Well Placement Optimization Using Firefly Algorithm and Crow Search Algorithm

. Optimization of well placement is one of the main di(cid:30)cult factors in the development process in the oil and gas industry. The well placement optimization is high dimensional, multi-modal and discontinuous. In previous research, conventional and non-conventional optimization techniques have been applied to resolve this problem. However, gradient-free optimization techniques such as genetic algorithm and particle swarm optimization which is considered as the most e(cid:30)cient algorithms in this area suffer from local optima. In this article, two new metaheuristic optimization techniques, namely, crow search algorithm and (cid:28)re(cid:29)y algorithm are applied to the well placement optimization problem and their applications to maximize the net pro(cid:28)t value are studied. To study the performance of the (cid:28)re(cid:29)y and crow search algorithm, Eclipse and MATLAB environment are used . The proposed techniques are compared to popular established methods for optimizing well placement. Results show that the (cid:28)re(cid:29)y algorithm is proved to be e(cid:30)cient and e(cid:27)ective compared to other established techniques. However, the standard crow search algorithm is not suited to this problem


Introduction
Well placement optimization has been a major issue in the eld development process for a while due to its high dimensional, discontinuous and multimodal objective function. For the successful supply and longevity of the operation, placement selection of such well is very important. In recent years, to deal with the increasing global energy demand the correct well placement optimization has been at the center of many studies [1]- [6]. There are several processes of nding out the best possible place for well to ensure uninterrupted oil supply. Still, due to the vast infrastructural work related to such placec 2020 Journal of Advanced Engineering and Computation (JAEC) 181 ment, it is very crucial to make sure that the well is going to be in the best available area.
In brief, net prot value should be kept under consideration while running the well placement process. In order to maximize net prot value, several works have already been done with dierent mathematical algorithms. Previous studies have established that algorithms based on gradient and derivatives remain quiet inconsistent and ineective due to the non-smooth, discontinuous and high dimensional objective function [1,7]. Some prior research was able to nd local optima using gradient-based algorithms [8].
Again, some other well-known techniques such as mixed-integer programming [9], multivariate interpolation algorithms [10], etc. were also used to nd the possible best location. However, these methods failed to live up to the expectation and hence researchers put more emphasis on non-classical methods. These methods are inspired by dierent natural phenomenon and showed more viability in nding global optimum value.
At rst, Guyaguler and Horne used GA to optimize well location and injection rate by considering the net present value (NPV) as the objective function [11]. Quality maps are incorporated as nonlinear constraints which helped to determine the correct location for the correct placement of the well [12]. Lyons and Nasrabadi reduced the calculation time by four times by combining pseudo-history matching with genetic algorithms (GA) [13]. Chang, Y. in his research work, showed that NSGA II is an effective and powerful method for solving multiobjective problems [14]. In another study, it is observed that the genetic algorithm is eective and robust for accurately solving the well placement problem [15].
Overall, GA is one of the most popular technique for optimization purposes in well location optimization problems. Generally speaking, GA can nd the global optimal value, but the convergence speed is slow. The genetic algorithm-based optimization method was used eectively, but doubt was cast upon issues like absolute convergence and stability [16]. Again, such algorithms have limitations like non-linearity and dis-continuity [17,18]. Onwunalu used PSO in his research, and has reduced the search space by gradually moving from an exploration mode to an exploitation mode [19].
Subjected to: here u n represents well coordinates, NPV presents net prot value, LB and UB are lower bound and upper bound of the reservoir respectively.
The prime motivation behind well placement optimization is to make sure that the expenditure remains minimum while maximizing the net prot. NPV changes randomly with the change of co-ordinates value hence well placement. The variables used in (4) are shown in Tab. 1 depicted from [6]. Eclipse simulation was used to calculate cumulative oil production, cumulative water production's value. Q 0 is cumulative oil production OPEX is the operational expenditure Q w is cumulative water production OPEX is the operational expenditure P 0 is oil price D is the discount rate C w is cost per unit volume of produced water T is the number of years passed since the production has started P g gas price Q g cumulative gas production 3.1. Firey algorithm  Figure 1 shows the ow diagram of rey algorithm. The brightness, β, and step length of each random movement, α t+1 , can be found as

Methodology
At the beginning of the algorithm a random feasible solution is generated and after each iteration, it compares the present calculated solution with the previous one. Finally, the brightest of the acquired solutions will take the lead. for example, rey i with less brightness will be attracted to a brighter rey j according to (7) given as below- where β 0 : brightness of X i at r = 0, γ: An algorithm parameter which indicates how much a distant rey is related with its nearest rey.
α: step length of each random movement.
ε: a random vector from uniform distribution between 0 and 1.

Crow search algorithm
Crow search algorithm (CSA) is a populationbased metaheuristic algorithm for solving optimizing solution of a practical problem. This algorithm was rst developed by Askarzadeh [34].    In [18] Askarzadeh explained such a scenario with a beautiful example. Suppose crow j wishes to visit its cache M j,itr at iter iteration. At the same time crow i starts following that crow to steal j 's food. At this point, there may appear two possible solutions: Case 1: When crow does not notice that it is being followed by crow i, crow j will lead crow i towards his hiding place. Position of crow i will be changed as follows : where r is a random number between 0 and 1 and f i,itr l denotes the ight length of crow i at iter iteration. Case 2: Where memory M i,itr+1 in (9) denotes the memory location of ith crow for itr+1 iteration. M i,itr+1 is updated based on the following equation: Crow j knows another crow i is following it.
Hence, to restrain its cache from being pilfered, crow j will y to another position to confuse crow i using the following equation:

Performance comparison criteria
To evaluate the performance of the algorithms several criteria was considered for this problem [36,37]. So, authors have considered several criteria mentioned below: Eectiveness is a simple measure of performance and is the average value between tests of the best solution found as a percentage of the global optimum or, where additionally, L 98 i is the number of unique function evaluations required to nd solution q such that f (q) ≥ 0.98f (p i ) for trial i (for minimization) and M is the total number of function evaluations per trial.     Each well has coordinates (x, y). The total number of variables to optimize in this experiment is therefore 2 × 2.  Fig. 7 shows that PSO has the lowest standard deviation compared to other algorithms and FF has the second-lowest standard deviation.

Advantage and disadvantage of proposed techniques
By analyzing the main characteristics of the FA, the following three points for its success can be highlighted: • FA can perform better than PSO, GA, and CSA to tackle highly nonlinear, multimodal optimization problem as FA can automatically subdivide its population into sub-groups, since local attraction is stronger than long-distance attraction.
• To avoid premature convergence as those in PSO and GA, FA does not update its location based on the personal best information, and there is no explicit global best either.
• FA can work as a DE, SA, and PSO so it has full advantage of these three algorithms [31]. Also, by controlling the scaling parameter FA can adapt to the problem landscape.
No free lunch theorem (NFL) noted that no single algorithm can be best for all problems [39]. "This is an Open Access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited (CC BY 4.0)."