich Vgel Standing s FE definitin Camach & Raghavan s FE definitin P/P Figue 14. Validatin f geneal IPR euatin (E.13) cnsideing FE=1. and using Vgel and Fetkvich s E vs. liuid FE definitin f Standing. 1.8 FE Fetkvich: ϑ= Vgel: ϑ= P/P Figue 15. Refeence cuves f Vgel and Fetkvich using E.(13) f diffeent FE with = and.1

14 14 SPE ϑ FE = /max P/P Figue 16. Refeence IPR (E.13) cuves, / max vs. P /P, f seveal values f and FE=1. /max EF ϑ= P/P Figue 17. Refeence IPR cuves (E.13), / max vs. P /P, f seveal values f FE and Fully Penetated Wells Φ(P)/Φ(P) Figue 18. Fit f data pints using nn-linea egessin t a cubic plynmial functin f FPW (E.) unde SGD and GS.

15 /max SPE Patially Penetated Wells Φ(P)/Φ(P) Figue 19. Fit f data pints using nn-linea egessin t a cubic plynmial functin f PPW (E.3) unde SGD and GS. 1.8 /max FE Fetkvich, n=1 FullyPenetated Well Patially Penetated Well Φ(P)/ Φ(P) Figue. Refeence cuves f FPW (E.) and PPW (E.3) in tems f ptential f diffeent FE unde SGD and GS. 1.9 ϑ= FE=.95 n=5.7.5 Seies P/P Figue 1. Fit f IPR cuve using E.(13) f the field test A.

16 16 SPE ϑ= FE=1. n=.7. /max P/P Figue. Fit f IPR cuve using E.(13) f the field test B.