On The Joint Optimization of Well Placement and Control
Authors: Awotunde Abeeb A. 
Published: June 2014

Abstract
Of concern, in the development of oil fields, is the problem of determining the optimal locations of wells and the optimal controls to place on the wells. Extraction of hydrocarbon resources from petroleum reservoirs in a cost effective manner requires that the producers and injectors be placed at optimal locations and optimal controls be imposed on the wells. While the optimization of well locations and well controls plays an important role in ensuring that the net present value of the project is maximized, optimization of other factors such as well type and number of wells also plays important roles in increasing the profitability of investments. Until very recently, improving the net worth of hydrocarbon assets has been focused primarily on optimizing the well locations or well controls, mostly manually. In recent times, automatic optimization using either gradient-based algorithms or stochastic (global) optimization algorithms has become increasingly popular.
We present a two-part series illustrating how to effectively optimize well placement and rates without dramatically increasing the size of the optimization problem. In this first part of the work, we present two approaches to reduce the number of design variables in well rate optimization. Polynomial and trigonometry models are proposed to model the change of well rates with time and these models are parameterized with coefficients which can be determined from any optimization method. Thus, instead of using well rates as the control variables in the optimization process, we use the parameters of equations that are able to model the variation of well rates with time. Each model has a specific number of coefficients to model the change in rate over time and each well has a distinct value for each coefficient in the model. This means that the number of variables is fixed regardless of the number of years of operating the wells. The total number of optimization parameters is thus the product of the number wells and the number of coefficients in the model. In this way the method allows us to find the average daily or monthly or annual well rate for each well. This significantly reduces the number of variables required and the time required in running the optimization algorithm. The method also makes it possible to use optimization algorithms that would otherwise break down when the number of design variables is too large.
In the second part, a joint optimization approach to estimating optimal well locations, well rates, well type and well number is proposed. Our approach uses a set of well coordinates and a set of well controls as the optimization parameters. The set of well controls, however, covers both the negative and positive parts of the real line. The search interval of the well controls is divided into three parts, one part denoting the region where the well is an injector, a second part denoting the region where there is no well, and a third part denoting the region where the well is a producer. By this, the optimization algorithm is able to match every member in the set of well coordinates to three possibilities within the search space of well controls: an injector, no well or a producer. The optimization is performed using differential evolution and one sample application is presented to show the effectiveness of the method. Results obtained show that the method is able to identify simultaneously, optimal well locations, optimal well controls, optimal well type and the optimum number of wells.