Nonlinear Risk Optimization Approach to Gas Lift Allocation

Article pubs.acs.org/IECR

Nonlinear Risk Optimization Approach to Gas Lift Allocation Optimization Mahdi Khishvand and Ehsan Khamehchi* Faculty of Petroleum Engineering, Amirkabir University of Technology, Tehran, Iran ABSTRACT: Gas lift allocation can be modeled as a nonlinear programming problem in which adjusting optimum gas injection rates and compressor pressure maximize the oil rate or other objective functions. This study describes a nonlinear programming approach to maximize daily cash flow of some gas-lifted wells in an uncertain condition for oil price. First, some solution points of each well are obtained by employing a production simulation software. Then, by use of nonlinear optimization, a model is developed for gas lift performance in each well. Then these functions are used to develop a model under capacity, pressure and other real constraints for the cash flow of production from these wells. Oil price is assumed as a triangle risk function in this model. Results show a significant increase in cash flow in comparison with old case due to appropriate gas injection parameters. Sensitivity analysis on this problem shows that oil price, compression cost and water−oil ratio variations should be considered in the long term optimization.

1. INTRODUCTION Reservoir pressure declines during the time and consequently, production rates decreases. Therefore, improved oil recovery or enhanced oil recovery methods are applied to increase the production rates and recovery factors of reservoirs. Continuous gas lift is one of the artificial lift methods increasing the oil production rate of low pressure reservoirs. Injecting gas through the tubing-casing annulus aerates fluid in the tubing. Therefore, reservoir pressure is able to lift the oil column and forces the fluid out of the wellbore. A schematic of the gas lift process is shown in Figure1.

limited available injected gas for injection, oil production constraints, and finally water handling constraints. In this study, to maximize oil production cash flow, gas injection rates are allocated to a group of 5 wells. The problem has uncertainty on oil price. Nonlinear programming is used to model oil production cash flow of the process. In this model, a risk function for oil price is assumed during project life. The problem is subjected to operational constraints that have rarely been investigated in the literature. These constraints are upper and lower oil production rates, water handling rate, pressure drop in pipe lines, and total gas injection rate. Additionally, this study makes a sensitivity analysis on some parameters which vary with time.

2. LITERATURE REVIEW Many studies have been conducted related to gas lift process optimization. Mayhill analyzed the relation between gas injection rate and oil production rate called gas lift performance curve (GLPC).1Gomez proposed a procedure to generate GLPC and developed a computer program to fit a second degree polynomial to it. He also proposed a procedure to obtain the optimum gas injection rate.2Hong employed a cubic spline interpolation technique for the estimation of the GLPCs.3 Kanu et al. established equal slope allocation method under both unlimited and limited gas supply.4 Lee presented a computer procedure and an optimal gas allocation model for field wide optimization.5 Camponogara and Nakashima solved the gas lift optimization problem subjected to constraints on the gas pipelines.6 Khamehchi studied nonlinear approach for oil field optimization based on gas lift optimization.7 Djikpesse described a novel approach to perform such optimizations involving nonsmooth models.8

Figure 1. A schematic of gas lift process.

Gas allocation to a group of gas lifted wells can strongly influence the performance of process, which is focused on this study. Gas lift allocation optimization problems have constraints, possibly nonlinear, such as limited allowable gas injection rates, © 2012 American Chemical Society

Received: Revised: Accepted: Published: 2637

March 8, 2011 November 2, 2011 January 11, 2012 January 11, 2012 dx.doi.org/10.1021/ie201336a | Ind. Eng.Chem. Res. 2012, 51, 2637−2643

Industrial & Engineering Chemistry Research

Article

defined as a triangle probability distribution function that is known “Risktriang”. A software known as “Risk Optimizer” is used that combines Monte Carlo simulation and genetic algorithm to perform a risk analysis on each possible solution generated during the optimization. In each simulation iteration, an oil price probability distribution function is sampled and a new value for the target cell is generated. At the end of simulation, the average of the trial solutions is the statistic which must be minimized for the distribution of target cell. This value is then returned to genetic algorithm optimizer to generate new and better trial.

Also, there exist various studies regarding a gas lift allocation problem. Research in gas lift optimization was devoted to optimization problems using either a single well model,9 or multiple wells model.10 Different methodologies were used in solving this problem such as linear programming,9 mixed integer linear programming,11 nonlinear programming,12 Mixed-Integer Nonlinear Formulation,13 piecewise linear formulations,14 dynamic programming,6 and so forth. In some other studies, a new function for GLPC was introduced or new techniques for solving the optimization problem have been presented. Most of the studies contain constraints which are less than those encountered in the real problems. In this work, additional real world constraints such as pressure drop constraints are included. Also, oil price is assumed as a risk function and sensitivity analysis on gas lift parameters are other novel aspects of this study that introduces a comprehensive knowledge of real case gas lift projects.

5. RESEARCH METHODOLOGY Reservoir and well characteristics collected from 5 wells located in one of the Iranian oilfields have been used in this study. Reservoir layers in this field involve “ASMARI” and “BANGESTAN” formations. Characteristics of each well are shown in Table1. Also, a schematic of well locations, distances and the compressor station are shown in Figure3. The aim of present study is to find optimum gas lift injection rates into each well. 5.1. Gas Lift Simulation. Using “Wellf lo”, a production simulation software, the required model is developed for production of the wells under gas lift operation. “Wellf lo” systems analysis software is a powerful and simpleto-use stand-alone application to design, model, optimize, and troubleshoot individual oil and gas wells, whether naturally flowing or artificially lifted. The input data are reservoir pressure, fluid properties, productivity index, and other production properties. The simulation uses “Glaso” correlations for solution gas ratio and oil formation volume factor, “Beal” correlation for oil viscosity and finally “Carr et al.” for gas viscosity prediction. The model uses a “Vogel” inflow performance relationship (IPR) to model reservoir pressure-flow rate behavior. Well vertical Flow Correlation is “Hagedorn and Brown” and the solution node is gas lift valve depth for each case. After simulating the gas lift process, some output data such as pressure, operating point, etc. at solution node is obtained. The main group of this data is operation points that indicate the oil production rate of well#i (Qo,i) at various gas injection rates (Qginj,i). Operation points, shown in Figure 4, are obtained by crossing IPR with outflow performance curve. The operating points of the gas lifted wells under various gas injection rates are shown in Table 2. 5.2. Gas Lift Performance Curve Fitting. Data of Table 2 are used to develop a GLPC for each well, by using a nonlinear optimization problem based on minimizing absolute relative error (ARE) for each well. The objective minimization function is shown as eq 4.

3. NONLINEAR PROGRAMMING Optimization is one of the most important areas of modern applied mathematics, with the general form shown as eqs 1−3.

min or max f (x)

(1)

Subject to the following:

gi(x) ≤ 0i = 1, 2....., m1

(2)

gi(x) = 0i = m1, m1 + 1, .....m

(3)

X ∈ Rn If the objective function or at least one of constraints is nonlinear, then the program is called a nonlinear optimization problem. These types of problems have local and global optimum points. A schematic of the local and global optimum point is shown in Figure2.

10

AREi =

∑ (|(Q o,i,j − f (Q ginj, i , j))|/Q o, i , j) j=1

i = 1, 2, 3, 4, 5 Figure 2. Local and global optimum for a nonlinear objective function.

(4)

where f(Qginj,i,j) is a function of gas injection rate and Subscript j indicates gas injection or oil production rate no. j. and well#i, respectively. Two types of functions are used to construct GLPC that was developed in previous works and are represented as follows.15

4. RISK OPTIMIZATION Risk optimization techniques are used to obtain optimum values in problems with uncertain parameters. In this process, a combination of simulation and optimization are applied to optimize model with uncertain factors. In this study, oil price is

A: Q o , i = C1 + C 2Q ging, i 0.5 + C3Q ging, i 2638

dx.doi.org/10.1021/ie201336a | Ind. Eng.Chem. Res. 2012, 51, 2637−2643


Article

Table 1. Data of Production Wells reservoir characteristics

well properties

parameter

unit

well no.1

well no.2

well no.3

well no.4

well no.5

reservoir top porosity Sw reservoir pressure oil gravity gas specific gravity PI water salinity GOR Pwf bottom hole temperature Pwh well head temperature mean perforation depth tubing ID operation valve depth injection gas gravity injection gas Z-factor (at 2500 psi) casing head pressure WC WOR

Ft % % psi °API air = 1.00 STB/D/psi ppm SCF/STB psi °F psi °F Ft inch Ft air = 1.00

7045 9.59 46.48 3737 30.20 0.75 1.56 2.0 × 105 853 2664 190 800 75 8789 4.5 3000 0.7 0.8 1300 5 0.053

7045 9.59 46.48 3737 30.20 0.75 2.00 2.0 × 105 853 2653 190 800 75 8789 4.5 3000 0.7 0.8 1300 6.5 0.070

7045 9.59 46.48 5776 28.04 0.65 1.80 1.8 × 105 940 4518 239 1500 75 11207 4 6890 0.7 0.8 2500 20 0.250

9685 6.22 45.53 5776 28.04 0.65 1.41 1.8 × 105 940 4304 239 1500 75 11207 4 6890 0.7 0.8 2500 10 0.111

9685 6.22 45.53 5776 28.04 0.65 1.52 1.8 × 105 940 4426 239 1500 75 11207 4 6890 0.7 0.8 2500 16 0.190

psi % %

B: Q o , i = C1 + C 2Q g ,inj + C3Q ginj, i 2 + C4 ln(Q ginj, i + 1) “GLPC”, a computer code, is developed for optimization and determining of the coefficients of the functions. Oil production rates of well#1 were fitted into these functions and are shown in Figure 5. The results of the program are represented in Tables 3 and 4. For each well, the best function, having lower objective value, is entered into the gas lift allocation model. Needless to say that, in all wells, function “B” is an appropriate function. 5.3. Mathematical Formulation of Allocation Problem. In this problem, gas injection rate and compressor outlet pressure must be determined in such a way that maximizes the daily production cash flow. A nonlinear risk optimization model was constructed for this goal. The objective function is the sum of three terms as follows:

Figure 3. Wells and compressor station locations. (Nodes 1 and 2 are conjunctions of pipelines. Straight lines indicate the pipelines).

5

CF =

5

∑ (CO × Q o, i) − Ccom − Cw × ∑ Q w , i i=1

i=1

(5)

Where Qw,i is water production rate of well#i. Co is the price of one barrel of oil and is a triangle risk function, Cw is water handling cost per barrel, and Ccom is the cost of gas compressing per day. As can be observed, the first term of the above equation is income of oil production. The second and third terms are compressing and water handling costs, respectively. The gas compressing cost is shown as eqs 6 and 7.16

C com = HHP × CHHP

(6)

0.2 ⎡⎛ ⎤ Pout ⎞ ⎢ HHP = 223 × ∑ Q ginj, i × ⎜ ⎟ − 1⎥ ⎢⎣⎝ Pin ⎠ ⎥⎦ i=1

(7)

5

Figure 4. A sample of operation point identification for well#1. 2639



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Table 2. Operating Points of the Gas Lifted Wells under Various Gas Injection Rates Qginj,i (MMSCF/D)

Qo,1 (STB/D)

Qo,2 (STB/D)

Qo,3 (STB/D)

Qo,4 (STB/D)

Qo,5 (STB/D)

0 0.40 0.80 1.20 1.60 2.00 2.40 2.80 3.20 3.60 4.000 4.400

1365.346 1609.902 1707.201 1769.622 1801.564 1814.326 1822.475 1829.117 1834.600 1839.172 1843.010 1846.239

1838.166 2020.316 2133.281 2211.707 2261.328 2288.595 2300.288 2306.088 2312.472 2318.734 2324.026 2328.512

1946.912 1946.912 1946.912 2298.994 2758.914 2856.810 2939.338 3008.834 3069.242 3124.152 3168.096 3204.352

1880.859 1880.859 2238.620 2346.387 2431.750 2500.481 2558.443 2606.905 2648.331 2682.023 2708.849 2730.081

1884.114 1884.114 1951.544 2421.707 2517.976 2596.500 2661.676 2716.976 2764.877 2804.284 2836.257 2863.615

Table 4. Gas Lift Allocation Input Data

Where CHHP is the daily cost per each horsepower used. HHP is required horsepower, Pout and Pin are outlet and inlet pressure into the compressor. 5.4. Constraints. After model development, constraints must be indicated. These constraints define feasible space of the NLP problem. 5.4.1. Gas Injection Constraint. The first constraint is gas availability and/or injection capacity. There is one compressor in this case which has a maximum rate of UQginj. This constraint is stated as eq 8.

3 1.3 15 5500 20000 7000 6000

⎡ ⎤0.51 P12 − P22 ⎢ ⎥ *D 2.53 Q g = 0.028*E* 1,2 ⎢ SG 0.961*Z*T*L ⎥ ⎣ g 1,2 ⎦

(9)

where E is efficiency factor and is equal to 0.95 in this case. Variables D1,2 and L1,2 are pipeline ID and length between nodes 1 and 2, respectively. In this study, D is constant and is 6 in. between all nodes. The temperature in the pipeline is assumed equal to surface temperature and compressibility factor of injection gas in operational condition is stated in Table 1.

5

(8)

i=1

risk triangle (57,63,70)

5.4.2. Pressure Constraints. The second group of constraints in this study are related to pressure drops between wells and the compressor station. In this case, the “Panhandle B” equation is used for gas frictional pressure drops in the horizontal pipe lines to calculate limits of injection rate into each well.17 The panhandle equation gives us an injection rate constraint for each well that is related to compressor pressure. The pressure drop between points 1 and 2 in the panhandle B equation is shown as eq 9.

Figure 5. The comparison of two surrogate models for GLPCs for well#1.

∑ Q ginj, i ≤ UQ ginj

oil price ($/bbl) water treatment cost ($/bbl) compressing cost ($/hhp/hour) UQginj(MMSCF/D) UQw(bbl/D) UQo(bbl/D) LQo(bbl/D) UPcom(psi)

Table 3. Results of Program for GLPCs well #

function

C1

C2

C3

1

A B A B A B A B A B

1365.346 1365.346 1838.166 1838.166 1946.912 1946.912 1880.859 1880.859 1884.114 1884.114

498.1971 −606.6954 458.206 −364.5831 −87.111 2283.555 462.052 −81.220 498.650 −39.683

−129.682 45.413 −106.999 22.101 372.923 −232.620 −20.977 0.520 −7.209 −4.390

2 3 4̀ 5

2640

C4 1353.169 988.278 −2604.086 711.151 733.865

ARE (type of optimum point) 0.3713080 0.2843990 0.5640331 0.1400242 0.5284333 0.2548542 0.1707766 0.9632071 0.3403804 0.2513803

× × × × × × × × × ×

10−2 (global) 10−2 (global) 10−2 (global) 10−2 (global) 10−1 (global) 10−1 (global) 10−1 (global) 10−2 (global) 10−1 (global) 10−1 (global)



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The gas injection rate must be limited, such that surface injection pressure becomes greater or equal to the required value. Pressure drop constraints add 7 nonlinear constraints to the problem. These constraints are between node no.1 and well#1 and well#2, node no. 2 and well#3 and 4, compressor station and well#3 and finally two more between compressor station and nodes 1 and 2. For example, constraint between point 1 and well#1 after simplification is stated as follows:

520.5Q ginj,1 < [p12 − 1690000]0.51

Table 5. Objective Values and Variables Pout Qginj,1 Qginj,2 Qginj,3 Qginj,4 Qginj,5 Qo,1 Qo,2 Qo,3 Qo,4 Qo,5 oil rate water rate CF

(10)

Note that in the case between compressor and node no. 1 or 2, the inequality changes to equality. Pressure drops in fittings of node no. 1 and 2 are neglected. 5.4.3. Capacity Constraints. The third, fourth, and fifth constraint sets are related to water and oil production rates. Due to separator limitation; producing an upper limit of water (eq 11) is permitted.

1

(11)

unit

CF Pout oil rate Qginj,1 Qginj,2 Qginj,3 Qginj,4 Qginj,5 Qg Qw

5 1

(12)

5

∑ Q o, i ≥ LQ o 1

4000 1 1 2 2 2 1744 2176 2865 2501 2596 9392 2036 630500

compressing cost ($/hhp/ hour)

oil price ($)

$ psi (STB/D) (MMSCF/D) (MMSCF/D) (MMSCF/D) (MMSCF/D) (MMSCF/D) (MMSCF/D) (STB/D)

565367 2675 11951 0.900 0.948 2.991 1.609 1.685 8.133 1767

730339 2804 12186 1.136 0.872 3.0868 1.962 2.377 9.435 18045

0.8 692470 2720 11899 0.956 1.3 3.575 2 1.4 9.231 1788

3

WOR 0.25

557936 644405 2600 2685 10868 12074 0.611 0.950 0.400 1.083 2.179 3.219 0.598 1.808 0.624 1.721 4.402 8.781 1592 3018

(13)

Increasing gas compressing cost concludes less gas injection, production rate, and finally cash flow. The lower gas injection rate, the less output pressure. Change in individual wells gas injection rates is not a general trend, and may decrease or increase, such that maximizes mean cash flow. Another key aspect is WOR (WC) that changes during production. For investigation, another model with WOR equal to 0.25 for each well is solved and compared to the base case. The results show that this parameter has no significant effect on total gas injection, but variation in WOR changes optimum individual injection rates into each well. On the basis of this study, gas lift operation can be optimized in other similar fields. More uncertain parameters, such as compressing cost, can be taken into account with a same procedure. Also, the procedure of the present study is a guideline to nonlinear continuous production optimization problems with uncertainty with real constraints.

The last constraint is the maximum compressor outlet pressure that is equal to UPcom psi. These constraints are stated as follows:

Pcom ≤ UPcom

old value

2700 1.001 1.001 3.165 2.008 1.857 1744 2177 3065 2501 2574 12121 1793 652410

risk triangle risk triangle (53,60,67) (63,70,77)

where UQw is an upper limit of water handling rate. The oil capacity of the pipe-line is also limited. This defines another constraint represented in eq 12. However, to satisfy oil demand, the production rate must be greater than a lower limit which is defined by LQo and is shown as eqs 12 and 13.

∑ Q o, i ≤ UQ o

optimum value

Psi (MSCF/D) (MSCF/D) (MSCF/D) (MSCF/D) (MSCF/D) (STB/D) (STB/D) (STB/D) (STB/D) (STB/D) (STB/D) (STB/D) $

Table 6. Sensitivity Analysis Results

5

∑ Q w , i ≤ UQ w

unit

(14)

6. RESULTS AND DISCUSSION The complex gas lift NLP is solved using “Risk-optimization” software that combines the Monte Carlo simulation technology with genetic algorithm optimization to allow the optimization of the models with uncertain values. Even if this algorithm could not find the global optimum, the solution will be near optimum. The population, cross over rate, and mutation rate of the GA is set to 500, 0.5, and 0.1, respectively. Results and output of software involve dependent and independent objective values. Optimum gas lift injection rates and compressor pressure are shown in Table 5. In the optimized case, by injection of optimum gas rates and adjusting compressor pressure added value of 8 million $/year is obtained compared to old one. A sensitivity analysis on oil price, gas compressing price and WOR is done in this study. The results are shown in Table 6. The greater value of the oil price, the higher optimum values of gas injection rate is achieved. Note that gas injection rates change in such a way that result in the highest possible cash flow.

7. VALIDITY OF SOLUTION POINTS For each well, the solution provided by the Risk optimizer can be fed into “Wellf lo” and cross checked. The solution points of Tables 5 and 6 are entered in “Wellf lo” and the result of “Wellf lo” are checked with the results of related GLPC (surrogate model) which is shown in Figure 6. Subsequently, the ARE between the values of surrogate model and “Wellf lo” is calculated for each well and the accuracy is checked. AREs for all wells are 2641



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Figure 6. A sample result of “Wellflo” crosschecking with surrogate model for well#1.

Table 7. Absolute Relative Error between Result of Wellf lo and Surrogate Model well no.

1

2

3

4

5

ARE

0.0013

0.0024

0.0215

0.0256

0.0309

(2) Gomez. V. Optimization of Continuous Flow Gas Lift Systems. M.S. Thesis. U. of Tulsa: Tulsa, Oklahama, USA, 1974. (3) Hong. H. T. Effect of the Variables on Optimization of Continuous Gas Lift System. M.S. Thesis, U. of Tulsa: Tulsa, Oklahama, USA, 1975. (4) Kanu, E. P.; Mach, J.; Brown, K. E. Economic Approach to Oil Production and Gas Allocation in Continuous Gas Lift. J. Pet. Tech., October, 1981: pp. 1887-1892. (5) Lee, H. K. Computer Design and Field Wide Optimization for Gas Lifted Wells. SPE Middle East Oil Technical Conference & Exhibition, Maname, Bahrain, April, 3−6, 1993. (6) Camponogara, E.; Nakashima, H. R. P. Solving a Gas Lift Optimization Problem by Dynamic Programming. Eur. J. Op. Res. 2006, 174, 1220−1246. (7) Khamehchi, E.; Rashidi, F. Karimi B. Nonlinear Approach for Oil Field Optimization Based on Gas Lift Optimization, Oil & Gas European Magazine, Issue 4, 2009. (8) Djikpesse, H. A.; Couët, B. Wilkinson, D. Gas Lift Optimization under Facilities Constraints. SPE 136977, 34th Annual SPE International Conference and Exhibition, Tinapa, Calabar, Nigeria, 31 July−7 August, 2010. (9) Fang, W. Y.; Lo, K. K. A Generalized Well-Management Scheme for Reservoir Simulation. SPE Res. Eng. 1996, 11, 116−120. (10) Alarcon, G.; Torres, C.; Gomez, L. Global optimization of Gas Allocation to A Group of Wells in Artificial Lift Using Nonlinear constrained Programming. J. Energy Resour. Tech. 2002, 124, 262− 268. (11) Kosmidis, V.; Perkins, J.; Pistikopoulos, E. A Mixed Integer Optimization Formulation for the Well Scheduling Problem on Petroleum Fields. Comput. Chem. Eng. 2005, 29, 1523−1541. (12) Nishikiori, N.; Redner, R. A.; Doty, D. R. Schmidt, Z. An Improved Method for Gas Lift Allocation Optimization.SPE19711, 84th Annual Technical Conference and Exhibition of the Society of Petroleum engineers, San Antonio, Texas, USA, October 8−11, 1989. (13) Rashid, K.; Demirel, L.; Couët, B. Gas-Lift Optimization with Choke Control using a Mixed-Integer Nonlinear Formulation. Ind. Eng. Chem. Res. 2011, 50 (5), 2971−2980. (14) Misener, R; Chrysanthos, E.; Floudas, A. Global Optimization of Gas Lifting Operations: A Comparative Study of Piecewise Linear Formulations. Ind. Eng. Chem. Res. 2009, 48 (13), 6098−6104.

shown in Table 7. As this table shows the error is not considerable and the results are acceptable.

8. CONCLUSIONS Gas lift allocation optimization causes a significant increase in the cash flow of oil production. Optimum value of gas injection rate and compressor pressure should be defined in any project. Oil price and compression cost have a significant effect on operational parameters determination in the optimization. Therefore, in long-term gas lift optimization, oil price and gas compression cost must be considered as the influencing parameters for either total or individual gas injection rates. Also, WOR changes can be considered for optimization of gas injection rates into each individual well.

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AUTHOR INFORMATION

Corresponding Author

*Tel: +98 912 287 6770.

SI METRIC CONVERSION FACTORS API 141.5/(131.5 + 0API) = g/cm3 bbl(STB) × 1.589 873 ×10−01 = m3 Ft × 3.048* ×10−01 = m °F (°F + 32)/1.8 = °C inch × 2.54 ×10−02 = Pascal psi × 6.895 ×10−03 = Pascal SCF × 2.83 ×10−02 = m3 0

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REFERENCES

(1) Mayhill, T. D. Simplified Method for Gas Lift Well Problem Identification and Diagnosis. SPE 5151, SPE 49th Annual Fall Meeting, Houston, Texas, USA, October 6−9, 1974. 2642



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(15) Hamedi, H.; Rashidi, F.; Khamehchi, E. A Novel Approach to the Gas-Lift Allocation Optimization Problem. Petroleum Science and Technology 2011, 29 (Issue 4), 418−427. (16) Economides, M.; Daniel Hill, A.Ehlig-EconommidesCh.Petroleum Production Systems; Prentice Hall Petroleum Engineering Series: New Jersey, USA, 1993. (17) Arnold, K. Stewart, M. Surface Production Operations; Gulf Publishing: Houston, Texas, USA, 1999.

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Nonlinear Risk Optimization Approach to Gas Lift Allocation

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