Production-System Optimization of Gas Fields

Production-System Optimization of Gas Fields Using Hybrid Fuzzy/Genetic Approach Hui-June Park, SPE, Korea National Oil Corporation; Jong-Se Lim, SPE, Korea Maritime University; Jeongyong Roh, Korea National Oil Corporation; and Joo M. Kang, SPE, and Bae-Hyun Min, Seoul National University

Summary The design of production systems of gas fields is a difficult task because of the nonlinear nature of the optimization problem and the complex interactions between each operational parameter. Conventional methods, which are usually stated in precise mathematical forms, cannot include the uncertainties associated with vague or imprecise information in the objective and constraint functions. This paper proposes a fuzzy nonlinear programming approach to accommodate these uncertainties and applies it to a variety of optimization processes. Specifically, the fuzzy ␭-formulation is combined with a hybrid coevolutionary genetic algorithm for the optimal design of gas-production systems. Both the multiple conflicting objective and constraint functions for production systems of gas fields are formulated in a feasible fuzzy domain. Then, the genetic algorithm is used as a primary optimization scheme for solving the optimum gas-production rates of each well and the pipeline segment diameters to minimize the investment cost with a given set of constraints in order to enhance the ultimate recovery. The synthetic-optimization method can find a global compromise solution and offer a new alternative with significant improvement over the existing conventional techniques. The reliability of the proposed approach is validated by a synthetic practical example yielding more-improved results. This method constitutes an offering of a powerful tool for cost savings in the planning and optimization of gas-production operations. Introduction The objective of production optimization is to determine the operational parameters, such as the production rates and the tubing or pipeline diameter, subject to all of the constraints in a given period of production for the purpose of increasing the ultimate recovery and reducing the production costs. In many mature fields, the typical oil- and gas-production systems have flowline networks with many wells and surface pipelines to gather the products for further separation or transportation (Fujii and Horne 1995). Because oil and gas production is constrained by the reservoir conditions, the flow capacity of the pipeline networks and surface facilities, and various safety and economic considerations, the implementation of the optimal control parameters is a difficult task. The main targets to optimize production system of gas fields are to deliver the contract rates over the period of the gas contract and to confirm the deliverability of the pipeline networks (Schneider et al. 2002). Generally, production-planning problems are stated in precise mathematical forms. It is practically impossible, however, to express the objectives and constraints of such optimization problems using crisp relations because of the fuzziness of the information, which might be vague, uncertain, and imprecise when collected from the field (Liu and Chen 1999). Some constraint violations may lead to a more adequate and practical solution to this problem (Xiong and Rao 2004). In this design problem, fuzzy mathematical programming can be a more efficient and flexible

Copyright © 2010 Society of Petroleum Engineers This paper (SPE 100179) was accepted for presentation at the SPE Europec/EAGE Annual Conference and Exhibition, Vienna, Austria, 12–15 June 2006, and revised for publication. Original manuscript received for review 15 April 2007. Revised manuscript received for review 30 July 2009. Paper peer approved 29 September 2009.

June 2010 SPE Journal

tool to accommodate uncertainty quantitatively in the presence of vague information. Since the concept of fuzzy logic was introduced by Zadeh (1965), it has been widely developed and applied to various areas where intuition and judgment still play a fundamental role. Fuzzy logic mimics the human decision-making and reasoning processes and thus is suitable for applications with high levels of complexity and uncertainty. Recently, fuzzy-logic systems have been used to solve a number of different problems in a variety of areas, including physical and chemical systems, production planning and scheduling, location and transportation problems, resource allocation in financial systems, and engineering design. In the petroleum industry, fuzzy logic has been put to work in many different areas since the early 1990s, including reservoir characterization (Hambalek and Gonzalez 2003; Lim and Kim 2004; Rafiei et al. 2009; Shokir 2006; Soto et al. 2001; Taghavi 2005; Zhou et al. 1993; Cuddy 2000; Kedzierski and Mallet 2006; Finol et al. 2002; Hajizadeh 2007a), optimal well operations (Alimonti and Falcone 2004; Garrouch and Lababidi 2003, 2005; Rivera 1994; Dumans 1995; Xiong et al. 2001), stimulation treatment (Xiong and Holditch 1995; Nitters et al. 2000), and economic analysis (Zolotukhin 2000; de Salvo Castro and Fereira Filho 2001; Agbon and Araque 2003; Chang et al. 2006). An extensive list of such applications is presented in Table 1. In particular, many applications are focused on engineering-design and control problems (Sengul and Bekkousha 2002; Mohaghegh et al. 2005; Widarsono et al. 2005; Nikravesh et al. 1997; Taheri 2008; Weiss et al. 2001; Cao et al. 2006; Zarei et al. 2008; Wu et al. 1997; Murillo et al. 2009; Kanj and Roegiers 1999; Hajizadeh 2007b; Garrouch and Al-Ruhaimani 2003) relating to the operation of production systems and facilities. Neuroth et al. (2000) applied fuzzy logic to the control of pump stations for oil- and gas-transport systems, and their method resulted in a reduction of the maintenance and operational costs. The design of the fuzzy-logic switchover control is based on the idea of manipulating one of the pressure-control signals so that the overall transition from one control mode to the other can be made more smoothly. Liu and Chen (1999) applied fuzzy-logic models to the optimization of surface pipeline systems in Daqing oilfield of China. In their approach for both single-objective and multiobjective cases in water-injection oilfields, an optimal solution of the design parameters is selected from the fuzzy-logic feasible domain. Sun et al. (2000) reported on a project, conducted jointly between SaskEnergy/Transgas and the University of Regina, which aims at developing an integrated decision-support system for the optimization of natural-gas-pipeline operations. A fuzzy programming model was used to determine the specific compressor unit to be turned on or off. Alimonti and Falcone (2004) applied artificial neural networks and fuzzy logic to multiphase-flow metering to provide a real-time monitoring of produced-flow rates and stream composition and to check the quality of this same information. However, previous works dealt mostly with the control of some part of a production systems or facilities. Only a limited work using the fuzzy nonlinear programming approach has been performed on production-system optimization, and the application of such techniques is still rare. In addition, conventional derivative-based optimization techniques for solving fuzzy-mathematical-programming problems result in suboptimal operations and require problemspecific algebraic manipulations. Therefore, the implementation of 417

TABLE 1—FUZZY-LOGIC APPLICATIONS IN THE PETROLEUM INDUSTRY Reservoir Characterization* • Virtual measurement (formation permeability and porosity prediction) • Lithofacies identification • Paleotopography modeling • Inversion of nuclear magnetic resonance log Optimal Well Operations and Stimulation Treatment** • Drilling process control (directional drilling, underbalanced drilling candidate selection, wellbore stability analysis) • Pressure control system • Integration of multiphase flowmetering • Mitigation of gas-hydrate-related drilling risks • Diagnosis of formation-damage mechanisms • Multilateral well completion • Well stimulation treatment design • Post-fracture deliverability prediction Optimization of Field Operations

†

• Infill drilling recovery forecast • Production data analysis • Injection optimization • Production-system optimization • Well-placement optimization • Pipe sticking prediction and avoidance • Corrosion prediction • Formation-damage assessment • Sand-management solutions Economic Analysis

††

• Resources and reserves determination • Risk assessment in petroleum development finance • Prediction of oil and gas spot prices *Abdulraheem et al. (2007); Hambalek and Gonzalez (2003); Lim and Kim (2004); Rafiei et al. (2009); Shokir (2006); Soto et al. (2001); Taghavi ( 2005); Zhou et al. (1993); Cuddy (2000); Kedzierski and Mallet (2006); Finol et al. (2002). ** Hajizadeh (2007a); Alimonti and Falcone (2004); Garrouch and Lababidi (2003, 2005); Rivera (1994); Dumans (1995); Xiong et al. (2001); Xiong and Holditch (1995); Nitters et al. (2000). † Sengul and Bekkousha (2002); Mohaghegh et al. (2005); Widarsono et al. (2005); Nikravesh et al. (1997); Taheri (2008); Weiss et al. (2001); Cao et al. (2006); Zarei et al. (2008); Wu et al. (1997); Murillo et al. (2009); Kanj and Roegiers (1999); Hajizadeh (2007b); Garrouch and Al-Ruhaimani (2003). †† Zolotukhin (2000); de Salvo Castro and Fereira Filho (2001); Agbon and Araque (2003); Chang et al. (2006).

a reliable and effective optimization method is crucial in engineering-design fields. This paper proposes a new hybrid fuzzy-logic optimization approach for single- or multiobjective gas-production systems with multiple constraints and provides a feasible solution by combining the fuzzy-logic ␭-formulation with a hybrid coevolutionary genetic algorithm. The proposed fuzzy-logic system can manage the resulting information in terms of reservoir and production network for an integrated approach to production-system analysis. The synthetic-optimization method can find a global compromise solution and offer a new alternative with significant improvement over the existing conventional techniques. The proposed method was applied to a practical case study for solving optimum allocation of the gas-production rates for each well and the pipeline segment diameters in order to minimize the investment cost with a given set of constraints. This case study demonstrates the feasibility of the proposed approach as a practical, cost-effective, and robust gasfield production management tool. Fuzzy Nonlinear Programming The conventional approach to design problems is to adopt a deterministic model formulated in a precise mathematical form. 418

Then, variations in the operating conditions, geometry, boundary conditions, etc. are considered in the optimization process by the use of simplifying hypotheses such as consideration of extreme or mean values or the application of safety factors (Schuëller and Jensen 2008). Uncertainties associated with variability and randomness in the process of observation and measurement may cause significant changes in the performance and reliability of the design variables. Furthermore, final solutions obtained by a deterministic model may become infeasible when the uncertainty in the system parameter is considered. Hence, a proper design procedure must accommoadate these types of uncertainties. The resulting system will perform within prescribed margins with a certain reliability (i.e., a quantitative measure of the system safety will be available) (Schuëller and Jensen 2008). The main approaches to optimization under uncertainty can be classified as stochastic programming (recourse models, robust stochastic programming, and probabilistic models) and fuzzy programming (flexible and possibilistic programming), depending on how uncertainty is modeled. Objective or random uncertainty has an intrinsically irreducible stochastic nature, which is often modeled through discrete or continuous probability functions in stochastic programming. By contrast, subjective or epistemic uncertainty, including model errors and errors introduced by the numerical solution of the corresponding problem, is related to a lack of information that basically could be reduced by additional information. Fuzzy programming is more appropriate for this type of uncertainty. The uncertainty associated with integrated gas-production systems is not random but rather involves the imprecise description of information collected from the gas field (Zadeh 1965). In such cases, it is more reasonable to define the optimization problem with some tolerance interval for imprecision, such as a membership function in fuzzy mathematical programming (Liu and Chen 1999). In general, the mathematical model of multiobjective nonlinear programming is stated as (Sahinidis 2004; Zimmermann 1991; Fletcher 1993) maximize or minimize f ( x ) = ⎡⎣ f1 ( x ) ,..., fk ( x ) ⎤⎦ gi ( x ) ≤ bi ,

subject to

i = 1,..., m

h j ( x ) = 0,

j = 1,..., n

li ≤ xi ≤ ui ,

i = 1,..., n , . . . . . . . . . . (1)

where x is an n-dimensional vector of decision variables, fi(x) are k conflicting objective functions, gj and hj denote inequality and equality constraints, bi indicates the allowable interval of the constraint function gj, and li and ui are the lower and upper bounds of the design variables, respectively. In fuzzy programming, objectives and constraints are treated as fuzzy sets. Some constraint violation is allowed, and the degree of satisfaction of a constraint including the range of uncertainty of the coefficients is defined as the membership function of the constraint (Sahinidis 2004). The fuzzy feasible region is defined by considering all the constraints as follows m

C = ∩ gi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2) i =1

and the membership degree of the decision vector x to the fuzzy feasible region C is given by

{

}

␮C ( x ) = min ␮ gi ( x ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) which is the minimum degree of satisfaction of the design vector x with respect to all of the constraints. For a given fuzzy goal G and fuzzy constraint C in a space of alternatives x, a decision domain D, which is a fuzzy feasible set resulting from the intersection of G and C , is expressed as

{

m

}

D = ␮G ( x ) ∩ ∩ ␮ gi ( x ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . (4) i =1

June 2010 SPE Journal

where ␮G ( x ) and ␮ gi ( x ) denote the membership functions of the objective and the constraint function, respectively. The intersection of fuzzy sets is defined by the min-operator, which is the minimum degree of satisfaction of the decision vector with respect to all of the objectives and constraints. Then, an optimal solution x* can be selected by maximizing the membership function of the fuzzy decision set D.

␮ D ( x * ) = max ␮ D ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5) The equivalent formulation of the max/min problem using the fuzzy ␭-formulation is stated as maximize ␭ subject to ␭ ≤ ␮G

␭ ≤ ␮ gi ( x ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

In fuzzy mathematical programming, objective functions are treated as constraints, with the lower and upper bounds defining the decisionmaker’s expectations. Some other types of membership function with additional statistical information have been proposed for the construction of objective and constraint membership functions. However, the linear membership function is commonly used because the optimization solution is relatively insensitive to the actual type of membership function (Sahinidis 2004). The details of this procedure are presented in Appendix A. To solve this nonlinear programming problem efficiently, a hybrid coevolutionary/genetic algorithm is proposed. Hybrid Coevolutionary/Genetic Algorithm Optimization processes require the evaluation of objective and constraint functions hundreds or thousands of times. Hence, the use of efficient optimization techniques that require fewer function calls and guarantee a global optimum is critical. A variety of simulation techniques can be used for the reliability estimation process. Generally speaking, optimization techniques can be classified as linear programming, dynamic programming, integer programming, geometric programming, heuristic method, Lagrangean multiplier method, etc. Genetic algorithms have been widely used as global methods for complex function optimization (Goldberg 1989). To handle the nonlinear constraints of general nonlinear programming problems, most genetic algorithms are based on penalty functions, which penalize infeasible solutions (Sakawa 2002). These methods have several impediments, such as distorting the evaluation function and increasing the number of searches. To overcome the many limitations of conventional genetic algorithms, a new hybrid coevolutionary/genetic algorithm is proposed for solving nonlinear optimization problems. This coevolutionary/genetic algorithm is based on a coevolution and repair strategy consisting of two separate populations, which are the population of the search points that satisfy the linear constraints of the problem, S, and the population of the reference points that satisfy all of the constraints of the problem, X. A reproduction in one population influences the evaluations of individuals in the other population. At least one initial reference point that is fully feasible is required to create an initial population of reference points. The random generation of initial reference points from individuals, however, is very difficult for most of the optimization problem. To avoid this difficulty, gradient-based sequential quadratic programming (SQP) is used to create an initial reference population in the initial phase of the search. Once an initial reference point is obtained by SQP, an initial reference population is created by using multiple copies of that point. Subsequently, the coevolution and repair processes are performed until the prescribed termination condition is satisfied. The repair process includes the following steps (Sakawa 2002), which are also presented in Fig. 1. 1. Generate two separate initial populations using the floatingpoint representation. 2. Apply closed crossover and mutation operators, in the sense that the resulting offsprings always satisfy the linear constraints, to the population of search points. The details of this procedure are presented in Appendix B. June 2010 SPE Journal

3. Generate new random points z = as+(1−a)r from a segment between search point s ∈ S and reference point r ∈ X using a random number a in the range of 0–1. 4. If the newly generated point z is feasible and its fitness value is better than r, then replace r with z as a new reference point. Also, replaces s by z with the replacement probability of pr. If generated point z is infeasible, repeat Step 3 until a feasible point is found or a preset number of generations is reached. 5. Repeat this repair process for one reference point and all search points. 6. After evaluating the individuals in the population of reference points, apply the selection operator to generate individuals of the next generation using the exponential ranking method. Consequently, two separate populations coevolve. Because the objective function is evaluated only for the reference points, the evaluation function is not distorted, as in the case of penalty-based methods. A flow diagram of the fuzzy optimization method is shown in Fig. 1. Interactive Fuzzy Multiobjective Programming (Sakawa 2002). For solving multiobjective-programming problems whose objective functions conflict with one another, the proposed optimization method is effectively applicable. After determining the membership function for each of the objectives and constraints, the decision maker specifies the reference membership values for all of the membership functions. For the specified reference membership levels ␮i , i = 1,..., k , the corresponding Pareto optimal solution (which is, in the minimum/maximum sense, nearest to the requirement or better than the requirement if the reference membership levels are attainable) is obtained by solving the minimum/maximum problem.

{

}

minimize max ␮i − ␮i ⎡⎣ f ( x ) ⎤⎦ . . . . . . . . . . . . . . . . . . . . . . (7) To circumvent the necessity of performing the Pareto optimality tests in the minimum/maximum problems, the augmented minimum/ k

{

}

maximum problem, in which the term ␳ ∑ ␮i − ␮i ⎡⎣ f ( x ) ⎤⎦ is i =1

added to the standard minimum/maximum problem, is adopted as follows k ⎧ ⎫ minimize max ⎨␮i − ␮i ⎡⎣ f ( x ) ⎤⎦ + ␳ ∑ ⎡⎣␮i − ␮i ( f ( x )) ⎤⎦ ⎬, i =1 ⎩ ⎭ . . . . . . . . . . . . . . . . . . . . . . . . (8)

User’s interface program

Contain fuzzy information

Yes

No Crisp mathematical modeling module

Fuzzy mathematical modeling module

Coevolutionary GA • Floating-point representation Reference populations

Search populations

• Exponential ranking selection • Genetic operation Satisfy termination condition

Repair process

Optimum solution

Fig. 1—Flow diagram of fuzzy nonlinear programming. 419

7

6

8

5

2

4

1

3

Well

P1

TABLE 2—TUBING STRING AND PIPELINE DATA FOR OPTIMIZATION EXAMPLES Number

Diameter (in.)

Length (ft)

T1

3

6,500

T2

3

6,600

T3

3

7,000

T4

3

6,600

T5

3

7,100

T6

5

7,500

T7

5

7,500

T8

5

7,500

T9

5

6,700

T10

3

7,200

T11

3

7,200

T12

3

7,100

T13

5

6,600

T14

5

6,600

T15

5

6,600

T16

5

6,600

T17

3

7,000

T18

3

7,000

T19

3

7,100

T20

3

7,200

P1

4

18,000

P2

5

14,000

P3

5

4,500

P4

5

12,500

If the current solution is not acceptable in the interaction processes, update the reference membership values on the basis of the decision condition and return to the corresponding augmented minimum/maximum problem.

P5

5

14,250

P6

5

15,000

P7

5

13,000

P8

5

14,500

Applications This section presents two examples of gasfield optimization. The first example compares the performance of fuzzy nonlinear programming with conventional crisp optimization method for the allocation of well rates to maximize the total gas production. The second example demonstrates the feasibility of interactive fuzzy multiobjective programming as a practical gasfield management tool. Fig. 2 shows a graphical representation of a synthetic gasgathering network with treelike and loop structures modifying Handley-Schachler et al.’s case study (2000). The network consists of 20 wells and nine manifolds connected to a single gathering center assumed to be operating at a fixed pressure of 2,300 psi. The reservoir block pressures of the wells and the permeability are in the range of 3,700–4,250 psi and 10–35 md, respectively. Table 2 lists the tubing string and surface pipeline data of the production system, which are denoted by T1–T20 and P1–P10, respectively. All of the tubing strings with diameters between 3 and 5 in. are connected to surface pipelines with diameters between 4 and 10 in. and lengths between 4,500 and 150,000 ft through chokes with a maximum diameter of ¾ in. The specific gravity of the gas is 0.85, and the temperature of the wellbore and wellhead are 130 and 40°F, respectively.

P9

5

18,700

P10

10

150,000

Manifold

P2

P3 9

P10 P4

Field gathering center

10

P5 11

P9 20

12

P8

P6

19

P7

13

14 15

16

17

18

Fig. 2—Configuration of gathering system for optimization examples.

where ␳ is a sufficiently small positive value. The proposed optimization technique can be adopted for each string s of the following fitness function. k ⎧ ⎫ f ( s ) = 1.0 + k ␳ − max ⎨␮i − ␮i ⎡⎣ f ( x ) ⎤⎦ + ␳ ∑ ⎡⎣␮i − ␮i ( f ( x )) ⎤⎦ ⎬ . i =1 ⎩ ⎭ . . . . . . . . . . . . . . . . . . . . . . . . (9)

Example 1. The objective of the problem is to maximize the total gas-production rate by optimally allocating the flow rate of each well. The optimization problem is formulated as follows: 20

maximize ∑ qi i =1

subject to 0 ≤ qi ≤ 20, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10) where qi is the gas-flow rate of well i in the range of 0–20 MMscf/D. 420

In this example problem, let the wellheads serve as the solution points so that the pressure of each node i can be calculated using two pressure-transverse calculations (Kumar 1987). If the pressure of wellhead i calculated from the reservoir side and from the gathering center are denoted as pir and pis , respectively, the system of equations for rate determination is pir = preservoir − ⌬pinflow − ⌬ptubing pis = pseparator + ⌬ppipeline + ⌬pchoke . . . . . . . . . . . . . . . . . . . . . . (11) In this synthetic example, a generic design of the production system was used in situations where the reservoir conditions for the wells, such as the reservoir static pressure and block permeability, are fixed. In the process of calculating the pressure transverse, the reservoir component refers to wellbore forms the boundary conditions for the gathering system. The pressure drop in the tubings and surface pipeline networks are evaluated using the Beggs and Brill (Kumar 1987) correlation from upstream and downstream, respectively. Short pipe sections onboard in the surface networks were excluded from the network simulation because the pressure drops through these pipes are very small and have little effect on the overall behavior of the production system. Sachdeva et al.’s model from Kumar (1987) was used for the calculation of the pressure drop through the chokes. On the basis of the calculation of the pressure transverse, this problem adopts the deliverability constraint suggested by Wang et al. (2002): pir − pis ≥ 0

i = 1,..., 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . (12) June 2010 SPE Journal

120

Objective Function

110 100 90 80 Average Best

70 60 0

10

20

30

40

50

60

70

80

90

100

Generation Number Fig. 3—Convergence of objective function by coevolutionary genetic algorithm.

The deliverability constraint ensures that the optimal rates identified for the problem lie within the deliverability capacity of the gathering system. According to the production rates and the status of deliverability constraint, the well status, which should be fully closed, be partially closed, or remain fully open, can be determined (Wang et al. 2002). To formulate the optimization problem in the fuzzy domain, an allowable deviation of 3 and 10% for each constraint is considered using a linear membership function. The decision variables are the gas-flow rates of each well. Fig. 3 shows the convergence performance obtained by using the proposed hybrid coevolutionary/genetic algorithm. Starting with the initial reference population obtained by SQP, the proposed approach significantly reduces the computational effort needed to generate fully feasible initial reference points. However, the

initial objective function value, 67.47 MMscf/D estimated by SQP, represents a small local optimum that is a relatively inferior value compared to the global optimum, 112.81 MMscf/D, because the search strategy is not extensive enough. Furthermore, the surface of the objective function is extremely complicated because some flow components in the system include discontinuities and flow interactions or a strong dependence between the decision variables. The coevolutionary/genetic algorithm is used mainly to find a small feasible region surrounding the global optimum point. The applied optimization algorithm reached the maximum possible solution in, on average, fewer than 60 generations without finding a local optimum solution. The proposed hybrid optimization approach efficiently embodies the overall framework of genetic operations and guarantees with higher degree of success the global optimum compared with the conventional derivative-based algorithm. Table 3 summarizes a comparison of the optimal solution obtained by the crisp and fuzzy optimization methods and presents the allocated production rates of each well with a fixed sink pressure. As can be seen from the solution, the fuzzy optimization approach provides an additional total production rate of 4.23 and 8.25 MMscf/D, which represent a 3.7 and 7.3% increase of total production rate, respectively, over the crisp optimization method. Conventional crisp optimization methods assume that constraints delimit as a crisp set of feasible decisions. This implies that, in a surface-pipeline-network system, for example, a wellhead pressure of 1,000 psi is acceptable while 1,001 psi is not. This is contrary to the human expert’s perception developed from many years of work experience. Fuzzy programming considers random parameters as fuzzy numbers, and constraints are treated as fuzzy sets. Some constraint violation is allowed, and the degree of satisfaction of a constraint is defined as the membership function by considering different degrees of weight or importance to violations of the different constraints. The proposed approach can achieve more reliable and practical solution to the problem by accommodating uncertainties related to the imprecise description of information collected in the petroleum-field-optimization process.

TABLE 3—ALLOCATED PRODUCTION RATES OF CRISP AND FUZZY OPTIMIZATION METHODS WITH A FIXED SINK PRESSURE OF 2,300 psi

Well Number

June 2010 SPE Journal

Crisp Result (Mscf/D)

Fuzzy Result With an Allowable Deviation for Each Constraints (Mscf/D) Deviation of 3%

Deviation of 10%

1

6,477

6,752

6,897

2

7,644

7,919

8,115

3

6,445

6,096

6,394

4

8,140

7,822

8,140

5

6,017

5,606

5,925

6

4,855

6,599

6,663

7

7,857

9,595

10,205

8

10,815

12,445

13,640

9

17,318

15,112

14,891

10

4,838

5,459

5,499

11

3,924

4,521

4,508

12

6,524

6,910

7,185

13

2,588

3,435

3,537

14

2,214

2,826

2,966

15

2,218

2,844

2,978

16

3,830

4,499

4,925

17

2,452

2,562

2,641

18

1,836

1,928

2,108

19

3,805

2,481

2,323

20

3,010

1,623

1,672

Total

112,807

117,035

121,064

421

Membership Degree

TABLE 4—INTERACTIVE PROCESSES OF MULTIOBJECTIVE PROBLEM

1

f1 (USD) f2 (MMscf/D)

0

Membership Degree

(a)

1,594,129

3,858,636

Pipeline Cost, USD

1

0 (b)

Interaction

112

380

Production Rate, MMscf/D

Fig. 4—Fuzzy objective function (a) pipeline cost, (b) total production rate.

Example 2. The multiobjective problem described in this example consists of two objective functions: minimizing the investment cost of pipeline networks ( f1) and maximizing the total gas-production rate or delivering the contract requirement over the period of the gas contract ( f2). The management options for achieving these objectives consist of optimal well-rate allocation and appropriate selection of the pipeline-segment diameters. The decision variables are the gas-flow rates of each well and the pipeline diameter of the surface production network. The constraints imposed on the system include the design ranges of the pipeline diameter, which are 4–12, 5–12, and 10–24 in. for P1, P2–P9, and P10, respectively. It is assumed that the construction cost of the pipeline is USD 4,000/in.-mile, regardless of the pipeline route. The membership

1

2

3

1

0.8

1

1

1

0.9

2,431,350

2,560,880

2,371,833

281.37

319.63

262.07

0.6303

0.5731

0.6566

0.6305

0.7731

0.5585

function of the objective is quantified by making use of the experienced designer’s technical knowledge and expectations and the data collected from domain experts. After calculating the individual minimum and maximum of the objective functions, the membership functions of the two objective functions are represented by the linear relationship shown in Fig. 4. To elicit a linear membership function of the pipeline cost, the minimum value of unacceptable levels and maximum value of totally desirable levels are accessed. In Fig. 4, the minimum level of the pipeline cost, USD 1,594,129, corresponds to the selection of all the lower bounds of the pipeline diameter and vice versa for the maximum level. Then, by taking account of the calculated minimum and maximum with respect to the total production rate, the membership function of the production rate is determined. Table 4 represents the interactive processes used for obtaining a Pareto optimal solution for maximizing the total gas-production rates and minimizing the investment cost of the pipeline network. The reference membership values of ( ␮1 , ␮2 ) from (1.0, 1.0) to (0.8, 1.0), and (1.0, 0.9) are applied sequentially. In the whole interaction process, the corresponding augmented minimum/ maximum problem is solved for the initial reference membership levels, and Tables 4 and 5 present the resulting optimal solutions and current membership values of the objective functions. If the decision maker is not satisfied with Interaction 1, Interactions 2 and 3 can be performed to improve the satisfaction levels for ␮1 and ␮2. For weighting ␮2 in Interaction 2, the construction cost and total production rate are increased by USD 2,560,880 and 319.63 MMscf/D, while, for weighting ␮1 in Interaction 3, they are decreased by USD 2,371,833 and 262.07 MMscf/D, respectively. The decision maker can efficiently obtain a satisfactory solution in this manner by updating the reference membership values. For the contract requirements problem, Fig. 5 shows the objective function of the gas contract rate. Table 6 lists the resulting optimal solution for the reference membership values of (1.0, 1.0). The optimized cost and production rate are USD 2,246,528 and 239.51 MMscf/D, respectively. The proposed approach applies effectively to various multiple decision problems.

TABLE 5—OPTIMIZATION RESULTS OF MULTIOBJECTIVE PROBLEM Interaction 1

422

Interaction 2

Interaction 3

Pipe Number

Rate

Number

Rate

Number

Rate

Number

P1

28.93

5.76

26.17

7.59

21.39

4.72

P2

72.72

8.19

79.16

8.71

66.35

8.78 8.64

P3

38.15

6.06

52.29

9.92

49.54

P4

153.92

11.24

169.08

10.40

128.20

8.51

P5

103.56

11.16

120.86

11.90

78.74

7.81

P6

67.38

8.43

73.06

8.99

48.59

9.30

P7

8.96

6.05

10.87

5.53

4.26

9.20

P8

6.70

7.25

11.61

5.73

11.52

8.61

P9

36.18

6.87

47.80

8.01

30.15

8.22

P10

281.37

14.84

319.63

15.59

262.07

14.19

June 2010 SPE Journal

TABLE 6—OPTIMIZATION RESULTS OF CONTRACT-REQUIREMENTS PROBLEM

Membership Degree

Interaction

Result

Pipe Number

Flow Rate (MMscf/D)

Diameter (in.)

1

P1

27.49

6.24

1

P2

60.50

7.11

f1 (USD)

2,246,528

P3

35.56

7.58

f2 (MMscf/D)

239.51

P4

128.98

10.14

0.7119

P5

91.05

7.67

0.9024

P6

37.03

5.95

P7

20.63

6.43

P8

32.07

7.09

P9

54.02

8.02

P10

239.51

13.72

1

0

235

240 245

Production Rate, MMscf/D Fig. 5—Fuzzy objective function for contract requirements problem.

Conclusions This study proposes a fuzzy nonlinear programming approach combined with a hybrid coevolutionary/genetic algorithm for the optimal design of gas-production systems. The reliability of the synthetic optimization method is validated by practical operation problems including well-rate allocation, the optimal design of pipeline networks, and multiple decision problems. Case studies demonstrated that more flexible and practical results were obtained with the proposed method by quantitatively accommodating uncertainties related to the field operation than with conventional methods. The proposed approach can provide an important element for future planning and optimization of the production operations of gas fields. References Abdulraheem, A., Sabakhi, E., Ahmed, M., Vantala, A., Raharja, I., and Korvin, G. 2007. Estimation of Permeability from Wireline Logs in a Middle Eastern Carbonate Reservoir Using Fuzzy Logic. Paper SPE 105350 presented at the SPE Middle East Oil and Gas Show and Conference, Bahrain, 11–14 March. doi: 10.2118/105350-MS. Agbon, I.S. and Araque, J.C. 2003. Predicting Oil and Gas Spot Prices Using Chaos Time Series Analysis and Fuzzy Neural Network Model. Paper SPE 82014 presented at the SPE Hydrocarbon Economics and Evaluation Symposium, Dallas, 5–8 April. doi: 10.2118/82014-MS. Alimonti, C. and Falcone, G. 2004. Integration of Multiphase Flowmetering, Neural Networks, and Fuzzy Logic in Field Performance Monitoring. SPE Prod & Fac 19 (1): 25–32. SPE-87629-PA. doi: 10.2118/87629-PA. Cao, D., Ni, Y., Yao, F., Weng, D., and Fu, G. 2006. Application and Realization of Fuzzy Method for Selecting Wells and Formations in Fracturing in Putaohua Oilfield: Production and Operations: DiagnosJune 2010 SPE Journal

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Appendix A—Fuzzy Objective Function In most fuzzy nonlinear programming approaches, the decision maker has two types of fuzzy goals (Zimmermann 1991; Sakawa 2002). In a minimum/maximum problem, the fuzzy goal (objective function) fi can be stated as achieving “substantially less than or equal to some value pi or greater than or equal to qi” (fuzzy minimium or fuzzy maximum). This type of statement can be quantified by eliciting a corresponding membership function ␮i[fi(x)] that is a strictly monotonically decreasing or increasing function with respect to objective function fi(x). For fuzzy goals expressed as “fi(x) should be in the vicinity of arbitrary value ri” (fuzzy equal), different functions for the left and right sides of ri can be used. When the fuzzy equal is included, it is desirable that fi(x) be as close to ri as possible. For the previously mentioned fuzzy ␭-formulation problem, the membership function of the objective ␮G ( x ) and constraints ␮ gi ( x ) can be defined as follows. Let R = fuzzy feasible region, R1 = ␣-level cut of R for ␣ = 1 and S R = support of R. The (crisp) set of elements that belong to the fuzzy set R, at least to the degree ␣ is called the ␣–level set (cut),

( )

R1 = { x | ␮ R ( x ) ≥ ␣} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A-1)

( )

In particular, the support of a fuzzy set R, S R is the crisp set of all elements whose degree of membership is larger than zero. The membership function of the objective function, given the solution space R using the linear variation, is then defined as ⎧0 ⎪ ⎪ f ⎪⎪ f ( x ) − sup R1 ␮G ( x ) = ⎨ f − sup f ⎪ sup R1 S R ⎪ ( ) ⎪1 ⎪⎩

⎫ ⎪ ⎪ ⎪⎪ if sup < f ( x ) ≤ sup f ⎬ . . . . . . . (A-2) R1 S( R) ⎪ ⎪ if sup f ≤ f ( x ) ⎪ S( R) ⎪⎭ if f ( x ) ≤ sup f R1

The sup f and sup f refer to the supremum (optimum) of objective ( )

R1

S R

function f values, which correspond to crisp and fuzzy optimization. The values of sup f and sup f are obtained by optimization in the R1

S ( R)

crisp and fuzzy feasible domains, respectively. For inequality constraints gi(x) ≤ bi, let the membership functions of the fuzzy sets be defined using the linear type of function over the allowable fuzzy transition intervals di, that is, if gi ( x ) > bi + di ⎧0 ⎫ ⎪ ⎪ ⎪ b + di − gi ( x ) ⎪ ␮ gi ( x ) = ⎨ i if bi < gi ( x ) < bi + di ⎬ . . . . . . . (A-3) d i ⎪ ⎪ ⎪⎩1 ⎪⎭ if gi ( x ) ≤ bi

June 2010 SPE Journal

Appendix B—Genetic Operators In the floating-point representation, each individual is coded as a vector of floating-point numbers of the same length as the solution vector, and each element is forced to be within the feasible region (Sakawa 2002). Selection Operator. As a selection operator, ranking selection is used, where the population is sorted from the best to the worst and the selection probability of each individual is assigned according to the ranking. Among the various linear and nonlinear ranking methods, the exponential ranking method is adopted. The selection probability pi for the individual of rank i is determined by pi = c ⋅ (1 − c )

i −1

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (B-1)

where c in the range of 0–1 represents the probability when an individual of rank 1 is selected. Observe that a larger value of c implies a stronger selective pressure. Crossover and Mutation Operators. The crossover and mutation operators used are closed operators in the sense that the resulting offsprings always satisfy the linear constraints S. For example, the arithmetic crossover for two points x , y ∈ S yields ␣ x + (1 − ␣ ) y ∈ S , 0 ≤ ␣ ≤ 1, and the resulting offspring always belongs to S when S is a convex set. As genetic operators for crossover and mutation, arithmetic crossover and uniform mutation are adopted, respectively. Hui-June Park is a reservoir engineer with the Korea National Oil Corporation in Anyang, Korea. email: [email protected]. His current interest is in production optimization in mature gas fields. Park holds a BS degree in mineral and petroleum engineering and MS and PhD degrees in civil, urban, and geosystem engineering, all from Seoul National University. Jong-Se Lim is a professor in the Department of Energy & Resources Engineering at Korea Maritime University. email: [email protected]. His research interests include production optimization and reservoir characterization using intelligent techniques. Lim holds BS, MS, and PhD degrees in mineral and petroleum engineering from Seoul National University. Jeongyong Roh is a reservoir engineer with the Korea National Oil Corporation in Anyang, Korea. email: [email protected]. His current interest is well testing. Roh holds BS and MS degrees in mineral and petroleum engineering and a PhD degree in civil, urban, and geosystem engineering, all from Seoul National University. Joo M. Kang is a professor in the Division of Energy System Engineering at Seoul National University. email: [email protected]. His research interests include well testing, reservoir characterization, and gas-production engineering. Kang holds BS and MS degrees in petroleum and mining engineering from Seoul National University and a PhD degree in petroleum and geological engineering from the University of Oklahoma. Bae-Hyun Min is a graduate student in the Division of Energy System Engineering at Seoul National University. email: [email protected]. His research interest includes well-placement optimization using artificial intelligence. Min holds BS and MS degrees in civil, urban, and geosystem engineering from Seoul National University.

425