A Monte Carlo Approach to Optimal Drilling
Authors: Reed Ronald L. 
Published: December 2007

Abstract
The variable weight-speed optimal drilling problem has been soloed rigorously using a Monte problem has been soloed rigorously using a Monte Carlo scheme to vary the drilling schedule toward the least cost per foot. Constraints are readily accommodated and the global nature of the optimum can he checked by starting with different initial paths. paths. Examples are given of optimal drilling schedules that lie partially interior to the feasible region as well as on upper weight or speed constraints.
Comparison with the variational method of Galle and Woods showed good agreement using drilling equations presented here. However, in all cases studied, it was found that variable weight-speed optimization offered very little advantage over the simpler constant weight-speed approach. In view of ibis, a last computer program was developed for the constant weight-speed case using the powerful conjugate-gradient technique. This method should be very effective in connection with field applications.

Introduction
Although numerous papers discuss drilling optimization or provide optimization techniques based on field experience, it appears that the first analytical approach to the problem was published by Moore. He used drilling equations that were so simple it was possible to calculate directly the optimum weight corresponding to a given rotary speed.
In 1959, Graham and Muench used what is sometimes called the "graphical" approach, together with more realistic drilling equations, to calculate optimum combinations of weight and speed to bearing failure. In their paper, cost per foot is computed vs weight for various depths at fixed speed. This is repeated for various speeds until the optimum is found for each depth.
The most significant contribution to the field appeared in 1960. This paper by Galle and Woods culminated years of drilling mechanics research, reducing it all to a concise set of drilling equations. Necessary conditions for the optimal variable weight-speed path are found using the classical calculus of variations with integrated drilling equations acting as constraints. The final result is obtained using a numerical procedure. One conclusion was that, in most cases, the variable weight-speed method would effect a saving of at least 10 percent over the best constant weight-speed schedule. Although we do not want to detract from the importance of this work in any way, it is subject to question on two counts. First, the method of solution requires weight and speed to vary continuously as a function of time. Second, it is recommended that bearing-wear-limited schedules be computed using tooth-wear-limited theory.
In 1962, Billington and Blenkarn used the equations and techniques developed by Galle and Woods to optimize the variable weight schedule when speed is fixed by rig limitations. They concluded there was little cost advantage in using a variable weight-speed schedule in preference to a constant speed program.
In 1963 the second contribution of Galle and Woods to optimal drilling appeared. In this paper ordinary constrained calculus is used to solve the two-dimensional best constant weight-speed problem. They also provide the best speed for a given weight and the best weight for a given speed. Charts are furnished so that hand calculations can be rapidly made. In the introduction, Galle and Woods note that, in certain cases, the procedures in which both weight and rotary speed are varied or where weight alone is varied result in only slightly lower cost than if the bit were properly operated at constant weight and rotary speed.
We have therefore seen a transition from the belief that in most cases variable weight-speed is at least 10 percent better than constant weight-speed to the opinion that there is very little difference between the two methods. At this point, one is not certain what to conclude.
In this paper a new, rigorous method of optimization is developed. It provides an independent check on the approximations involved in the variational approach of Galle and Woods and compares optimal variable weight-speed schedules to the corresponding optimal constant weight-speed programs.