Multivariate Optimization of Production Systems
Authors: Carroll James A. , Horne Roland N. 
Published: December 2007

Summary
Traditional analysis of production systems treats individual nodes one at a time, calculating a feasible, but not necessarily optimal, solution. Multivariate optimization enables determination of the most profitable configuration, including all variables simultaneously. The optimization can also find the optimal recovery over a period of time, rather than just at a single instant, as in traditional methods.

Introduction
Mathematically, optimization involves finding the extreme values of a function. Given a function of several variables, Equation 1
an optimization scheme will find the combination of these variables that produces an extreme value in the function, whether it is a minimum or a maximum value. Many examples of optimization exist. For instance, if a function gives an investor's expected return on the basis of different investments, numerical optimization of the function will determine the mix of investments that will yield the maximum expected return. This is the basis of modern portfolio theory. If a function gives the difference between a set of data and a model of the data, numerical optimization of the function will produce the best fit of the model to the data. This is the basis for nonlinear parameter estimation. Similar examples can be given for network analysis, queueing theory, decision analysis, etc.
Historically, the petroleum industry has used optimization to allocate production through pipeline networks, to schedule transoceanic shipments of petroleum from supply sites to demand sites, to model refinery throughput, and to determine the best use of limited capital. Lasdon et al.1 point out that the production sector of the petroleum industry has seen few successful applications of optimization methods.
The objective of this study is to demonstrate the effectiveness of multivariate optimization techniques applied to the performance of hydrocarbon wells. The study consists of two primary phases: the development of a well model that determines the economic benefit of a well and the optimization of a well's economic benefit with multivariate optimization techniques.

Optimization Studies in Petroleum Engienering
Optimization methodologies are the focus of operations research, a field that began in the 1940's. Operations-research concepts were not adopted by the petroleum industry until the early 1950's. The bulk of the literature since then discusses linear programming techniques applied to reservoir management on a macrolevel. Aronofsky2 provides a detailed treatment of optimization methods in petroleum engineering.
Aronofsky and Lee3 developed a linear programming model to maximize profit by scheduling production from multiple single-well reservoirs. Aronofsky and Williams4 extended that model to investigate the problems of scheduling production for a fixed drilling program and scheduling drilling for a fixed production schedule. Charnes and Cooper5 used linear programming to develop a reservoir model that minimized the cost of wells and facilities subject to a constant production schedule. Attra et al.6 developed a linear programming model to maximize flow rate subject to several production constraints. All these models used linear reservoir models that were based on material-balance considerations, and the reservoirs generally were assumed to be uniform and single phase.
Rowan and Warren7 demonstrated how to formulate the reservoir management problem in terms of optimal control theory. Bohannon8 used a mixed-integer linear programming model to optimize a pipeline network. O'Dell et al.9 developed a linear programming model to optimize production scheduling from a multireservoir system. Huppler10 developed a dynamic programming model to optimize well and facility design, given the delivery schedule and using a material-balance reservoir model. Kuller and Cummings11 developed an economic and investment for petroleum reservoirs.