Gas Deliverability Model with Different Vertical Wells Properties

Transcription

1 PROC. ITB En. Scence Vol. 35 B, No., 003, Gas Delverablty Model wth Dfferent Vertcal Wells Proertes L. Mucharam, P. Sukarno, S. Srear,3, Z. Syhab, E. Soewono,3, M. Ar 3 & F. Iral 3 Deartment of Petroleum Enneern ITB Deartment of Mathematcs ITB 3 Research Grou for Industral & Aled Mathematcs, - ITB Abstract. We resent here a as delverablty comutatonal model for snle reservor wth mult wells. The questons of how lon the as delvery can be sustaned and how to estmate the lateau tme are dscussed here. In order to answer such a queston, n ths case, a couln method whch conssts of materal balance method and as flow equaton method s develoed by assumn no water nflux n the reservor. Gven the rate and the mnmum ressure of as at the rocessn lant, the as ressure at the wellhead and at the bottom hole can be obtaned. From here, the estmaton of the as delverablty can be done. In ths aer we obtan a comutatonal method whch ves drect comutaton for ressure dro from the rocessn lant to the wells, takn nto account dfferent well behavor. Here AOF technque s used for obtann as rate n each well. Further Tan & Adewum correlaton s aled for ressure dro model alon vertcal and horontal es and Rune-Kutta method s chosen to comute the well head and bottom hole ressures n each well whch then ben used to estmate the lateau tmes. We obtan here drect comutatonal scheme of as delverablty from reservor to rocessn lant for snle reservor wth mult-wells roertes. Comutatonal results ve dfferent rofles (.e. as rate, lateau and roducton tme, etc) for each well. Further by selectn roer flow rate reducton, the flow dstrbuton after lateau tme to sustan the delvery s comuted for each well. Keywords: Absolute Oen Flow (AOF); as delverablty; roducton lateau. Introducton Comutaton of as delverablty s an nterestn and comlcated roblem n as ndustres snce t contans the queston of how lon a well(s) can stll delver as to the sales ont. Wth ths calculaton, the lateau tme of the delverablty can be redcted whch s n turn, f the as delvery has reached that ont whle the contract tme has not yet fnshed, then there are otons to sustan the as delvery by nstalln comressors or establshn a new well(s). Several aers have shown comutaton for as delverablty for snle well roertes (see for examle n Fevan & Whtson (995) and Mott (999)). In

2 6 L. Mucharam, et al. ths aer, we resent drect comutaton for delverablty from reservor to the rocessn lant wth mult wells roertes. Here, as rate and ressure at the rocessn lant are consdered as the constrant of the calculaton. Ths s used as a smlfcaton of the roblem. Provded a number of data, we can fnd out the ressure at the wellhead and at the bottom hole of each well. After comutn the as ressure at the bottom hole, the calculaton of the as delverablty can be started. However, t should be notced that a as well can only delver certan ercentae rate from ts Absolute Oen Flow (AOF). Ths wll determne whether the as rate from exstn well(s) can meet the customer need. The method of Rune-Kutta n solvn the Tan & Adewum [3] correlaton can be found n secton. In secton 3, we dscuss about as rate allocaton that can be taken from each well, whle AOF and materal balance are talked n the next sectons. Calculaton results of delverablty and numercal analyss & further dscusson are resented n secton 6 and 7 resectvely. Gas Delverablty n a Vertcal Well Gas delvery from a reservor to the bottom hole and then to the rocessn lant and to the sales ont wll exerence some ressure dro. Althouh the comutaton of the ressure dstrbuton alon these dfferent elnes should necessarly be calculated, we refer to smlfy the roblem for the delvery u to the rocessn lant. Further calculaton extendn u to the sales ont can be done smlarly. In order to delver as from the bottom hole to the rocessn lant satsfyn a ven outlet ressure, the mnmum ressure at the bottom hole (P wflmt ) must be comuted frst usn an arorate ressure dro equaton. Bascally, there are many equatons avalable to estmate ressure dro of dry as. The most oular equatons for as flow n horontal and slhtly nclned elnes are Weymouth and Panhandle (A and B) equatons. For bottom hole ressure calculaton, Cullender-Smth and Sukkar-Cornel methods [4] can be aled. However, all of these equatons do not nclude the knetc enery term n t. Recently, Tan & Adewum [3] develoed an analytcal equaton for as elnes, whch was derved from comressble flud flow model n es wthout nelectn the knetc enery term. Ths equaton can be aled to redct the bottom-hole ressure for as wells and also for ressure dstrbuton n lon es. Ths s ben used here and can be descrbed as follows

3 Gas Delverablty wth Dfferent Vertcal Wells Proertes 7 m ZRT dp dp f m ZRT = + + A P dx dx DM A P M Rearrann (), we et dp dx M P sn α. () ZRT f m ZRT M Psnα + DM A P ZRT. () m ZRT M A P = Further by nteratn () and assumn constant temerature and comressblty factor taken from the averae values we obtan f m ZRT D + ln DA sn M α f f m DA M P + Z R M P + Z R snα T snα T D f P ln P + L = 0. (3) To obtan the desred ressure, the Newton-Rahson method s used. Another smlar aroach n [5] wth dfferent technque of averan can also be used to derve a ressure dro equaton. Ths averan technque to nterate the ressure dro equaton () s naturally far from accurate for relatvely lon e. Here n ths aer we use the full equaton () for ressure dro calculaton, recsely the fourth order Rune-Kutta method that s known wth hher order accuracy s used for numercal comutaton. The local truncaton error of ths method s of order O (h 5 ) and ts lobal error s of order O (h 4 ) [6]. Ths method s descrbed below.. Rune-Kutta Method Consder equaton () as an ntal value roblem. To aly the Rune-Kutta method, we need all roertes on the rht sde of the equaton at startn ont. In short, the method can be descrbed below wth Pn + = Pn + ( K + K + K 3 + K 4 ), (4) 6

4 8 L. Mucharam, et al. Inut Data Desred Pressure? Outlet Pressure Inlet Pressure L = 0 L = Pe Lenth P = P nlet P = P outlet h ostve h neatve Condton: Condton: L < Pe Lenth L > 0 Whle Condton = TRUE K = h f (P, L) K = h f (P + 0.5K, L + 0.5h) K 3 = h f (P + 0.5K, L + 0.5h) K 4 = h f (P + K 3, L + h) Pnew = K + K + K3 + P + 6 K4 P = Pnew L = L + h Desred Pressure = P Fure Rune-Kutta rocedure n determnn desred ressure.

5 Gas Delverablty wth Dfferent Vertcal Wells Proertes 9 ( ) K = h f L n, P n h K K = h f Ln +, Pn + h K K 3 = h f Ln +, Pn + K 4 = h f ( Ln + h, Pn + K 3 ). Take equaton () as the f (L, P) functon. If we would lke to estmate the nlet ressure, then h must be a neatve value and L must be equal to the lenth of e. On the other hand, f the desred ressure s the outlet, then h must be a ostve value, and L must be set equal to 0. As an llustraton of the Rune-Kutta method, see the flow chart on fure. 3 Rate Allocaton Before on on to the delverablty calculaton, t should be assured frst that as rate taken from the reservor can satsfy the customer needs throuh the wells. Ths s because a as well can only delver certan ercentae of ts Absolute Oen Flow (AOF). However, n ths aer the value of AOF durn the roducton tme s estmated from the ntal value of AOF. To determne as rate dstrbuton from each well, a wehted AOF calculaton s used as dected n fure. Q : Q : : Q n = AOF : AOF : : AOF n Yes Q x %AOF No Q : Q : : Q n : Q n+ = AOF : AOF : : AOF n+ Gas rate = Q Q = x% x AOF No Add Well? Yes Fure Determnn as rate from each well.

6 30 L. Mucharam, et al. Shortly, to calculate the delverablty, we can use the follown equaton [4] ( ) n Q = C P P. (5) r wf 4 Absolute Oen Flow (AOF) Absolute Oen Flow s as flow rate that could be obtaned f the bottom hole ressure reduced to ero s. Thus, the value of AOF can be wrtten as follows ( 4.73 ) n, AOF = C (6) P r where C can be wrtten as follows C = x 0 6 kh r T µ ln 0.47 r e w. (7) Snce ressure of reservor and the value of C chane wth the tmes, the AOF wll also chane wth the roducton tmes. Hence, we could consder t as a functon whch deends on reservor ressure and tme. The estmaton of AOF s very mortant here, because t wll determne as rate that can be roduced from a well. However, t should be notced that the value of AOF may be dfferent for each well. 5 Materal Balance In calculatn delverablty, we modfy materal balance method to estmate tme roducton of reservor. It s shown here a comutaton of roducton tme whch s derved from the materal balance equaton. For as volumetrc reservor, the materal balance can be reresented as follows where n = n - n f, (8) n n n f = amount of as mole roduced = amount of ntal as mole at reservor = remaned as mole at reservor. By substtutn real as equaton nto equaton (8), we wll have scg V V = (9) T T T sc sc f f

7 Gas Delverablty wth Dfferent Vertcal Wells Proertes 3 and fnally we et G TV sc T sc V = + sct f (0) sc T f Because V = IGIP B and B can be smlfed nto G = T f ft scf then equaton (0) IGIP = + IGIP, () where IGIP stands for Intal Gas In Place n scf unt. Suosed that ntal ressure, devaton factor, and IGIP (Intal Gas In Place) are already known. Because tme (t) can be obtaned from dvdn G by constant Q, then we can derve recursvely functon of tme deend on as follows IGIP G G G G Fure 3.a Illustraton of P over vs IGIP. Fure 3.b Illustraton of P over vs G. G IGIP t = = + Q Q IGIP Q

8 3 L. Mucharam, et al. t t G G = Q IGIP = Q G3 G = Q IGIP = Q And fnally we et functon of tme (t), whch deends on only ressure varable as follows: t n Gn Gn IGIP n = = n Q n n Q () An nterated calculaton of flow erformance from the reservor to the rocessn lant s shown n fure 4. Pressure of Wellhead (Pwh Lmt) Calculatn Pressure Dro Processn Plant (P mn) Calculatn Pressure Dro Bottom Hole Flown Pressure Lmt Gas Delverablty Calculaton Determne Reservor Performance Q, tme, Pwf, condensate, etc Fure 4 Flow charts reresent calculaton rocedure n redctn as delverablty as functon of tme.

9 Gas Delverablty wth Dfferent Vertcal Wells Proertes 33 No Pwf = Pwf lmt Calc Qas Calc. Pwf from Rawlns- Schellhard eq. Pwf > Pwf lmt Calc. as roertes:,µ ; F(,T) Inut, P R = P x Yes Pwf = Pwf lmt Calc tme t = G/Qas P/ Gas Delverablty Sto untl Q <= Q lmt G 6 Samle Calculaton Fure 5 Flow chart of as delverablty calculaton. Consder we have to delver as from snle reservor wth three wells to the rocessn lant whch can be dected as follows W R W Processn Plant W 3 Fure 6 Illustraton of the case.

10 34 L. Mucharam, et al. The detal data are shown below. Processn Plant: o Gas rate = 00 MMscfd o Pressure = 300 sa. Well Well C Value N value The tubn well roertes Deth (ft) Dameter (nch) Rouhness (nch) α (de.) Well.87x x Well.85x x Well 3.80x x Table Well roertes. Pelnes Well Well Processn lant Lenth Dameter Rouhness α Tem. (ft) (nch) (nch) (de.) ( o F) Well x Well x Well x Table Well-rocessn lant roertes. Reservor o IGIP = 300 Bscf o Pressure = 00 sa o Tem. = 70 o F o SG = 0.70 o CO = 0% mole o HS = 0% mole o N = 0% mole v. AOF rato = 5% After calculatn the delverablty, we have the follown results. Well Gas rate Wellhead ress. BHP (MMscfd) (sa) (sa) Well Well Well Table 3 Calculaton result. Tem. ( o F)

11 Gas Delverablty wth Dfferent Vertcal Wells Proertes 35 Well Plateau Tme (year) Prod. Tme (year) Cumulatve Plateau (BSCF) Cumulatve Prod. Tme (BSCF) Well Well Well Total Result Table 4 Gas delverablty result To et detaled results, see fures 7, 8, and 9. Fure 7 Profle of as rate versus tme. 7 Numercal Analyss and Further Dscusson Here, we determne as rate allocaton that can be taken from each well by a wehted ntal AOF value. Wthn the comutaton, the ntal AOF value of each well s chaned due to the chane of reservor ressure. After calculatn the rate, we must check whether as from all wells satsfes the customer needs or not. If t doesn t, then we should establsh a new well. Otherwse, the demand cannot be fulflled.

12 36 L. Mucharam, et al. Fure 8 Profle of ressure versus tme. Fure 9 Profle of cumulatve as versus tme.

13 Gas Delverablty wth Dfferent Vertcal Wells Proertes 37 In short, t can be sad that we dvde the calculaton nto three staes. Frst, we should estmate as rate that can be roduced from each well and then evaluate whether t can satsfy the demand. Secondly, we should redct ressure dro for each well so that the entre bottom holes ressures can be found. And fnally, we calculate the delverablty for all wells smultaneously. However, we must be careful because there are some roertes of wells that could be dfferent one another and would always chane for every dfferent tme. In the samle case above, t can be seen that f we have mult-wells wth dfferent roertes, then we wll have dfferent rofles for each well (.e. as rate, lateau tme, roducn tme, as cumulatve, etc). We should also ont out that n estmatn ressure dro, Rune-Kutta method s used to solve the Tan-Adewum correlaton. Ths method s a well-known method to solve ordnary dfferental equaton roblem because of ts accuracy. By ths method, the best aroach soluton of the roblem can be obtaned. 8 Summares. The as delverablty model n ths study s bascally develoed based on the materal balance method n whch vertcal wells wth dfferent roertes can be modeled and mlemented for redctn as delverablty of a as feld.. For better ndustry alcaton, future develoment of ths model wll nclude multle reservor system, comlex well comletons, and varous surface searator condtons Acknowledement The authors would lke to thank the Research Consortum OPPINET for fundn the research. Nomenclature A = cross-sectonal area of elne (L ) D = dameter of elne (L) f = frcton factor (dmensonless) = ravtatonal acceleraton (L/T ) h = ste se L = lenth of elne (L) m M = mass flow rate of as (M/T) = as molecular weht

14 38 L. Mucharam, et al. P = ressure (M/LT ) R = unversal as constant T = temerature v = as velocty (L/T) x = axal coordnate (L) Z = as comressblty factor (dmensonless) a = anle of elne (deree) References. Fevan, O. & Whtson C. H., Modeln Gas Condensate Well Delverablty, Proc. Of 995 SPE Annual Techncal Conference (Oct. 5, 995) Mott, R., Calculatn Well Delverablty n Gas Condensate Reservors, EAGE 0 th Euroean Symosum on Imroved Ol Recovery, Brhton UK, Auust 8-0 (999). 3. Tan, Shfen & Adewum, M. A., Develoment of Analytcal Desn Equaton for Gas Pelnes, Proc. Of 99 SPE Annual Techncal Conference, (Oct. 4 7, 99) Ikoku, C. U., Natural Gas Producton Enneern, John Wley & Sons Inc., New York (984). 5. Zhou, Junyan & Adewum, M. A., The Develoment and Testn of a New Flow Equaton, Proceedn of PSIG Meetn, Houston, TX (995). 6. Hoffman, J. D., Numercal Methods for Enneers and Scentsts, McGraw-Hll Internatonal Edtons (993). 7. Ahmed, T., Hydrocarbon Phase Behavor, Gulf Publshn Comany, Houston, Texas (989).