A Stochastic Inventory Model for Scheduling Development Drilling
Authors: Hartsock James H. , Greaney William A. 
Published: December 2007

Abstract
Employing the methodology of operations research, a mathematical model is formulated to schedule the development drilling of an oil field. The objective of the model is to minimize the total discounted cost of the development operation, and the decision variable is the number of wells drilled as a function of time. The model is analogous to an inventory problem in which the penalty assessed for failing problem in which the penalty assessed for failing to meet the demand for crude oil is balanced against the cost of drilling, completing, and equipping wells and the cost of transporting rigs and personnel to and from the development area. Because of the requirement that a single decision maker controls the scheduling, the model is particularly applicable to the development of fields in foreign countries. Since the production rates from future wells are unknown and some uncertainty exists in the projected crude oil demand, neither the field producing capacity nor the demand can be treated deterministically, but must be considered as random variables. Assuming that the production rates are independent, log-normally distributed random variables and the demand is normally distributed; the expected shortage is derived as a function of the number of wells drilled and completed. An option is also provided to include a uniformly distributed demand. provided to include a uniformly distributed demand. Once the field is placed on production, the mean and variance of the production rates will decline in proportion to the remaining reservoir energy. The proportion to the remaining reservoir energy. The model is structured so that the mean and variance of the production rates can be adjusted for any type of producing mechanism; therefore, the drilling schedule is planned not only to satisfy a projected demand rate but also to offset the loss in producing capacity when the field is placed on production. The total cost function results in a nonlinear unconstrained minimization problem, which is solved by the pattern search technique of Hooke and Jeeves. The solution to the model demonstrates the sensitivity of the drilling policy to the unit shortage cost and shows that there exists a unit shortage cost below which the field cannot be developed economically.

Introduction
The exploration and production of crude oil and natural gas are classic examples of decision making under uncertainty. irrespective of the technological advances in the fields of geology, geophysics, geochemistry, and reservoir engineering, the presence of commercial hydrocarbon deposits can presence of commercial hydrocarbon deposits can only be proved by a substantial investment in the drilling of one of more wells. Geologists have long recognized the stochastic nature of hydrocarbon deposits and several interesting prospecting models have been derived. Dowds prospecting models have been derived. Dowds proposed that the rock-hydrocarbon distribution proposed that the rock-hydrocarbon distribution within a geographical area could be studied as a Markov chain. using this approach, he found that the Markov chains from well-to well and area-to area tended toward statistical equilibrium. By contouring the statistical entropy, he was able to isolate areas with the highest probability of finding oil. An interesting application of discriminant analysis was presented by Wignall, in which well and field data were analyzed in order to define a discriminant function that could be used to differentiate between producing and nonproducing wells. producing and nonproducing wells. Grayson, in a famous application of utility theory, proposed that alternative drilling locations be examined on the basis of their expected utility. Kaufmann later improved on this work by showing that, within a given sedimentary basin, reserves were log-normally distributed. Once oil has been discovered and responsibility for developing these reserves has been transferred to the production department, petroleum engineers historically have ignored risk and uncertainty and have considered the development drilling problem as a deterministic one. The first approach toward scheduling the development drilling of a field was given by Aronofsky and Williams. They formulated the problem as a linear programming model with the objective of maximizing the present worth of future net income.