pubs.acs.org/iecr Nonlnear Rsk Optmzaton Approach to Gas Lft Allocaton Optmzaton Mahd Khshvand and Ehsan Khamehch* Faculty of Petroleum Engneerng, Amrkabr Unversty of Technology, Tehran, Iran ABSTRACT: Gas lft allocaton can be modeled as a nonlnear programmng problem n whch adjustng optmum gas njecton rates and compressor pressure maxmze the ol rate or other objectve functons. Ths study descrbes a nonlnear programmng approach to maxmze daly cash flow of some gas-lfted wells n an uncertan condton for ol prce. Frst, some soluton ponts of each well are obtaned by employng a producton smulaton software. Then, by use of nonlnear optmzaton, a model s developed for gas lft performance n each well. Then these functons are used to develop a model under capacty, pressure and other real constrants for the cash flow of producton from these wells. Ol prce s assumed as a trangle rsk functon n ths model. Results show a sgnfcant ncrease n cash flow n comparson wth old case due to approprate gas njecton parameters. Senstvty analyss on ths problem shows that ol prce, compresson cost and water ol rato varatons should be consdered n the long term optmzaton.. INTRODUCTION Reservor pressure declnes durng the tme and consequently, producton rates decreases. Therefore, mproved ol recovery or enhanced ol recovery methods are appled to ncrease the producton rates and recovery factors of reservors. Contnuous gas lft s one of the artfcal lft methods ncreasng the ol producton rate of low pressure reservors. Injectng gas through the tubng-casng annulus aerates flud n the tubng. Therefore, reservor pressure s able to lft the ol column and forces the flud out of the wellbore. A schematc of the gas lft process s shown n Fgure. Fgure. A schematc of gas lft process. Gas allocaton to a group of gas lfted wells can strongly nfluence the performance of process, whch s focused on ths study. Gas lft allocaton optmzaton problems have constrants, possbly nonlnear, such as lmted allowable gas njecton rates, lmted avalable njected gas for njecton, ol producton constrants, and fnally water handlng constrants. In ths study, to maxmze ol producton cash flow, gas njecton rates are allocated to a group of wells. The problem has uncertanty on ol prce. Nonlnear programmng s used to model ol producton cash flow of the process. In ths model, a rsk functon for ol prce s assumed durng project lfe. The problem s subjected to operatonal constrants that have rarely been nvestgated n the lterature. These constrants are upper and lower ol producton rates, water handlng rate, pressure drop n ppe lnes, and total gas njecton rate. Addtonally, ths study makes a senstvty analyss on some parameters whch vary wth tme. 2. LITERATURE REVIEW Many studes have been conducted related to gas lft process optmzaton. Mayhll analyzed the relaton between gas njecton rate and ol producton rate called gas lft performance curve (GLPC). Gomez proposed a procedure to generate GLPC and developed a computer program to ft a second degree polynomal to t. He also proposed a procedure to obtan the optmum gas njecton rate. 2 Hong employed a cubc splne nterpolaton technque for the estmaton of the GLPCs. 3 Kanu et al. establshed equal slope allocaton method under both unlmted and lmted gas supply. 4 Lee presented a computer procedure and an optmal gas allocaton model for feld wde optmzaton. Camponogara and Nakashma solved the gas lft optmzaton problem subjected to constrants on the gas ppelnes. 6 Khamehch studed nonlnear approach for ol feld optmzaton based on gas lft optmzaton. 7 Djkpesse descrbed a novel approach to perform such optmzatons nvolvng nonsmooth models. 8 Receved: March 8, 20 Revsed: November 2, 20 Accepted: January, 202 Publshed: January, 202 202 Amercan Chemcal Socety 2637 dx.do.org/0.02/e20336a Ind. Eng.Chem. Res. 202,, 2637 2643
Also, there exst varous studes regardng a gas lft allocaton problem. Research n gas lft optmzaton was devoted to optmzaton problems usng ether a sngle well model, 9 or multple wells model. 0 Dfferent methodologes were used n solvng ths problem such as lnear programmng, 9 mxed nteger lnear programmng, nonlnear programmng, 2 Mxed-Integer Nonlnear Formulaton, 3 pecewse lnear formulatons, 4 dynamc programmng, 6 and so forth. In some other studes, a new functon for GLPC was ntroduced or new technques for solvng the optmzaton problem have been presented. Most of the studes contan constrants whch are less than those encountered n the real problems. In ths work, addtonal real world constrants such as pressure drop constrants are ncluded. Also, ol prce s assumed as a rsk functon and senstvty analyss on gas lft parameters are other novel aspects of ths study that ntroduces a comprehensve knowledge of real case gas lft projects. 3. NONLINEAR PROGRAMMING Optmzaton s one of the most mportant areas of modern appled mathematcs, wth the general form shown as eqs 3. mn ormax f ( x) () Subject to the followng: g ( x) 0 =, 2..., m (2) g ( x) = 0 = m, m +,... m (3) X R n If the objectve functon or at least one of constrants s nonlnear, then the program s called a nonlnear optmzaton problem. These types of problems have local and global optmum ponts. A schematc of the local and global optmum pont s shown n Fgure2. Fgure 2. Local and global optmum for a nonlnear objectve functon. 4. RISK OPTIMIZATION Rsk optmzaton technques are used to obtan optmum values n problems wth uncertan parameters. In ths process, a combnaton of smulaton and optmzaton are appled to optmze model wth uncertan factors. In ths study, ol prce s defned as a trangle probablty dstrbuton functon that s known Rsktrang. A software known as Rsk Optmzer s used that combnes Monte Carlo smulaton and genetc algorthm to perform a rsk analyss on each possble soluton generated durng the optmzaton. In each smulaton teraton, an ol prce probablty dstrbuton functon s sampled and a new value for the target cell s generated. At the end of smulaton, the average of the tral solutons s the statstc whch must be mnmzed for the dstrbuton of target cell. Ths value s then returned to genetc algorthm optmzer to generate new and better tral.. RESEARCH METHODOLOGY Reservor and well characterstcs collected from wells located n one of the Iranan olfelds have been used n ths study. Reservor layers n ths feld nvolve ASMARI and BANGE- STAN formatons. Characterstcs of each well are shown n Table. Also, a schematc of well locatons, dstances and the compressor staton are shown n Fgure3. The am of present study s to fnd optmum gas lft njecton rates nto each well... Gas Lft Smulaton. Usng Wellf lo, a producton smulaton software, the requred model s developed for producton of the wells under gas lft operaton. Wellf lo systems analyss software s a powerful and smpleto-use stand-alone applcaton to desgn, model, optmze, and troubleshoot ndvdual ol and gas wells, whether naturally flowng or artfcally lfted. The nput data are reservor pressure, flud propertes, productvty ndex, and other producton propertes. The smulaton uses Glaso correlatons for soluton gas rato and ol formaton volume factor, Beal correlaton for ol vscosty and fnally Carr et al. for gas vscosty predcton. The model uses a Vogel nflow performance relatonshp (IPR) to model reservor pressure-flow rate behavor. Well vertcal Flow Correlaton s Hagedorn and Brown and the soluton node s gas lft valve depth for each case. After smulatng the gas lft process, some output data such as pressure, operatng pont, etc. at soluton node s obtaned. The man group of ths data s operaton ponts that ndcate the ol producton rate of well# ( o, ) at varous gas njecton rates ( gnj, ). Operaton ponts, shown n Fgure 4, are obtaned by crossng IPR wth outflow performance curve. The operatng ponts of the gas lfted wells under varous gas njecton rates are shown n Table 2..2. Gas Lft Performance Curve Fttng. Data of Table 2 are used to develop a GLPC for each well, by usng a nonlnear optmzaton problem based on mnmzng absolute relatve error (ARE) for each well. The objectve mnmzaton functon s shown as eq 4. ARE = ( ( f( )) / ) 0 j= =, 2, 3, 4, o,,j gnj, j, oj,, where f( gnj,,j ) s a functon of gas njecton rate and Subscrpt j ndcates gas njecton or ol producton rate no. j. and well#, respectvely. Two types of functons are used to construct GLPC that was developed n prevous works and are represented as follows. 0. A: o, = C + C2gng, + C 3 gng, (4) 2638 dx.do.org/0.02/e20336a Ind. Eng.Chem. Res. 202,, 2637 2643
Table. Data of Producton Wells parameter unt well no. well no.2 well no.3 well no.4 well no. reservor characterstcs reservor top Ft 704 704 704 968 968 porosty % 9.9 9.9 9.9 6.22 6.22 S w % 46.48 46.48 46.48 4.3 4.3 reservor pressure ps 3737 3737 776 776 776 ol gravty API 30.20 30.20 28.04 28.04 28.04 gas specfc gravty ar =.00 0.7 0.7 0.6 0.6 0.6 PI STB/D/ps.6 2.00.80.4.2 water salnty ppm 2.0 0 2.0 0.8 0.8 0.8 0 GOR SCF/STB 83 83 940 940 940 well propertes P wf ps 2664 263 48 4304 4426 bottom hole temperature F 90 90 239 239 239 P wh ps 800 800 00 00 00 well head temperature F 7 7 7 7 7 mean perforaton depth Ft 8789 8789 207 207 207 tubng ID nch 4. 4. 4 4 4 operaton valve depth Ft 3000 3000 6890 6890 6890 njecton gas gravty ar =.00 0.7 0.7 0.7 0.7 0.7 njecton gas Z-factor (at 200 ps) 0.8 0.8 0.8 0.8 0.8 casng head pressure ps 300 300 200 200 200 WC % 6. 20 0 6 WOR % 0.03 0.070 0.20 0. 0.90 2 o, 2 g,nj 3 gnj, B: = C + C + C + C ln( + ) 4 gnj, Fgure 3. Wells and compressor staton locatons. (Nodes and 2 are conjunctons of ppelnes. Straght lnes ndcate the ppelnes). GLPC, a computer code, s developed for optmzaton and determnng of the coeffcents of the functons. Ol producton rates of well# were ftted nto these functons and are shown n Fgure. The results of the program are represented n Tables 3 and 4. For each well, the best functon, havng lower objectve value, s entered nto the gas lft allocaton model. Needless to say that, n all wells, functon B s an approprate functon..3. Mathematcal Formulaton of Allocaton Problem. In ths problem, gas njecton rate and compressor outlet pressure must be determned n such a way that maxmzes the daly producton cash flow. A nonlnear rsk optmzaton model was constructed for ths goal. The objectve functon s the sum of three terms as follows: CF = ( C ) C C O o, com w = = w, Where w, s water producton rate of well#. C o s the prce of one barrel of ol and s a trangle rsk functon, C w s water handlng cost per barrel, and C com s the cost of gas compressng per day. As can be observed, the frst term of the above equaton s ncome of ol producton. The second and thrd terms are compressng and water handlng costs, respectvely. The gas compressng cost s shown as eqs 6 and 7. 6 C = HHP C () com HHP (6) Fgure 4. A sample of operaton pont dentfcaton for well#. 0.2 HHP = 223 Pout gnj, P n = (7) 2639 dx.do.org/0.02/e20336a Ind. Eng.Chem. Res. 202,, 2637 2643
Table 2. Operatng Ponts of the Gas Lfted Wells under Varous Gas Injecton Rates gnj, (MMSCF/D) o, (STB/D) o,2 (STB/D) o,3 (STB/D) o,4 (STB/D) o, (STB/D) 0 36.346 838.66 946.92 880.89 884.4 0.40 609.902 2020.36 946.92 880.89 884.4 0.80 707.20 233.28 946.92 2238.620 9.44.20 769.622 22.707 2298.994 2346.387 242.707.60 80.64 226.328 278.94 243.70 27.976 2.00 84.326 2288.9 286.80 200.48 296.00 2.40 822.47 2300.288 2939.338 28.443 266.676 2.80 829.7 2306.088 3008.834 2606.90 276.976 3.20 834.600 232.472 3069.242 2648.33 2764.877 3.60 839.72 238.734 324.2 2682.023 2804.284 4.000 843.00 2324.026 368.096 2708.849 2836.27 4.400 846.239 2328.2 3204.32 2730.08 2863.6 Table 4. Gas Lft Allocaton Input Data ol prce ($/bbl) rsk trangle (7,63,70) water treatment cost ($/bbl) 3 compressng cost ($/hhp/hour).3 U gnj (MMSCF/D) U w (bbl/d) 00 U o (bbl/d) 20000 L o (bbl/d) 7000 UP com (ps) 6000 Fgure. The comparson of two surrogate models for GLPCs for well#. Where C HHP s the daly cost per each horsepower used. HHP s requred horsepower, P out and P n are outlet and nlet pressure nto the compressor..4. Constrants. After model development, constrants must be ndcated. These constrants defne feasble space of the NLP problem..4.. Gas Injecton Constrant. The frst constrant s gas avalablty and/or njecton capacty. There s one compressor n ths case whch has a maxmum rate of U gnj. Ths constrant s stated as eq 8. = gnj, U gnj (8).4.2. Pressure Constrants. The second group of constrants n ths study are related to pressure drops between wells and the compressor staton. In ths case, the Panhandle B equaton s used for gas frctonal pressure drops n the horzontal ppe lnes to calculate lmts of njecton rate nto each well. 7 The panhandle equaton gves us an njecton rate constrant for each well that s related to compressor pressure. The pressure drop between ponts and 2 n the panhandle B equaton s shown as eq 9. 0. = * * P 2 P2 2 g 0.028 E * D SG * Z* T* L g 0.96,2,2 2.3 (9) where E s effcency factor and s equal to 0.9 n ths case. Varables D,2 and L,2 are ppelne ID and length between nodes and 2, respectvely. In ths study, D s constant and s 6 n. between all nodes. The temperature n the ppelne s assumed equal to surface temperature and compressblty factor of njecton gas n operatonal condton s stated n Table. Table 3. Results of Program for GLPCs well # functon C C 2 C 3 C 4 ARE (type of optmum pont) A 36.346 498.97 29.682 0.373080 0 2 (global) B 36.346 606.694 4.43 33.69 0.2843990 0 2 (global) 2 A 838.66 48.206 06.999 0.64033 0 2 (global) B 838.66 364.83 22.0 988.278 0.400242 0 2 (global) 3 A 946.92 87. 372.923 0.284333 0 (global) B 946.92 2283. 232.620 2604.086 0.24842 0 (global) 4 A 880.89 462.02 20.977 0.707766 0 (global) B 880.89 8.220 0.20 7. 0.963207 0 2 (global) A 884.4 498.60 7.209 0.3403804 0 (global) B 884.4 39.683 4.390 733.86 0.23803 0 (global) 2640 dx.do.org/0.02/e20336a Ind. Eng.Chem. Res. 202,, 2637 2643
The gas njecton rate must be lmted, such that surface njecton pressure becomes greater or equal to the requred value. Pressure drop constrants add 7 nonlnear constrants to the problem. These constrants are between node no. and well# and well#2, node no. 2 and well#3 and 4, compressor staton and well#3 and fnally two more between compressor staton and nodes and 2. For example, constrant between pont and well# after smplfcaton s stated as follows: 2 0. gnj, 20. < [ p 690000] (0) Note that n the case between compressor and node no. or 2, the nequalty changes to equalty. Pressure drops n fttngs of node no. and 2 are neglected..4.3. Capacty Constrants. The thrd, fourth, and ffth constrant sets are related to water and ol producton rates. Due to separator lmtaton; producng an upper lmt of water (eq ) s permtted. w, U w () where U w s an upper lmt of water handlng rate. The ol capacty of the ppe-lne s also lmted. Ths defnes another constrant represented n eq 2. However, to satsfy ol demand, the producton rate must be greater than a lower lmt whch s defned by L o and s shown as eqs 2 and 3. o, o, U L o o (2) (3) The last constrant s the maxmum compressor outlet pressure that s equal to UP com ps. These constrants are stated as follows: Pcom UPcom (4) 6. RESULTS AND DISCUSSION The complex gas lft NLP s solved usng Rsk-optmzaton software that combnes the Monte Carlo smulaton technology wth genetc algorthm optmzaton to allow the optmzaton of the models wth uncertan values. Even f ths algorthm could not fnd the global optmum, the soluton wll be near optmum. The populaton, cross over rate, and mutaton rate of the GA s set to 00, 0., and 0., respectvely. Results and output of software nvolve dependent and ndependent objectve values. Optmum gas lft njecton rates and compressor pressure are shown n Table. In the optmzed case, by njecton of optmum gas rates and adjustng compressor pressure added value of 8 mllon $/year s obtaned compared to old one. A senstvty analyss on ol prce, gas compressng prce and WOR s done n ths study. The results are shown n Table 6. The greater value of the ol prce, the hgher optmum values of gas njecton rate s acheved. Note that gas njecton rates change n such a way that result n the hghest possble cash flow. Table. Objectve Values and Varables unt optmum value old value P out Ps 2700 4000 gnj, (MSCF/D).00 gnj,2 (MSCF/D).00 gnj,3 (MSCF/D) 3.6 2 gnj,4 (MSCF/D) 2.008 2 gnj, (MSCF/D).87 2 o, (STB/D) 744 744 o,2 (STB/D) 277 276 o,3 (STB/D) 306 286 o,4 (STB/D) 20 20 o, (STB/D) 274 296 ol rate (STB/D) 22 9392 water rate (STB/D) 793 2036 CF $ 6240 63000 Table 6. Senstvty Analyss Results unt ol prce ($) rsk trangle (3,60,67) compressng cost ($/hhp/ hour) WOR rsk trangle (63,70,77) 0.8 3 0.2 CF $ 6367 730339 692470 7936 64440 P out ps 267 2804 2720 2600 268 ol rate (STB/D) 9 286 899 0868 2074 gnj, (MMSCF/D) 0.900.36 0.96 0.6 0.90 gnj,2 (MMSCF/D) 0.948 0.872.3 0.400.083 gnj,3 (MMSCF/D) 2.99 3.0868 3.7 2.79 3.29 gnj,4 (MMSCF/D).609.962 2 0.98.808 gnj, (MMSCF/D).68 2.377.4 0.624.72 g (MMSCF/D) 8.33 9.43 9.23 4.402 8.78 w (STB/D) 767 804 788 92 308 Increasng gas compressng cost concludes less gas njecton, producton rate, and fnally cash flow. The lower gas njecton rate, the less output pressure. Change n ndvdual wells gas njecton rates s not a general trend, and may decrease or ncrease, such that maxmzes mean cash flow. Another key aspect s WOR (WC) that changes durng producton. For nvestgaton, another model wth WOR equal to 0.2 for each well s solved and compared to the base case. The results show that ths parameter has no sgnfcant effect on total gas njecton, but varaton n WOR changes optmum ndvdual njecton rates nto each well. On the bass of ths study, gas lft operaton can be optmzed n other smlar felds. More uncertan parameters, such as compressng cost, can be taken nto account wth a same procedure. Also, the procedure of the present study s a gudelne to nonlnear contnuous producton optmzaton problems wth uncertanty wth real constrants. 7. VALIDITY OF SOLUTION POINTS For each well, the soluton provded by the Rsk optmzer can be fed nto Wellf lo and cross checked. The soluton ponts of Tables and 6 are entered n Wellflo and the result of Wellflo are checked wth the results of related GLPC (surrogate model) whch s shown n Fgure 6. Subsequently, the ARE between the values of surrogate model and Wellf lo s calculated for each well and the accuracy s checked. AREs for all wells are 264 dx.do.org/0.02/e20336a Ind. Eng.Chem. Res. 202,, 2637 2643
Fgure 6. A sample result of Wellflo crosscheckng wth surrogate model for well#. Table 7. Absolute Relatve Error between Result of Wellf lo and Surrogate Model well no. 2 3 4 ARE 0.003 0.0024 0.02 0.026 0.0309 shown n Table 7. As ths table shows the error s not consderable and the results are acceptable. 8. CONCLUSIONS Gas lft allocaton optmzaton causes a sgnfcant ncrease n the cash flow of ol producton. Optmum value of gas njecton rate and compressor pressure should be defned n any project. Ol prce and compresson cost have a sgnfcant effect on operatonal parameters determnaton n the optmzaton. Therefore, n long-term gas lft optmzaton, ol prce and gas compresson cost must be consdered as the nfluencng parameters for ether total or ndvdual gas njecton rates. Also, WOR changes can be consdered for optmzaton of gas njecton rates nto each ndvdual well. AUTHOR INFORMATION Correspondng Author *Tel: +98 92 287 6770. SI METRIC CONVERSION FACTORS 0 API 4./(3. + 0 API) = g/cm 3 bbl(stb).89 873 0 0 =m 3 Ft 3.048* 0 0 =m F ( F + 32)/.8 = C nch 2.4 0 02 = Pascal ps 6.89 0 03 = Pascal SCF 2.83 0 02 =m 3 REFERENCES () Mayhll, T. D. Smplfed Method for Gas Lft Well Problem Identfcaton and Dagnoss. SPE, SPE 49th Annual Fall Meetng, Houston, Texas, USA, October 6 9, 974. (2) Gomez. V. Optmzaton of Contnuous Flow Gas Lft Systems. M.S. Thess. U. of Tulsa: Tulsa, Oklahama, USA, 974. (3) Hong. H. T. Effect of the Varables on Optmzaton of Contnuous Gas Lft System. M.S. Thess, U. of Tulsa: Tulsa, Oklahama, USA, 97. (4) Kanu, E. P.; Mach, J.; Brown, K. E. Economc Approach to Ol Producton and Gas Allocaton n Contnuous Gas Lft. J. Pet. Tech., October, 98: pp. 887-892. () Lee, H. K. Computer Desgn and Feld Wde Optmzaton for Gas Lfted Wells. SPE Mddle East Ol Techncal Conference & Exhbton, Maname, Bahran, Aprl, 3 6, 993. (6) Camponogara, E.; Nakashma, H. R. P. Solvng a Gas Lft Optmzaton Problem by Dynamc Programmng. Eur. J. Op. Res. 2006, 74, 220 246. (7) Khamehch, E.; Rashd, F. Karm B. Nonlnear Approach for Ol Feld Optmzaton Based on Gas Lft Optmzaton, Ol & Gas European Magazne, Issue 4, 2009. (8) Djkpesse, H. A.; Coueẗ, B. Wlknson, D. Gas Lft Optmzaton under Facltes Constrants. SPE 36977, 34th Annual SPE Internatonal Conference and Exhbton, Tnapa, Calabar, Ngera, 3 July 7 August, 200. (9) Fang, W. Y.; Lo, K. K. A Generalzed Well-Management Scheme for Reservor Smulaton. SPE Res. Eng. 996,, 6 20. (0) Alarcon, G.; Torres, C.; Gomez, L. Global optmzaton of Gas Allocaton to A Group of Wells n Artfcal Lft Usng Nonlnear constraned Programmng. J. Energy Resour. Tech. 2002, 24, 262 268. () Kosmds, V.; Perkns, J.; Pstkopoulos, E. A Mxed Integer Optmzaton Formulaton for the Well Schedulng Problem on Petroleum Felds. Comput. Chem. Eng. 200, 29, 23 4. (2) Nshkor, N.; Redner, R. A.; Doty, D. R. Schmdt, Z. An Improved Method for Gas Lft Allocaton Optmzaton.SPE97, 84th Annual Techncal Conference and Exhbton of the Socety of Petroleum engneers, San Antono, Texas, USA, October 8, 989. (3) Rashd, K.; Demrel, L.; Coueẗ, B. Gas-Lft Optmzaton wth Choke Control usng a Mxed-Integer Nonlnear Formulaton. Ind. Eng. Chem. Res. 20, 0 (), 297 2980. (4) Msener, R; Chrysanthos, E.; Floudas, A. Global Optmzaton of Gas Lftng Operatons: A Comparatve Study of Pecewse Lnear Formulatons. Ind. Eng. Chem. Res. 2009, 48 (3), 6098 604. 2642 dx.do.org/0.02/e20336a Ind. Eng.Chem. Res. 202,, 2637 2643
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