Application of the Complex Method for Constrained Optimization to Oilfield Problems
Authors: Shum Y.M. , Konopnicki D.T. 
Published: December 2007

Abstract
In the application of optimization techniques, considerable difficulty is often encountered in determining the most applicable or effective method for solving a particular problem. In an attempt to solve this dilemma, M. J. Box developed the Complex Method, which can be used in solving a broad spectrum of problems. The method is a multivariable, direct-search technique that is efficient and convenient in optimizing problems with nonlinear objective functions subject to inequality constraints on explicit or implicit variables. This method has been applied successfully to many problems in the chemical industry; however, the applicability of this method for solving oilfield problems apparently has not been recognized. In this paper, two oilfield problems, a push-pull steam stimulation process problems, a push-pull steam stimulation process and a well-drilling schedule, were optimized using the Complex Method. The results were compared with those obtained by other optimization techniques reported in the literature. For both problems, the Complex Method was as efficient as other methods and probably was more efficient in obtaining the global optimum, both in terms of the number of objective-function evaluations required and of computing time. The solution to the constrained push-pull steam stimulation process was obtained push-pull steam stimulation process was obtained in one-fourth the number of evaluations required with the Hooke and Jeeves search technique; the solution to the drilling problem with 48 variables was obtained in only 2,123 objective-function evaluations, or 29 seconds of computing time on an IBM 370/165. Through the solution of two problems, the Complex Method was demonstrated to be easily applied and effective in obtaining optimum solutions as compared with other methods. This method should be implemented easily in obtaining optimum solutions as compared with other methods. This method should be implemented easily in obtaining optimum solutions to other oilfield problems.

Introduction
In recent years, the application of optimization techniques to oilfield problems has become widespread because of the possibility of increased profits or reduced costs with implementation of such studies. Several types of oilfield problems have been optimized using a wide variety of optimization techniques. E. L. Dougherty conducted an extensive review of the major applications of optimization techniques to oilfield problems. in these studies, the authors used a variety problems. in these studies, the authors used a variety of optimization techniques because of the difficulty of selecting an effective method for a particular problem. The Complex Method is a general optimization technique that can be applied directly to a variety of nonlinear problems with inequality constraints. The method is a problems with inequality constraints. The method is a multivariable direct-search technique; thus, the application of this method is especially convenient when derivatives of the objective function are difficult to obtain. The broad applicability of the Complex Method has been demonstrated in the chemical industry by successfully optimizing problems ranging from absorber-strippers to nonisothermal polymerization reactors. The method is implemented easily and is efficient in obtaining the global optimum while imposing inequality constraints on explicit and implicit variables. The objective of this paper is to demonstrate the usefulness and effectiveness of the Complex Method in obtaining the optimum values for oilfield problems, as compared with other methods. Two oilfield problems that previously have been solved by other methods were optimized using the Complex Method. The sample problems were the following:the optimum cycle length and the amount of steam injected in successive cycles are determined for a push-pull steam stimulation process such that the cumulative discounted net income is maximized; andthe optimum set of rotary speeds and weights on a bit is determined such that the drilling cost per foot is minimized in a drilling operation. These problems were selected to illustrate the wide range of mathematical formulations for which the Complex Method is applicable. The variable weight-speed schedule was selected for optimization rather than the constant weight-speed schedule because of the complexity of the variable weight-speed problem.
SPEJ
p. 123