A Method for Determining the Optimum Location of Wells in a Reservoir Using Mixed-Integer Programming
Authors: Rosenwald Gary W. , Green Don W. 
Published: December 2007

Abstract
This paper presents a mathematical modeling procedure for determining the optimum locations of procedure for determining the optimum locations of wells in an underground reservoir. It is assumed that there is a specified production-demand vs time relationship for the reservoir under study. Several possible sites for new wells are also designated. possible sites for new wells are also designated. The well optimization technique will then select, from among those wellsites available, the locations of a specified number of wells and determine the proper sequencing of flow rates from Those wells so proper sequencing of flow rates from Those wells so that the difference between the production-demand curve and the flow curve actually attained is minimized. The method uses a branch-and-bound mixed-integer program (BBMIP) in conjunction with a mathematical reservoir model. The calculation with the BBMIP is dependent upon the application of superposition to the results from the mathematical reservoir model. This technique is applied to two different types of reservoirs. In the first, it is used for locating wells in a hypothetical groundwater system, which is described by a linear mathematical model. The second application of the method is to a nonlinear problem, a gas storage reservoir. A single-phase problem, a gas storage reservoir. A single-phase gas reservoir mathematical model is used for this purpose. Because of the nonlinearity of gas flow, purpose. Because of the nonlinearity of gas flow, superposition is not strictly applicable and the technique is only approximate.

Introduction
For many years, members of the petroleum industry and those concerned with groundwater hydrology have been developing mathematical reservoir modeling techniques. Through multiple runs of a reservoir simulator, various production schemes or development possibilities may be evaluated and their relative merits may be considered; i.e., reservoir simulators can be used to "optimize" reservoir development and production. Formal optimization techniques offer potential savings in the time and costs of making reservoir calculations compared with the generally used trial-and-error approach and, under proper conditions, can assure that the calculations will lead to a true optimum. This work is an extension of the application of models to the optimization of reservoir development. Given a reservoir, a designated production demand for the reservoir, and a number of possible sites for wells, the problem is to determine which of those sites would be the best locations for a specified number of new wells so that the production-demand curve is met as closely as possible. Normally, fewer wells are to be drilled than there are sites available. Thus, the question is, given n possible locations, at which of those locations should n wells be drilled, where n is less than n? A second problem, that of determining the optimum relative problem, that of determining the optimum relative flow rates of present and future wells is also considered. The problem is attacked through the simultaneous use of a reservoir simulator and a mixed-integer programming technique. There have been several reported studies concerned with be use of mathematical models to select new wells in gas storage or producing fields. Generally, the approach has been to use a trial-and-error method in which different well locations are assumed. A mathematical model is applied to simulate reservoir behavior under the different postulated conditions, and then the alternatives are postulated conditions, and then the alternatives are compared. Methods that evaluate every potential site have also been considered. Henderson et al. used a trial-and-error procedure with a mathematical model to locate new wells in an existing gas storage reservoir. At the same time they searched for the operational stratagem that would yield the desired withdrawal rates. In the reservoir that they studied, they found that the best results were obtained by locating new wells in the low-deliverability parts of the reservoir, attempting to maximize the distance between wells, and turning the wells on in groups, with the low-delivery wells turned on first. Coats suggested a multiple trial method for determining well locations for a producing field.