Abstract

In this paper, we discuss a practical oil production planning optimization problem. For oil wells with insufficient reservoir pressure, gas is usually injected to artificially lift oil, a practice commonly referred to as enhanced oil recovery (EOR). The total gas that can be used for oil extraction is constrained by daily availability limits. The oil extracted from each well is known to be a nonlinear function of the gas injected into the well and varies between wells. The problem is to identify the optimal amount of gas that needs to be injected into each well to maximize the amount of oil extracted subject to the constraint on the total daily gas availability. The problem has long been of practical interest to all major oil exploration companies as it has the potential to derive large financial benefit. In this paper, an infeasibility driven evolutionary algorithm is used to solve a 56 well reservoir problem which demonstrates its efficiency in solving constrained optimization problems. Furthermore, a multi-objective formulation of the problem is posed and solved using a number of algorithms, which eliminates the need for solving the (single objective) problem on a regular basis. Lastly, a modified single objective formulation of the problem is also proposed, which aims to maximize the profit instead of the quantity of oil. It is shown that even with a lesser amount of oil extracted, more economic benefits can be achieved through the modified formulation.

1.  Introduction

Petroleum, either oil or gas, is a finite and scarce resource upon which modern society heavily depends. Hence, mankind is forced to optimize oil production and consumption. In this paper, we consider a crude oil production system. In the system, there is an underground oil reservoir and the reservoir has a number of wells. There are two basic methods of extracting oil from such reservoirs (Kosmidis et al., 2005): (i) naturally flowing and (ii) gas lift. In the first method, the oil flows naturally to the surface, while in the second method, high pressure gas (e.g., CO2) is injected into the well to facilitate oil extraction. Gas lift is considered to be the most economical method for artificial lifting of oil (Ayatollahi et al., 2004; Camponogara and Nakashima, 2006).

In this study, we consider the gas lift extraction method. As will be discussed later, for a given well, the oil production per day can be expressed as a nonlinear function of the amount of gas injected into the well per day. The oil production per day increases with the increase of gas used to a certain level and then decreases. This means an excess of gas may increase the gas cost, as well as the production cost, without providing any benefit in terms of oil production volume. For a given amount of gas used, the amount of oil extraction significantly varies from well to well. As a result, an inappropriate gas allocation to different wells, under limited gas availability, will reduce the overall production and hence profitability from the entire reservoir. So the single objective gas lift optimization problem is to allocate a limited amount of gas to a number of wells in a reservoir while maximizing the total oil production per day. However, the amount of gas may vary from day to day. This requires the management to solve the problem again on a daily basis, depending upon the amount of available gas for the particular day.

While single objective formulation has so far been the usual practice adopted in the industry, a bi-objective formulation can provide solutions for a wide range of gas availability values in a single run, hence eliminating the need to solve the problem on a daily basis. The bi-objective formulation is presented as:

1. Maximize oil production.

2. Minimize gas used.

Prior research in gas lift optimization focused on a single objective optimization problem using either a single well model (Fang and Lo, 1996) or a multiple well model (Dutta-Roy and Kattapuram, 1997). A range of methodologies has been used, including the equal slope method (Nishikiori et al., 1989), linear programming (Fang and Lo, 1996), mixed integer linear programming (Kosmidis et al., 2005), quadratic programming (Dutta-Roy and Kattapuram, 1997), dynamic programming (Camponogara and Nakashima, 2006), and others. In this research, we consider a 56 well problem and illustrate the benefits of solving single and bi-objective formulations of the problem using the infeasibility driven evolutionary algorithm (IDEA). Evolutionary algorithms (EAs) have been used to solve a number of multi-objective optimization problems from the domain of operations research in recent years (Sarker et al., 2002). An excellent comprehensive review of evolutionary multi-objective optimization appears in Coello Coello (1999).

In this paper, the oil production planning problem is modeled as both a single and a bi-objective integer linear programming problem, where the nonlinear oil production functions are transformed into piecewise linear functions. Considering the complexity, the time required, and the need to solve both single and multi-objective formulations of the problem, we have introduced an IDEA to solve the problem. We propose a chromosome representation that significantly reduces the number of variables and constraints required as opposed to the representation required in solving the problem using conventional approaches. Such a representation clearly justifies the use of EAs for such problems. A set of representative data was initially received from Shell system personnel for problem formulation and analysis. The single objective results were compared with the best known results reported by Buitrago et al. (1996; see Section 5.1 for details). Past study (Ray and Sarker, 2007) indicated that NSGA-II is able to identify reasonably good solutions for this problem. In this study, we present the results of IDEA and compare it with those obtained using NSGA-II and more recent algorithms (MOEA/D and the Liu Li algorithm) for multi-objective formulations of the problem. Furthermore, a more practical formulation considering profit maximization instead of maximization of oil extraction is presented and analyzed.

This paper is organized as follows. Following this introduction, we present a mathematical model of the problem in Section 2. The details of the solution approach using the IDEA is given in Sections 3 and 4, followed by detailed numerical experiments in Section 5. In Section 6, an alternative formulation of the problem is proposed, which maximizes economic benefits instead of the oil extracted. Finally, Section 7 provides concluding remarks.

2.  Mathematical Model

As discussed earlier, oil production per day from a given well is a function of gas injected into the well on that particular day (Buitrago et al., 1996). However, there is no standard function which can be used to determine the production level for all wells. The practice is to collect production data at a number of discrete points of gas use for each well. Using the discrete data points, researchers in the field generate the corresponding function either as a piecewise linear function (Camponogara and Nakashima, 2006) or as a quadratic function. Although both functions have drawbacks in estimating the production level accurately, we use a piecewise linear function as this is widely used in the industry. The datasets for a few wells are shown in Figure 1. As is clear from the figure, the dataset for different wells can have different characteristics : monotonically increasing (a, b), with a peak (c, d), with one or more flat regions (e,f) or with minimum gas threshold (g, h).

Figure 1:

Fifty-six oil well problem: Left column shows oil extraction versus gas injection data for the oil well. Right column shows the linear interpolation of data.

Figure 1:

Fifty-six oil well problem: Left column shows oil extraction versus gas injection data for the oil well. Right column shows the linear interpolation of data.

The mathematical model of the problem is formulated as follows.

2.0.1.  Parameters

Five parameters have to be specified in order to set up the problem, which are:

• N: The number of wells.

• In: The number of line segments (gas used, x axis) in the function for well n (as shown in Figure 1(b); joining any two points is considered to be a line segment).

• Gni: The slope of the function (oil produced per unit of gas used) at the line segment i in well n.

• GL: The limit of gas usage in all wells per day.

• Uni: The upper limit of gas usage rate at the individual line segment i in the well n.

2.0.2.  Variables

The problem contains the following variables:

• Yn: The amount of gas used in well n.

• xni: The gas usage in segment i in well n.

• Sni: 0 if xni<Uni, 1 if xni=Uni.

The relationship between Yn and xni is , where

In addition, for a given value of Yn (which is in segment i in well n), all xn(i+1) will be equal to zero and all xn(i-1) will be at the upper bound. However, the value of xni will be greater than zero and less than or equal to the upper bound.

2.1.  Single Objective Model

We first present a single objective model for the above problem. The objective of the problem is to maximize the total daily production. The variable of the optimization problem, that is, xni, represents the gas used in segment i of oil well n. The value of Gni, that is, the slope of the oil-extraction–gas-injection curve is obtained from the given data.

2.1.1.  Constraints

The following constraints must be satisfied for the problem.

2.1.1.1 Constraint 1

Calculating the gas usage level in each well:

2.1.1.2 Constraint 2

To make sure that constraint 1 is meaningful for a piecewise linear curve, the following condition must hold:
Here, Sn0=1 and .

2.1.1.3 Constraint 3

Gas limitation: The total use of gas in all wells in a given day must not be greater than the available amount of gas:

2.1.1.4 Constraint 4

Non-negativity constraints:

2.2.  Multi-Objective Model

In reality, we want to maximize oil production using the minimum possible gas. So the problem can be defined as a bi-objective problem where the objectives are:

1. Maximization of oil production.

2. Minimization of gas use.

The corresponding objective functions are as follows:
The constraints are the same as those for the single objective model.

3.  Solution Approach

To transform the nonlinear function into piecewise linear functions, as shown in the mathematical model, for each well n, we need (In–1) binary variables and In constraints. In addition, we would require real variables for xni and N real variables for Yn. If we assume that every well consists of In line segments, then constraint 2 will generate N × (In–1) binary variables and constraints. So the mathematical model would contain (2) variables and (N × (In+1)+1) constraints. For any feasible solution, one needs to satisfy (N × (In+1)+1) constraints. As an example, for In = 10 and N = 56, the mathematical model would contain 1,120 variables (of which 504 are binary variables) and 617 constraints.

In solving the above problem using an EA, we need to design a chromosome representing the necessary variables. In conventional chromosome design, the chromosome representation usually contains all (2 × ) variables as shown in Table 1. Our proposed alternative chromosome representation (as shown in Table 2) would consist of only N real variables Yn, each of which would denote the amount of gas injected into the well. Because of this alternative chromosome representation, as discussed earlier, only a single constraint is required, arising out of the total gas availability per day.

Table 1:
Conventional Representation
Number12
Variables x11 x12  Y1  YN
Number12
Variables x11 x12  Y1  YN
Table 2:
Alternative Representation Used for IDEA
Number12N
Variables Y1 Y2   YN
Number12N
Variables Y1 Y2   YN

4.  Infeasibility Driven Evolutionary Algorithm (IDEA)

In this section, we discuss our algorithm for solving the oil production planning problem. The algorithm has been proposed and studied on a number of benchmark constrained optimization problems earlier (Singh et al., 2008; Ray et al., 2009). For convenience of explanation, we start with a general formulation of a multi-objective optimization problem as shown in Equation (1).
1
where is the design variable vector bounded by lower and upper bounds . A single objective optimization problem follows the same formulation with k=1.
The optimal solutions of the constrained optimization problems often lie along the constraint boundary, a fact also previously emphasized in the literature (Hedar and Fukushima, 2006; Koziel and Michalewicz, 1999). To effectively search along the constraint boundary, the original k objective constrained optimization problem is reformulated as a k+1 objective unconstrained optimization problem as given in Equation (2). The first k objectives are the same as in the original constrained problem. The additional objective represents a measure of constraint violation, which is referred to as the violation measure.
2
The main steps of IDEA are outlined in Algorithm 1. As in NSGA-II, an offspring population is evolved from parents selected by binary tournament using crossover and mutation. For crossover, IDEA uses simulated binary crossover (SBX, Deb and Agrawal, 1995) operator, as given in Equation (3).
3
where is calculated as
4
and where ui is a uniform random number in the range [0, 1) and is the user defined parameter.

4.0.1.  Distribution Index for Crossover

The subscript i in the parent(x) and child(y) solutions refer to the ith decision variable.

For mutation, a polynomial mutation Deb and Goyal (1996) operator, as defined in Equation (5), is used.
5
where is calculated as
6
where ri is the uniform random number in the range [0, 1) and is the user defined parameter.

4.0.2.  Distribution Index for Mutation

For preserving diversity, IDEA uses crowding distance sorting (see Algorithm 2), which ranks the solution based on their distance from the neighboring solutions (with the same nondominated ranks) in the front.

The main difference between NSGA-II and the IDEA is the mechanism for elite preservation. In IDEA, a few infeasible solutions are retained in the population at every generation. Individual solutions in the population are evaluated as per the original problem definition (Equation (1)) and marked infeasible if any of the constraints are violated. The solutions of the parent and the offspring population are divided into a feasible set (Sf) and an infeasible set (Sinf). The solutions in the feasible and the infeasible sets are both ranked using nondominated sorting and crowding distance sorting of k+1 objectives. NSGA-II, on the other hand, uses nondominated sorting and crowding distance for ranking feasible solutions and ranks infeasible solutions in the increasing value of maximum constraint violation. For the feasible solutions, nondominated sorting using k+1 objectives is equivalent to the nondominated sorting the original k objectives, as the additional objective value (which is based on the constraint violations) for feasible solutions is always 0.

The next step is to choose the solutions that form the population for the next generation. In IDEA, a user-defined parameter is used to identify the proportion of the infeasible solutions to be retained in the population. The numbers Nf () and Ninf () denote the number of feasible and infeasible solutions in the population respectively, where Np is the population size. If the infeasible set Sinf has more than Ninf solutions, then first Ninf solutions are selected based on the rank; otherwise all the solutions from Sinf are selected. The rest of the solutions are selected from the feasible set Sf, provided there are at least Nf number of feasible solutions. If Sf has fewer solutions, all the feasible solutions are selected and the rest are filled with infeasible solutions from Sinf. The solutions are ranked from 1 to Np in the order they are selected. Hence, the infeasible solutions that get selected first (at most Ninf), get higher rank than the feasible solutions.

As an example, assuming a population size of 100, during any given generation, 100 child solutions will be created. In the pool of 200 (parent + child) solutions, if there are 40 infeasible and 160 feasible solutions, then NSGA-II will select the best 100 feasible solutions for the next generation, hence preferring all feasible solutions over all infeasible solutions. On the other hand, assuming we use for IDEA, it would select 20 best infeasible solutions (based on nondominated + crowding distance sorting of k+1 objectives), and 80 best feasible solutions. Hence, good infeasible solutions are preferred over feasible solutions during the course of evolution.

In NSGA-II, the elite preservation mechanism weeds out the infeasible solutions from the population. To retain the infeasible solutions in the population, an alternate mechanism is required. In IDEA, the infeasible solutions are ranked higher than the feasible solutions, thus adding selection pressure to generate better infeasible solutions. The presence of infeasible solutions with higher ranks than the feasible solutions translates into an active search through the infeasible space. This feature of IDEA accelerates the movement of solutions toward the constraint boundary. With the modified problem definition and ranking of the infeasible solutions higher than the feasible solutions, IDEA can find the solutions to the original problem more efficiently.

4.1.  Constraint Violation Measure

The additional objective in the modified problem formulation is based on the amount of relative constraint violation among the population members. The constraint violation measure of a solution is based on the constraint violation levels for all constraint values for that solution. Consider one of the constraints (gi). All the solutions in the population are sorted in ascending order based on the value of the constraint violation for gi. The solutions that do not violate the constraint gi are assigned constraint violation value of 0 (and gi does not contribute to the violation measure of those solutions). The rest of the solutions are assigned a constraint violation level for constraint gi based on the sorted list, starting with rank 1 for the solution with the least constraint violation. Solutions with the same value of constraint violation get the same rank. This ranking procedure is repeated for all the constraints. The constraint violation measure for each solution is then calculated as the sum of the ranks (based on the constraint violation level) obtained for all the constraints. The calculation of the violation measure is illustrated in Table 3 for a sample population of 10 solutions, for a problem with three constraints.

Table 3:
Calculation of Constraint Violation Measure
ViolationsRelative ranks
SolutionC1C2C3C1C2C3Violation measure
3.50 90.60 8.09 18
5.76 7.80 6.70 15
0.00 3.40 7.10 10
1.25 0.00 0.69
13.75 90.10 5.87 17
100.70 2.34 3.20 12
0.00 5.09 4.76
1.90 0.00 0.00
0.00 0.56 0.00
10 8.90 2.30 9.80 15
ViolationsRelative ranks
SolutionC1C2C3C1C2C3Violation measure
3.50 90.60 8.09 18
5.76 7.80 6.70 15
0.00 3.40 7.10 10
1.25 0.00 0.69
13.75 90.10 5.87 17
100.70 2.34 3.20 12
0.00 5.09 4.76
1.90 0.00 0.00
0.00 0.56 0.00
10 8.90 2.30 9.80 15

The violation measure favors solutions with good ranks for most (or all) of the constraints. As a result, a solution with a large violation in only one of the constraints has roughly the same preference as a solution with marginal violations of multiple constraints. The violation measure is used as the additional objective in IDEA to rank the infeasible solutions using nondominated sorting. As a result, the final population consists of the solutions with marginal constraint violations.

A visualization of how the IDEA expedites the search by searching through both feasible and infeasible solutions is given in Figure 2, based on studies presented in Ray et al. (2009). Figure 2(a) shows the constraints and the feasible space for the two-variable constrained problem g06 (Koziel and Michalewicz, 1999). Figure 2(b) shows final populations obtained using IDEA and NSGA-II across 30 runs on the problem. It can be seen from the distribution of the solutions that NSGA-II tries to approach the optimum solution through the feasible regions only. On the other hand, IDEA solutions are observed to approach toward the optimum solutions from various directions through infeasible regions. The improvement in performance can be gauged from the fact that the feasible solutions obtained using IDEA are all concentrated very close to the optimum, unlike NSGA-II solutions.

Figure 2:

(a) Constraints for the g06 problem, where the optimum solution is (14.095, 0.84296). (b) Final solutions obtained for g06 (across multiple experiments).

Figure 2:

(a) Constraints for the g06 problem, where the optimum solution is (14.095, 0.84296). (b) Final solutions obtained for g06 (across multiple experiments).

5.  Numerical Experiments

5.1.  Single Objective Formulation

For the single objective formulation of the problem, 30 independent runs of NSGA-II and IDEA are done. The crossover and mutation parameters used for both the algorithms are the same, and are listed in Table 4.

Table 4:
Parameters Used for IDEA and NSGA-II for Studies on Oil Well Problem
ParameterValue
Population size 100
Maximum function evaluations 10,000
Crossover probability 0.9
Mutation probability 0.1
Crossover distribution index 10
Mutation distribution index 20
Infeasibility ratio (for IDEA) 0.05
ParameterValue
Population size 100
Maximum function evaluations 10,000
Crossover probability 0.9
Mutation probability 0.1
Crossover distribution index 10
Mutation distribution index 20
Infeasibility ratio (for IDEA) 0.05

The summary of results obtained using both these algorithms is shown in Table 5. It is seen that since the optimum value is bounded by the constraint (on the amount of available gas), IDEA is able to achieve better solutions than NSGA-II by preserving some infeasible solutions during the search. The presence of these infeasible solutions results in a focused search near the constraint boundary. The median runs of both the algorithms are plotted in Figure 3, which shows a faster rate of convergence using IDEA as compared to NSGA-II.

Table 5:
Performance of NSGA-II and IDEA on Single Objective 56 Oil Well Problem (Superior Objective Values Shown in Bold)
NSGA-IIIDEA
Best 22326.0 22348.7
Median 21947.6 22094.8
Worst 21673.6 21816.3
NSGA-IIIDEA
Best 22326.0 22348.7
Median 21947.6 22094.8
Worst 21673.6 21816.3
Figure 3:

Comparison of convergence of NSGA-II and IDEA algorithms for the median run.

Figure 3:

Comparison of convergence of NSGA-II and IDEA algorithms for the median run.

To test the statistical significance of the results, the nonparametric Mann-Whitney U test was performed on the solutions obtained using NSGA-II and IDEA for 30 runs. The U and z values obtained for the test are 160 and 4.28, respectively, which indicates the probability of p< .0001 for the directional test, indicating that objective values obtained using IDEA are statistically lower than those obtained using NSGA-II.

It is clear that the best result obtained using IDEA, or 22,348.7 barrel per day (BPD), is better than the best result obtained from NSGA-II, 22,326.0 BPD, and significantly better than the result obtained by Buitrago et al. (1996), 21,789.9 BPD, and Ray and Sarker (2007), 22,033.4 BPD. The results indicate that IDEA is able to identify a solution with 315.3 BPD more than the previously known best solution to the problem (by Ray and Sarker, 2007). Needless to mention, such a saving translates to $8.6 million per year assuming a crude oil price of$75 per barrel.

The gas injection values (solution vector) corresponding to best solution using IDEA are 524.533, 510.114, 488.864, 12.1068, 66.1525, 188.099, 246.647, 286.949, 1295.88, 33.759, 956.782, 1450.4, 97.9226, 217.637, 643.664, 510.579, 645.138, 277.762, 219.168, 983.439, 1,016.48, 404.29, 23.1851, 1,091.18, 360.634, 89.1857, 21.0864, 4.74104, 34.3172, 63.0425, 1.61972, 248.457, 509.636, 116.482, 264.189, 116.106, 3.33286, 286.71, 341.892, 101.805, 58.9932, 8.39999, 1,256.69, 21.3252, 7.90973, 313.914, 26.4506, 2,448.83, 1,507.3, 33.4525, 2.9478, 20.3695, 46.2917, 0.591372, 11.9429, and 1799.98 with an oil extraction of 22,348.70 BPD.

5.2.  Multi-Objective Formulation

The multi-objective formulation of the problem is solved using four different MOEA algorithms, namely NSGA-II (Deb et al., 2002), IDEA (Singh et al., 2008; Ray et al., 2009) MOEA/D (Zhang and Li, 2007), and the Liu Li algorithm (Liu and Li, 2009).

As mentioned before, NSGA-II employs nondominated sorting of the solutions in the population for convergence, and maintains diversity by a secondary ranking process called crowding distance. IDEA utilizes marginally infeasible solutions in addition to nondominated sorting to expedite the convergence. MOEA/D, on the other hand, is an algorithm based on decomposition. It decomposes a multi-objective problem into a number of scalar optimization subproblems and optimizes them simultaneously. The Liu Li algorithm operates by dividing the decision space into smaller regions and determined weights. The Liu Li algorithm was recently judged among the top three algorithms in the special session on MOEA competition (for constrained problems) in the 2009 IEEE Congress on Evolutionary Computation (Zhang and Suganthan, 2009).

The crossover and mutation parameters for NSGA-II and IDEA are kept the same as that used for experiments on the single objective problem.1 Also, since MOEA/D is an algorithm designed to solve unconstrained multi-objective optimization problems, MOEA/D is run for the unconstrained oil well problem disregarding the constraint on the amount of available gas. Thereafter, from the final population obtained using MOEA/D, the solutions that violate the constraint are omitted while comparing the performance with other algorithms.2 The parameters used for the Liu Li algorithm are determined using the guidelines given in the earlier studies (Liu and Li, 2009). A population size of 100 has been used for all algorithms, and the maximum number of function evaluations is set to 10,000.

Two performance measure metrics were used to compare the performance of the algorithms. These are displacement, as introduced by Bandyopadhyay et al. (2008), and hypervolume, from Zitzler and Thiele (1998). The details of the metrics have been omitted here for brevity. A lower value of displacement, and a higher value of hypervolume, indicates a better approximation of the front. (For more details of the metrics, readers are referred to Bandyopadhyay et al., 2008 and Zitzler and Thiele, 1998.) For calculating the displacement, the reference Pareto front has been constructed by accumulating solutions obtained using each algorithm for all the runs, and then doing a nondominated sorting on this collection of solutions.3

Unlike the single objective formulation, the optimum solution (Pareto front) for the multi-objective problem does not lie on the constraint boundary, as in this case the constraint is only the upper bound on one of the objectives. Hence, in this case, IDEA is not expected to give significant benefit over NSGA-II in terms of convergence. However, it increases the probability of getting solutions near the constraint (maximum bound on gas injection). In terms of overall convergence measures, the performance of IDEA for the oil well problem is actually found to be worse than NSGA-II, because when using the same population size, less feasible solutions are available to cover the Pareto front.

Since the population of IDEA primarily focuses on solutions on the constraint boundary, the performance of IDEA can be improved by posing certain artificial constraints to guide the search in addition to the original constraints. In the present work, studies introduced an additional constraint into the problem. The constraint was formulated as explained in the following paragraph.

To formulate the problem so that all objectives have to be minimized, f1 is the negative of total oil extracted, and f2 is the amount of gas injected. Now, given that the two objectives are in conflict, it is understood that on the Pareto front, the solution with maximum value of f1 will have the least value of f2. Using the original constraint (which is the upper bound on f2), IDEA is able to get solutions close to this end of the curve. There are, however, no constraints to spread the solutions to the other extreme of the curve (where f1 is at a maximum). Hence, an additional constraint is posed by using the maximum value of f1 achieved during the run at any given time. Among the solutions that satisfy the original constraint, the solution with maximum value of f1 is stored. If this maximum value is F1, max, then an additional constraint is posed as . This way, whenever a solution is found which has an f1 value greater than the existing maximum value F1, max, then this solution is evaluated as infeasible. Thereafter, the F1, max value is updated with that of the new solution. By imposing these artificially created infeasible solutions (which satisfy the original constraint), the search is directed to find more and more solutions toward the unconstrained end of the front (i.e., where f1 is at a maximum).

The results of these five experiments, that is, NSGA-II, IDEA, IDEA with additional constraints (IDEA (AC)), MOEA/D, and the Liu Li algorithm are summarized in Table 6. It is seen that the performance of IDEA is inferior to NSGA-II and MOEA/D if the original problem formulation is used. However, with additional constraints, there is significant improvement in the performance of IDEA on the problem, resulting in a better median performance as compared to NSGA-II and MOEA/D. Also, the median performance of MOEA/D is inferior to NSGA-II, possibly because MOEA/D was run on an unconstrained problem. In the process, the algorithm achieved solutions violating the constraint on available gas, which are not considered for calculating the metrics. The Liu Li algorithm outperforms all of the above algorithms in terms of both indicators.

Table 6:
Performance on Multi-Objective 56 Oil Well Problem
AlgorithmBestMedianWorst
Hypervolume
NSGA-II 0.822 0.7265 0.607
IDEA 0.772 0.6765 0.581
IDEA (AC) 0.81 0.743 0.655
MOEA/D 0.856 0.7305 0.581
Liu Li 0.99 0.9455 0.855
Displacement
NSGA-II 140.038 204.995 294.556
IDEA 159.995 256.869 349.383
IDEA (AC) 129.219 194.72 285.559
MOEA/D 124.356 205.773 316.781
Liu Li 94.3328 136.565 194.682
AlgorithmBestMedianWorst
Hypervolume
NSGA-II 0.822 0.7265 0.607
IDEA 0.772 0.6765 0.581
IDEA (AC) 0.81 0.743 0.655
MOEA/D 0.856 0.7305 0.581
Liu Li 0.99 0.9455 0.855
Displacement
NSGA-II 140.038 204.995 294.556
IDEA 159.995 256.869 349.383
IDEA (AC) 129.219 194.72 285.559
MOEA/D 124.356 205.773 316.781
Liu Li 94.3328 136.565 194.682

Values in boldface indicate the best performing algorithm's metric values.

For visualization of performance, the best runs obtained using each algorithm (based on displacement metric) are plotted in Figure 4, and the median runs are plotted in Figure 5. The figures echo similar trends as were suggested by the hypervolume and the displacement measures. It is also interesting to observe that the spread of the nondominated solutions obtained using the Liu Li algorithm is much higher compared to the other algorithm, which is reflected in the superior metric values obtained. However, within the range of solutions obtained by the other algorithms (corresponding to oil extraction > 16,000 BPD), it is observed that the solutions obtained using all four algorithms (NSGA-II, IDEA, IDEA(AC), and MOEA/D) dominate those obtained using the Liu Li algorithm.

Figure 4:

Comparison of best runs (based on displacement metric) obtained for the multi-objective formulation of the 56 oil well problem.

Figure 4:

Comparison of best runs (based on displacement metric) obtained for the multi-objective formulation of the 56 oil well problem.

Figure 5:

Comparison of median runs (based on displacement metric) obtained for the multi-objective formulation of the 56 oil well problem.

Figure 5:

Comparison of median runs (based on displacement metric) obtained for the multi-objective formulation of the 56 oil well problem.

In addition to the solutions obtained using the multi-objective formulation, Figures 4 and 5 also show solutions obtained using single objective maximization of oil extracted, with upper bounds on the total gas available for injection. Four values of maximum gas available have been chosen here, that is, 22,500, 17,000, 11,000, and 5,000 thousand standard cubic feet of gas (MSCF). The results obtained using these four single objective instances (best out of 30 runs) are plotted in both these figures. It is seen that the multi-objective formulation gives good quality solutions and a large number of solutions for the decision maker in a much smaller number of evaluations. For example, in this case, to get four near optimal solutions using single objective optimization, 4 × 10,000 evaluations were used, compared to just 10,000 for multi-objective formulation.

It is worthwhile to mention here how the infeasibility ratio affects the performance of IDEA for the problems studied in this paper. For the experiments reported, a value of was used (5% population is infeasible) consistently. Experiments with other values of suggest different trends for single objective and multi-objective optimization problems. For single objective problems, the algorithm does not need to spread the population along the Pareto front, but instead needs to focus the population on the solution with the optimum objective value. Therefore, increasing the number of infeasible solutions can aid the algorithm in getting even better solutions. For example, using IDEA with for a single objective problem, among 30 runs, a best value of 22,418.3 and a median value of 22,177 were achieved, which is better than the performance reported in Table 5. On the other hand, using a higher proportion of infeasible solutions for the multi-objective problem deteriorates the performance of IDEA for this problem, as comparatively less feasible solutions are left to cover the Pareto front.

6.  A More Practical Formulation

In the single objective formulation studied in the previous sections, maximization of the oil extracted was considered. In the solutions found for the problem, while the amount of gas injected into some of the wells was high (more than 1000 MSCF/day), the amount injected into a few is very low (less than 5 MSCF/day). However, in real life situations, there is additional cost associated (such as fixed setup cost plus fixed and/or variable operational cost) with the extraction of oil in any of the oil wells (e.g., labor cost, administration, logistics). Consequently, it is not profitable to invest resources into oil wells that are eventually used to extract very little oil. Therefore, a modified objective formulation was considered in this section, where the profit was maximized rather than the volume of oil extraction. The idea is to eliminate wells which are used for extracting small quantities of oil, thus saving on fixed costs.

The objective considered is the maximization of daily profit (in $US), given by where Po is the oil price, Pc is the price of injecting gas, Fc is the fixed cost of operation associated with each well, and Nu is the number of wells used for oil extraction. Here, the price of oil has been assumed to be$75 per barrel, the gas injection cost is $1.5 per MSCF, and the fixed cost which includes labor, supervision, and administration cost, has been taken as$6700 per day.4

Using the modified formulation, 30 independent runs of IDEA were done, and the results were compared to those obtained using the original formulation. The results are shown in Table 7. Even though the total oil extracted using the modified formulation is marginally smaller than for the original formulation, there is a greater economic benefit. This is because in the modified formulation, the addition of overhead costs (labor, supervision, administration, etc.) results in a preference for solutions where least number of wells are used. The profits obtained using the modified formulation are significantly higher as compared to the original formulation, as seen from Table 7. On a median run, an increase of $68,000 per day (approximately 5.5%) is achieved. This improvement can be explained based on the number of wells used for oil extraction. For the case where oil extraction is maximized, it is seen that all wells are being used for extraction. However, on average, about 11 wells are injected with less than 25 MSCF/day, resulting in a correspondingly small amount of oil extracted. On the other hand, when the profit is being maximized, on average, 11 wells are left completely unused, thereby saving any further investment into those wells. Table 7: Results Using IDEA for Two Different Formulations, the First Maximizing Oil Extracted, and the Second Maximizing the Profit Formulation 1 (maximize oil)Formulation 2 (maximize profit) Oil extracted (barrels)Profit (million$)Oil extracted (barrels)Profit (million $) Best 22,348.7 1.276 22,246.6 1.321 Median 22,091.8 1.240 21,986.9 1.308 Worst 21,816.3 1.118 21,649.7 1.286 Formulation 1 (maximize oil)Formulation 2 (maximize profit) Oil extracted (barrels)Profit (million$)Oil extracted (barrels)Profit (million $) Best 22,348.7 1.276 22,246.6 1.321 Median 22,091.8 1.240 21,986.9 1.308 Worst 21,816.3 1.118 21,649.7 1.286 The results obtained using IDEA and NSGA-II for formulation 1 (oil extraction maximization), results using IDEA for formulation 2 (profit maximization), and the Matlab code for both these formulations of the problem are available for download at http://seit.unsw.adfa.edu.au/research/sites/mdo/codes_frameworks.htm 7. Summary In this paper, we have presented studies on a practical gas lift optimization problem involving oil extraction from a reservoir consisting of 56 wells. The contributions made through the studies are summarized below. • IDEA is used to study the conventional single objective formulation of the problem. The results obtained using IDEA are consistently better than NSGA-II as shown from average objective values and statistical tests. The best solution obtained using IDEA indicates an increase of 315.3 BPD in oil extraction compared to the previous best solution reported. This implies a potential savings of$8.6 million in a year. The benefit of preserving marginally infeasible solutions for the constrained optimization problem is clearly evident from the study.

• A multi-objective formulation of the problem is solved, which provides solutions for the whole range of gas availability in a single run. The performances of five different algorithms are studied on the multi-objective formulation. In its basic form, IDEA is outperformed by NSGA-II since the optimum solution (Pareto front) does not lie on a constraint boundary, unlike the single objective formulation. However, to deal with such a case, the imposition of artificial constraints was proposed, which leads to considerable improvement in performance. With the artificial constraints, IDEA is found to perform better than NSGA-II. However, the overall best solutions are obtained using the Liu Li algorithm, which tends to achieve a much more diverse (but not necessarily as converged) set of solutions as compared to the rest of the algorithms. It is to be noted here that the comparison among IDEA and NSGA-II is of particular interest because both of these algorithms use exactly the same kind of evolution operators, and hence the improvements in the results can be solely attributed to the modified ranking scheme. The MOEA/D and Liu Li algorithms, on the other hand, use different evolution schemes (compared to NSGA-II/IDEA), and therefore a direct comparison may not hold significant implications. For a more meaningful comparison, infeasibility based ranking can be implemented in these algorithms and then comparisons could be made with base algorithms to demonstrate benefits.

• Last, a modified single objective problem formulation is proposed, in which maximization of profit is considered in lieu of the volume of oil extracted. In real life, there are overhead costs involved in exploring and operating any well. Therefore, it is imperative that the quantity of oil extracted from such wells be commensurately high, a factor which is not considered in the original formulation. It is demonstrated that even with a relatively smaller amount of oil extracted, significantly higher economic benefits can be derived using the modified formulation. On a median run, as much as a \$68,000 increase in profit was achieved in this study.

Acknowledgments

The first author's work is funded by Research Publication Fellowship from the Research and Research Training Office (RRTO), University of New South Wales at Australian Defence Force Academy (UNSW@ADFA), Canberra ACT, Australia.

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