Multi-objective drilling trajectory optimization using decomposition method with minimum fuzzy entropy-based comprehensive evaluation

Highlights

• A drilling trajectory optimization problem with complex constraints is considered.

• MOEA/D is combined with an adaptive penalty function to solve the MOP.

• A comprehensive evaluation method based on minimum fuzzy entropy is proposed.

Abstract

This paper is concerned with the optimization problem of drilling trajectory design, which plays a vital role in ensuring the safety and enhancing the efficiency of an industrial drilling process. Due to complex geological environment and limited capacity of equipment, the problem to be addressed features multi-objective and multi-constraint, and consists of two challenges: (1) how to formulate a proper optimization scheme and efficiently solve it for a group of solutions; and (2) how to pick out a desired result from the obtained Pareto solutions according to certain requirements. In this paper, to meet drilling practice, three objective functions are introduced regarding trajectory length, well-profile energy, and target error, respectively. Constraints are the range of decision parameters, non-negative constraints and bound of the target area. As a result, a comprehensive optimization model for the design of drilling trajectory is established. A novel optimization algorithm is devised to deal with the contradictory objectives and multiple nonlinear constraints, which combines an adaptive penalty function with multi-objective evolutionary algorithm based on decomposition. A fuzzy-entropy-based evaluation approach is further employed to determine a satisfactory solution from the group of obtained ones. A case study illustrates that (1) our optimization solution is indeed beneficial to the optimization of drilling trajectory; and (2) the optimization algorithm and the decision method therein outperform some existing ones, which shows both practical and theoretical significance.

Introduction

Horizontal wells are the main type of resource exploitation, such as oil, natural gas, and shale gas. Pre-drill trajectory design is one of the key points to improve the safety and efficiency of drilling. Measuring every point on a trajectory in drilling practice is very difficult, so a drilling trajectory’s actual shape is always unknown. Therefore, it is necessary to raise assumptions to calculate the coordinates of points on a drilling trajectory [1]. The commonly used mathematical models are curvature radius method [2], circular arc method [3], minimum curvature method [4], and nature curve method [5]. Based on these models, the optimization of a drilling trajectory tries to find the optimal trajectory under some specific indicators and constraints.

In a definite drilling technology and geological environment, decreasing the trajectory length can reduce the cost of a drilling project. The description parameters (coordinates, inclination angles, and azimuth angles, etc.) of a trajectory model always have nonlinear relationships, leading to nonlinearity in constraint functions. Comparing with methods that need gradient information of the mathematical model, evolutionary algorithms only need the value of objective functions to guide the search. They are popular in drilling trajectory optimization problems to minimize a trajectory length [6], [7], [8].

However, a small trajectory length sometimes refers to a complex well structure, such as a rapid variation of inclination angles and azimuth angles, and significant dog-leg severity. They will increase the risk of drilling accidents. Thus, drilling trajectory optimization should be considered as a multi-objective optimization problem (MOP).

Some researchers introduced the target error as the second objective [9], [10], [11]. However, they did not study the complexity of a trajectory and only considered the range of decision parameters as constraints.

Mansouri et al. [12] introduced drill-string torque as the second objective. Furthermore, they applied the multi-objective genetic algorithm to solve the MOP. Based on this MOP model, Zheng et al. [13] introduced well-profile energy as the third objective. Although both of these indicators can reflect the complexity of the drilling trajectory, the target error was not considered. Besides the constraint of decision parameters’ range, only some non-negative constraints are studied.

Wang et al. [14] proposed a MOP of a sidetracking horizontal well trajectory. Their study considered the trajectory length, the complexity of the trajectory, and the target error. Limiting the location of the trajectory endpoint is set as a constraint. To solve the MOP, they transformed tri-objective functions into a single objective function by a linear weighted sum method. On the one hand, the transformation may lose some information of the MOP. On the other hand, the algorithm cannot obtain non-dominated solutions in a single run. Furthermore, decision-makers have to intervene in the optimization process. In our previous work [15], we aimed at these disadvantages and tried to solve the MOP proposed by Wang et al. [14] without considering the target error. Table 1 shows the objectives and constraints used in previous studies.

For the MOP mentioned above, constraint handling is a critical part. Some multi-objective optimization algorithms deal with constraints and objectives separately, such as constrained NSGA-II [16] and C-MOEA/D [17]. In these two algorithms, the feasible individuals are always dominating infeasible ones. Infeasible individuals are sorted by their constraint violation. These methods may lose fitness information of infeasible individuals, which affects the population convergence. As an improvement of this, an epsilon constraint handling method is studied [18], [19]. In these methods, the domination relationship depends on the epsilon value and the constraint violation. Penalty methods are convenient to combine with different optimization algorithms and effectively applied to engendering problems [20]. Especially adaptive penalty function (APF) methods can direct the population to the feasible area according to the current population status [21], [22]. It is appropriate to combine APF methods with MOEA/D (multi-objective evolutionary algorithm based on decomposition) [23] to handle constraints in MOPs. The combination can take advantage of applying APF methods to scalar optimization algorithms.

Multi-objective evolutionary optimization algorithms can obtain a set of solutions in a single run. But applying every solution to drilling practice is impossible. Therefore, it is necessary to provide preference solutions to decision-makers. Many researchers used multiple criteria decision making (MCDM) algorithms such as TOPSIS (technique for order preference by similarity to an ideal solution) [24], LINMAP (the linear programming technique for multidimensional analysis of preference) [25], and Shannon Entropy [26] to address these problems. These methods rank solutions and select the best one as the final solution [27], [28]. However, the preference of a decision-maker is a qualitative evaluation. A fuzzy comprehensive evaluation method transforms the qualitative evaluation into the quantitative evaluation, and its application in engineering problems was studied [29]. The membership degree theory of fuzzy mathematics can quantify the preferences of a decision-maker.

Considering the characteristics of the MOP and the strength and the weakness of existing algorithms, the main contribution of this work includes three parts:

• The nonlinear, multi-objective, and multi-constraint of a drilling trajectory optimization problem are taken into consideration. Combining an adaptive penalty method with MOEA/D, adaptive penalty function based MOEA/D (APF-MOEA/D) is proposed. The proposed algorithm is effective for MOP in drilling practices and has a higher hypervolume (HV) comparing with other methods.

• Because of the ambiguity of decision-maker’s preference, a comprehensive evaluation method based on minimum fuzzy entropy (CE/MFE) is proposed. The membership functions are established, and solutions are ranked according to an evaluation matrix. CE/MFE can obtain the solutions that have a high membership degree in good under different weight vectors.

• A drilling trajectory optimization problem is studied to ensure the efficiency and safety of the drilling process. The application of MOEA/D with CE/MFE on this MOP could give control parameters that satisfying different preference situations in drilling practice.

A parameter setting is done to obtain better performance on this MOP. The performance of APF-MOEA/D is compared with NSGA-II, C-MOEA/D. We also combined an epsilon constraint handling method [19] with MOEA/D and named it SaE-MOEA/D as a comparison. To illustrate the parameter sensitively, we studied the other two engineering problems in drilling practice. Some popular decision-making methods are used as a comparison: TOPSIS, LINMAP, Shannon Entropy.

The paper is organized as follows. Section 2 formulates the multi-objective drilling trajectory optimization problem of sidetracking horizontal wells. Section 3 describes the optimization method and the decision-making method to solve the MOP. Section 4 studies the application of the proposed method on drilling practice problems and presents a comparison of proposed methods with some existing methods. Finally, conclusions are made in Section 5.