Suleiman O. Agboola, Dulu Appah and Oriji A. Boniface

Transcription

1 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May A Compaative analysis of pessue Gadient Models fo Vetical Multiphase Flow in Poducing Oil Wells in Nige Delta Suleiman O. Agboola, Dulu Appah and Oiji A. Boniface Abstact Gas-liquid flow commonly occus in oil and gas poduction and pocessing stuctue. The pediction of pessue gadient in multiphase flow fo vetical pipes is of inteest fo the oil industy and also a citical vaiable fo the best design of suface facilities. Models like empiical coelations and mechanistic models ae available to calculate the multiphase flow pessue gadient, holdup and phases distibution. In this study, simple and easy-tocompute pessue gadient models wee developed using MS Excel and VB.Net. The models wee validated with pessue gadient measued at the field. The models wee developed fo accuate measuement of pessue gadient in selected Nige-Delta oil wells as thee is limited validity of existing coelations that ae based on quality, egion and scope of data upon which they wee developed. Due to the exteme complexity of two-phase flow, the total pessue gadient was assumed to be dependent only on the no-slip liquid holdup (fo ease of estimation) by neglecting the acceleation and fictional components since the elevation component accounts fo not less than 90-99% of the total pessue gadient and neglecting the fictional tem accounts fo not moe than 10% eo. Pessue gadients got with the Okiszewski coelation was used to model fo slug flow and it gave an aveage pecent eo of 12.13% when compaed with field measuement. Aziz et al coelation was used to model fo both Mist and Bubble flow egimes; it gave aveage eo pecentages of 9.79% and 6.87% espectively when compaed with actual field measuement. Beggs and Bill coelation was used to model fo intemittent flow and it gave an aveage eo pecentage of 8.41% when compaed to actual field measuement. These models would be helpful in quick accuate estimation of pessue gadient which aids; in selecting coect tubing sizes; pedicting when a well quit flowing and hence pedict time fo atificial lift; designing atificial lift installations; detemining Pwf and PIs of wells; and pedicting maximum flow ates. Index Tems keywods: Multiphase flow, Nige Delta, Intemittent flow, Bubble flow, Slug flow, Mist flow, liquid holdup, pessue gadient flow equies an undestanding of the physical system [11]. 1 INTRODUCTION AND BACKGROUND Among the empiical coelations, Aziz et al coelation in When two o moe phases flow simultaneously in pipes, [8] seems to have some theoetical justification and with the flow behavio is much moe complex than fo singlephase flow. Shea stesses at the pipe walls ae diffeent fo pefomance inside poduction pipelines moe accuately. some modification it could pedict multiphase flow each phase as a esult of thei diffeent densities and Pessue losses encounteed duing vetical flow of two viscosities [1]. In spite of this, the ability to pedict it in phases ente into a wide ange of design calculations tuly new situations is not vey good. Diffeences ae (which may include tubing size and opeating well head pimaily as a esult of the vaiety of flow egimes that one pessue of a flowing well). Thee ae publications on ties to bidge with a single coelation scheme. Anothe seveal coelations that ae used to pedict pessue dop poblem is the lage numbe of dimensionless vaiables that in tubings and pipes fo the simultaneous, upwad, ae noticeably impotant, at least at some conditions [2]. continuous flow of wate, oil/gas. These coelations ae Pessue dop of seveal fluids have been investigated both empiical because of the exteme complexity of multiphase theoetically and expeimentally by seveal authos [3]. flow and so thee s a limited validity of the coelations Because of the highly complex and unpedictable natue of based on the quality, egion and scope of the data upon multiphase flow, most ealy investigatos used laboatoy which they ae based. Theefoe some coelations fail fo and/o field data to develop empiical coelations fo othe applications asides pefoming well fo cases in the evaluating pessue dop duing multiphase flow [3], [4], ange of data used in developing the coelation [12]. The [5], [6], [7], [8], [9] and [10]. The validity of these empiical aim of this study is to compae existing models fo coelations is to some degee limited to the quality and pedicting multiphase flow pessue gadient in vetical scope of the data and type of expeimental measuements flow fo poducing wells in the Nige-Delta povince. Some used in thei development. Theefoe, a bette appoach is authos have investigated the total pessue dop in twophase vetical flow in pipes. Chieci et al coelation in [3] to attempt to model the flow system and then to test the pedicted pessue dop in a two-phase vetical flow with mass tansfe between the flowing phases and the aveage Suleiman O. Agboola is cuently pusuing mastes degee pogam in density of the flowing fluid and the fiction losses wee petoleum engineeing at the Univesity of Pot Hacout, Nigeia, calculated accoding to the locally pevailing flow egime. talk2sahola@yahoo.com Pof. Dulu Appah is cuently a pofesso of petoleum poduction Giffith and Wallis, and Duns and Ros coelations in [13] Engineeing, Univesity of Pot Hacout, Nigeia and [14] wee used to evaluate the egions of existence of D. Oiji Boniface is cuently the HOD petoleum engineeing the vaious flow egimes. A new elationship was poposed Univesity of Pot Hacout, Nigeia. aloiji2000@yahoo.com (This infomation is optional; change it accoding to you need.) fo extapolating at bubble Reynolds numbe, (NRe)b>6,000 model against actual data. Pope modeling of multiphase and oveall Reynolds numbe (NRe)t>6,000, the Giffith 2016

2 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May and Wallis coelation in [13] was used to calculate the aveage fluid density in the slug-foth flow egime. Thei method was tested fo validity on 31 actual oil well cases coveing a boad ange of API gavity, GOR, oil flow ate, and pessue dop. The validity of the method was shown by efeing to the deviation between pedicted and actual pessue dops. This showed an aveage eo of 0.12 %, aveage absolute eo of 4.36 % and a standad deviation 5.42 % [15]. Sandip et al. in [16] developed a new model to pedict the pessue dop unde foam flow conditions. By using a simple dift flux model and using some of the available liteatue data on foam, a foam pessue dop model was developed, and compaed with 570 data points collected fom many gas wells. The match between the model and the actual obsevations was easonable. Khasanov et al. in [17] developed a new mechanistic model fo two-phase flow in vetical and inclined pipes based on Dift-flux appoach. Unlike the othe mechanistic models in [18] and the unified mechanistic model, the developed model incopoated a system of nonlinea equation to solve using an explicit equation fo liquid hold up pediction. The simple fom of liquid hold up pediction fomula enables analytical integation of pessue gadient in two phase flow along the pipe. The model was evaluated using TUFFP databank and Rosnef field data. Evaluation showed that in compaison with mechanistic models, the poposed model enables calculating pessue gadient with compaable accuacy, and less calculation esouces equied (Poceedings of Intenational Oil Confeence and Exhibition in Mexico IOCEM, 06/2007). Coelations used in solving multiphase flow poblems ae categoized into two goups, namely, empiical and mechanistic models. The empiical appoach was the ealy method used by eseaches to solve multiphase flow poblems in the past. Duing this time the investigatos employed the use of simplifying assumptions and physicals methods based on field and expeimental data fom the laboatoy. Mechanistic model on the othe hand was a late appoach, based on full desciption of the elementay mechanism happening in multiphase flow. Empiical coelations ae categoized into thee goups accoding to [7]; Goup 1 Coelations in this goup do not conside slip between phases and flow egime. The thee coelations in this goup ae as follows: Poettmann and Capente in [4], Baxendell and Thomas in [5], and Fanche and Bown in [6]. Goup 2 Fo this goup, slip is consideed, but no flow egime is consideed. Thee coelations ae pesented in this categoy, they ae: Hagedon and Bown in [19], Gay and Asheim coelations. Goup 3 This categoy consides both slip and flow egime Coelations in this goup includes: Duns and Ros in [14], Okiszewski in [7], Aziz et al. in [8], Chieiciet al in [3], Beggs and Bill in [9] and Mukhejee and Bill in [10]. Beggs and Bill in [10] developed a coelation to compute pessue gadient in all ange of pipe inclination. They used data obtained fom small-scale test facility expeiment compising of 1.0 and 1 ½ in section made fom acylic pipe, 90ft. long to develop the coelation. The fluid system used was ai and wate. Fom the expeiment, they vaied the gas flow ate between 0 and 300 Mscf and fo the liquid flow ate, 0 and 30gal/ min. The aveage pessues of the system wee between 35 and 95psia. Aziz et al. in [8] developed a mechanistic coelation fo vetical two phase flow. The aim of thei model involved the pediction of actual flow pattens based on simplified flow patten map. The two phase popeties and vaiables such as density, fictional facto and pessue gadient wee calculated fom boad equations accuate fo each flow configuation. They developed new coelations fo slug and bubble flow pattens. They used the Duns and Ros coelation in [14] fo mist flow patten but used intepolation method poposed by Duns and Ros in [14] fo tansition flow. Taitel et al in [20] developed a mechanistic model fo flow patten detemination. They poposed a physical mechanism fo the tansition bounday in between flow pattens and modeled each tansition bounday on the basis of the mechanism by which it occus. Fom thei model, fou distinctive flow pattens, namely bubbly, chun and slug flow pattens wee obseved. Hassan and Kabi in [21] and [22] developed a mechanistic model fo multiphase flow in vetical tube. Thei model focused on tansition boundaies individually. Valid citeia fo each tansition bounday wee developed. The fou diffeent tansition boundaies obseved wee, bubble-slug flow, the tansition-dispesed bubble flow, the slug-chun flow and the tansition-annula tansitions boundaies. Expeimental data fom liteatue that occued at a void faction of 0.25 was used to detemine the bubble-slug flow tansition. They pesumed that since a tansition is a slow pocess, it was bette to use a teminal velocity of Taylo bubbles in slug flow fo detemining the bubble-slug flow tansition bounday. Fo dispesed bubbles flow patten, the tansition is ascibed to the beakdown of lage bubbles in the liquid as a esult of high flow ates. They used Taitel et al. equation in [20] fo mixtue velocity and linked it to the maximum bubble diamete possible unde tubulent condition. An expession fo gas supeficial velocity was deived fo tansition to annula flow patten because void faction tends to unity. Ansai et al. in [18] developed a mechanistic model that pedicts a tansition bounday of diffeent flow egime, pessue dop fo bubble-slug flow and annula flow egime. They developed an implicit equation fo calculating liquid hold up in the bubble flow patten and deived slug flow patten to fully developed and developing slug flow.

3 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May METHODOLOGY eo pecent will be estimated fo each of the coelations used fo the espective flow egimes. 2.1 Model Development A model will be developed using MS Excel, and validated using Visual Basic softwae. Fo simplicity of esult, the acceleation and fiction components wee neglected (assumed zeo) in this study. Pessue gadients will be estimated using the Okiszewski in [7], Aziz et al in [8], and the Beggs and Bill in [9] coelations fo the diffeent flow egimes. The pessue gadients that will be got fom the individual coelations will be compaed to the actual pessue gadient got fom the field measuement. The 2.2 Data Collection Measued field data (such as oil FVF, gas FVF, oil gavity, gas gavity, liquid suface tension, viscosities of oil/gas, flowate of oil/gas, BHFP, WHP, solution GOR, flowing tempeatue and pessue well depth, tubing size, measued pessue gadient etc.) wee obtained fom poducing oil wells in the Nige-Delta. TABLE 1 COLLECTED FLUID PROPERTIES (FP) OF THE 11 OIL WELLS FP q l q g API B o B g T P Ö D H µ o µ g R s

4 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May RESULTS 3.1 Gaphic Use Inteface of VB.Net Figue 1 - The inteface of the visual basic model that was developed Figue 2 - analysis of well 1 with the Aziz et al model (fo mist flow) 2016

5 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May Figue 3 - analysis of well 1 with the Beggs and Bill model (fo intemittent flow) Figue 4 - analysis of well 1 with the Okiszewski model (fo slug flow) 2016

6 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May Figue 5 - analysis of well 2 with the Aziz et al model (fo mist flow) Figue 6 - analysis of well 2 with the Okiszewski model (fo slug flow) 2016

7 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May Figue 7 - analysis of well 3 with the Aziz et al model (fo Bubble flow) Figue 8 - analysis of well 3 with the Okiszewski model (fo Bubble flow) 2016

8 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May Figue 9 - analysis of well 4 with the Aziz et al model (fo mist flow) Figue 10 - analysis of well 4 with the Okiszewski model (fo slug flow) 2016

9 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May Figue 11 - analysis of well 5 with the Aziz et al model (fo mist flow) Figue 12 - analysis of well 5 with the Beggs and Bill model (fo intemittent flow) 2016

10 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May Figue 13 - analysis of well 5 with the Okiszewski model (fo slug flow) The same analysis caied out fo well 1 to well 5 as seen above, wee also done fo wells 6 to Discussion of esults and validity of the eos of 2.73%, 4.44%, 4.04% espectively while wells 9, 10, Models 11 wee ove-pedicted with aveage pecent eo of 14.99%. The oveall standad deviation was Fo Figues 14 to 17 ae epesentations on using an existing bubble flow, once again the Aziz et al coelation was empiical coelation (Aziz et al. in [8], Beggs and Bill in utilized in estimating the pessue gadients fo wells 8, 9, [9], and Okiszewski in [7]) on all the eleven wells. Fo mist 10 and 11, these esults wee compaed to the actual flow, pessue gadient got fom utilizing Aziz et al pessue gadients in the espective wells. Fom figue 17, coelation fo wells 1 to 7 wee compaed with actual field wells 8, 9, 10, 11 wee ove-pedicted with highest pecent pessue gadients in the espective wells. Fom figue 14, eo of 16.9% occuing at well 8 due to the lage tubing wells 1 and 6 wee almost accuately pedicted with 0.79% size (6 ) because Pwf deceases as tubing size inceases and 0.69% eos espectively howeve well 2 is geatly while wells 9, 10, 11 have an aveage pecent eo of 3.52%. unde-pedicted (30% eo) pobably due to the high age The oveall standad deviation was Fom the of the field. In contast, wells 3,4,5,7 wee slightly undepedicted with aveage pecent eo of 9%. The oveall fo diffeent flow egimes (using the empiical coelations foegoing; the models developed pefomed satisfactoily standad deviation was Fo Slug flow, pessue as basis fo establishing the egimes), having aveage gadients got fom utilizing the Okiszewski's coelation in pecent eo anging fom 6.87% (bubble flow), 8.41% [7] fo wells 1 to 7 wee compaed with the actual field (Intemittent flow), 9.79% (mist flow), to 12.13% (Slug flow) pessue gadients in the espective wells. Fom figue 15, when compaed (validated) against pessue gadient of well 2 (0.68% eo) was obseved to be almost accuate each well measued at the field which is concodant with while well 7 was unde-pedicted in contast to wells ange ecoded in liteatues. It was obseved that the 1,3,4,5,6 which wee ove-pedicted with aveage pecent pessue gadient was ove-pedicted (highe than eo of 15.56%. The oveall standad deviation was measued) in many cases with few exceptions of undepediction. Howeve, excessive pecent eo noted in some Fo intemittent flow, pessue gadients got fom utilizing the Beggs and Bill's coelation in [9] fo wells 1, 5. 6, 7, 8, wells could be as a esult of any of the following factos; 9, 10 and 11wee compaed with the actual pessue Assumptions made in developing the models such as gadients in the espective wells. Fom figue 16, wells 1 neglecting both fictional and acceleation components, age and 5 wee obseved to be slightly unde-pedicted with of the well, fluid popety coelations, wate cut, and pecent eos of 7.77% and 3.30% espectively. In contast, sanding etc. wells 6, 7, 8 wee slightly ove-pedicted with pecent 2016

11 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May P R² = G a d ( p s i / f t ) R² = Well numbe Measued Pedicted Figue 14(a) - Compaison of pedicted PG and the measued PG (Mist flow) P G a d ( p s i / f t ) Measued Pedicted Well numbe Measued Pedicted Figue 14(b) - Compaison of pedicted PG and the measued PG (Mist flow) 2016

12 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May R² = P G a d ( p s i / f t ) R² = Measued Pedicted Well numbe Figue 15(A) - Compaison of Pedicted PG And The Measued PG (Slug Flow) P G a d ( p s i / f t ) Measued Pedicted Well numbe Measued Pedicted Figue 15(b) - Compaison of pedicted PG and the measued PG (Slug flow) 2016

13 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May P G a d ( p s i / f t ) R² = R² = Well numbe Measued Pedicted Figue 16(a) - Compaison of pedicted PG and the measued PG (Intemittent flow) P G a d ( p s i / f t ) Measued Pedicted Well numbe Measued Pedicted Figue 16(b) - Compaison of pedicted PG and the measued PG (Intemittent flow) 2016

14 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May P R² = G a d ( p s i / f t ) R² = Well numbe Measued Pedicted P G a d Figue 17(a) - Compaison of pedicted PG and the measued PG (Bubble flow) ( p s i / f t ) Measued Pedicted Well numbe Measued Pedicted Figue 17(b) - Compaison of pedicted PG and the measued PG (Mist flow) 4 CONCLUSIONS Based on this study, the following conclusions wee made: Simple and quick-yield models fo pedicting pessue 2016

15 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May gadient in selected Nige Delta wells wee developed, evey well have a unique pessue gadient coelation indices which epesent its popensity to having diffeent flow egime at diffeent stage of its life, the developed model is eliable as the aveage pecent diffeence/eo between obseved and pedicted PG agees with that ecoded in liteatue fo othe wells investigated in diffeent egions, compaed to existing models, the developed models yielded bette esults, being tailoed fo selected wells within the Nige Delta and finally, pedictions with this model will be useful in: (i) Selecting coect tubing sizes; (ii) Envisaging when a well quit flowing & hence time fo atificial lift; (iii) Designing atificial lift installations; (iv) Detemining Pwf& PIs of wells; (v) Pedicting maximum flow ates. 5 RECOMMENDATIONS The following ecommendations ae suggested to highlight aeas of additional eseach to impove the fomulation of the model developed in this wok: [2] James P. Bill, H. Dale Beggs, (1994): Two-Phase Flow in Pipes. 6th ed. Univesity of Tulsa, Oklahoma, USA. [3] Chieci,G.L. Gucci, G.M., Sclocchi, G. (1974): Two-Phase Vetical Flow in Wells - Pediction of Pessue Dop. SPE-4316-PA. [4] Poetmann, F.H, and Capente P.G. (1952): "Multiphase Flow of Gas, Oil and Wate Though Vetical Flow Stings With Application to The Design of Gas-Lift Installations". Dilling and Poduction Pactice, pp [5] Baxendell P.B., and Thomas, R. (1961): "The Calculation of Pessue Gadients In High-Rate Flowing Wells". SPE-2-PA, Jounal of Petoleum Technology, 13(10); , Octobe [6] Fanche, G.H, J., and Bown, K.E. (1963): "Pediction of Pessue Gadients fo Multiphase Flow in Tubing", SPE Jounal,Mach 1963, pp [7] Okiszewski, J. (1967): "Pedicting Two Phase Pessue Dop In Vetical Pipes. Jounal of Petoleum Technology, 19(6); , SPE PA. [8] Aziz, K., Govie, G.W and Fogaasi, M. (1972): "Pessue Dop in wells poducing oil", and gas", J.cdn.pet tech. (July to spptembe) [9] Beggs, H.D, and Bill J.P. (1973): "A Study of Two Phase Flow In Inclined Pipes", Jounal of Petoleum Technology, 25(5); , May [10] Mukhejee, H., and Bill, J.P., (1985): "Pessue Dop Coelation Fo Inclined Two-Phase Flow", Jounal of Enegy Resouces Technology, Vol 107, pp , Decembe [11] Rashid, A. H., Kabi, S. C., (1988): "A study of multiphase flow behavio in vetical wells", Society of Petoleum Enginees Jounal Pape, SPE PA, pp i Inceased numbe of poducing well/field within [12] Lawson, J.D. and Bill, J.P. (1974): A Statistical the Nige-Delta should be used in both model fomulation Evaluation of Methods used to pedict pessue and development. losses fo multiphase flow in vetical oil well ii In ode to impove tubing. SPE_AIME, Univesity of Tulsa, SPE the accuacy of two-phase PA. flowing pessue gadient, effots should be made to impove fluid popety coelations. [13] Giffith, P., and Wallis, G. (1961): "Two-Phase Slug Flow", ASME J. Heat Tansfe, 83, pp iii Futue study should incopoate hoizontal and inclined wells. [14] Duns, H., J., and Ros, N.C.J. (1963): "Vetical Flow of Gas and Liquid Mixtues In Wells", Poceedings 6 REFERENCES of 6th Wold Petoleum Congess, pp [1] [15] Jounal of Petoleum Technology, 08/1974. [16] Sandip S., Mohan, K. and Cio, P. (2009): Pessue Dop Pedictions in Tubing in the Pesence of Sufactants.SPE [17] Khasanov, M. M., Kasnov, V., Khabibullin, R., Pashali, A., Guk, V. (2007): "A simple mechanistic model fo liquid holdup and pessue gadient pediction in vetical and inclined gas-liquid flow", SPE MS, Pesented at the intenational oil confeence and exhibition in Mexico, June, Veacuz, Mexico. [18] Ansai, A., and Behbahaninia, A.R. (1997): PVT Model Chaacteization of Highly Vaiable Oil Popeties With Depth, 15th Wold Petoleum Congess, Octobe 1997, WPC [19] Hegedon, A.R and Bown, K.E. (1965): Expeimental Study of Pessue Gadients Occuing Duing Continuous Two-Phase Flow in Small Diamete Vetical Conduits". Jounal of Petoleum Technology, Apil 1965, pp

16 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May [20] Taitel, Y., Bane, D., and Dukle, A.E. (1980): "Modeling Flow Patten Tansitions Fo Steady Upwad Gas-Liquid Flow in Vetical Tubes", AlChE Jounal, Vol 26, pp [21] Hassan, A.R. and Kabi, C.S. (1986): A Study of Multi-Phase Flow Behaviou in Vetical Wells: Pat I-Theoetical Teatment. Univesity of Noth Dakota, and Schlumbege Oveseas S.A espectively, SPE APPENDIX 7.1 Geneal Multiphase Pessue gadient equation [22] Hassan, A.R. and Kabi, C.S. (1989): A Study of Multi-Phase Flow Behaviou in Vetical Wells: Pat II- Field Application. Univesity of Noth Dakota, and Schlumbege Oveseas S.A espectively, SPE dz Total = dp dz elevation + dp dz fictional + dp dz acceleation 1 But; dz acceleation = Aziz et al Coelation Bubble flow: = ρ dz LH L + ρ L (1 H L ) 3 elevation Whee; H L = 1 V sg 1.2V m 4 +V bs V bs = 1.4 σ 1 Lg ρ L ρ g 4 ρ 2 5 L = fρ 2 sv m 6 dz fictinal 2g c d Mist flow: = ρ V SL V sg dz L + ρ elevation V g = ρ m V m = ρ L λ L + ρ g λ g 7 m = fρ 2 gv g dz fictional 2g c d Beggs and Bill coelation Intemittent flow: dz elevation = ρ LH L + ρ g (1 H L ) 9 Whee; H L = H L (φ)ψ 10 = f tpρ n V2 m dz fictional 2g c d 11 = dz elevation + dp dz fictional 12 dz total 1 E k E k = ρ LV m V sg g c ρ

17 Intenational Jounal of Scientific & Engineeing Reseach, Volume 7, Issue 5, May Okiszewski Coelation Slug flow: dz elevation = ρ L(V SL +V b )+ρ g V sg V m +V b + ρ s δ = ρ s 14 = fρ 2 LV m dz fictional Bubble flow 2g c d V SL+V b V m +V b + δ = ρ f 15 dz elevation = ρ s = ρ L H L + ρ g (1 H L ) 16 Whee; H L = V m 1 + V m 2 V s Vs 2 4 V g Vs 17 dz fictional = fρ L V SL HL 2 2g c d 18 8 NOMENCLATURE ρ L = Liquid density V m = Mixtue Velocity d = Pipe Diamete V sg = Supeficial gas velocity V b = Bubble point velocity V SL = Supeficial liquid velocity λ L = no-slip liquid holdup λ g = no slip gas holdup q L = Liquid flow ate q g = gas flow ate B o = Oil fomation volume facto B g = Gas fomation volume facto T = Tempeatue P = Pessue μ o = Oil Viscosity μ g = gas viscosity H L = Liquid holdup 2016