Optimization Techniques For Weight-On-Bit And Rotary Speed. Part I: Point And Interval Optimization
Authors: Bentsen R.G. , Wilson D.C. 
Published: March 2010

Abstract
Currently available techniques for optimizing weight-on-bit and rotary speed are somewhat limited in scope. Theydetermine optimum policy in. the light of a part of the drilling process, rather than in the light of the entire drilling process. The object of this two-part investigation was to develop techniques which are less limited in scope, and to demonstrate the importance of optimizing the drilling process as a whole, further than optimizing separate parts of that process.
Three optimization methods have been developed. The methods differ primarily in the scope over which the optimization takes place. That is, the first method seeks to minimize the cost per foot drilled during a given bit run, the second method minimizes the cost of a selected interval and the third method minimizes the cost over a series of intervals. The methods are listed in order of increasing scope and complexity, the first two being examined in Part I, and the third treated in Part II. Results and conclusions regarding all three methods are to be found in Part II.
It was found that each of the methods gave a worthwhile cost saving, and that the saving increased as the scope and complexity of the method increased. The need for better data also increases as the scope and complexity of the method increases. Hence, the method used should be chosen according to the amount and quality of the data available.

Introduction
Although many investigations have taken place over the years, in 1958 Speer(1) was the first to propose a comprehensive method for determining optimum drilling techniques. He demonstrated empirically the interelationships of penetration rate, weight-an-bit, rotary speed, hydraulic horsepower and formation drillability. Speer combined these five relationships into a chart for determining optimum drilling techniques from a minimum of field test data.
Moore(2) was the first to suggest an analytical method for finding the optimum weight and speed. His equations for drilling rate and bearing wear were such that it was possible to calculate directly the optimum weight corresponding to a given rotary speed.
The optimization technique suggested by Moore has been used extensively by other investigators on the more sophisticated drilling equations developed later.
A graphical approach, together with more realistic drilling equations, was used by Graham and Muench(2) to calculate the optimum combinations of weight and speed to minimize drilling cost per foot. The cost per foot was computed versus weight for various depths at constant rotary speed. This procedure was repeated for various speeds until the optimum rotary speed was found for each depth.
The next, and probably most significant, contribution to optimal drilling came in 1960 when Galle and Woods(4) presented a technique designed to minimize drilling costs by varying bit weight and rotary speed throughout a hit run. The authors established a set of empirical equations for drilling rate, dulling rate and bearing wear rate as functions of the drilling parameters. The calculus of variations was used to find the manner in which weight and rotary speed should be varied to minimize drilling cost.