Linear Programming Optimizes Unit Negotiations
Authors: Douglass Tom R. 
Published: December 2007

A method that applies linear programming to the problem of negotiating unit equity is described, developed, and demonstrated. The unit negotiation process is discussed, and two linear programming models are compared in a simple example problem. The appendix presents in detail the derivation of an algorithm to solve the linear programming models with bidirectional constraints and negative constraint bounds.

Introduction

Unitization is a widely accepted practice of combining multiple leases under the supervision of one operator for the purposes of improved reservoir management.

Linear programming is a mathematical technique to optimize complex systems when those systems can be modeled by a series of linear expressions. The solution method for linear programming, known as the simplex method, was developed in 1947 by the American mathematician George B. Dantzig. Linear programming has found wide use in the refinery and chemical businesses as well as in other forms of business planning. This paper will demonstrate the use of the linear programming paper will demonstrate the use of the linear programming technique as a tool for unit negotiations.

The mathematical development of a special version of the simplex method is provided in the Appendix for completeness; an understanding of it is not essential for the use of the linear programming models discussed.

The Unitization Process

When unitization of producing leases is proposed for the purpose of secondary recovery or for other reasons, the purpose of secondary recovery or for other reasons, the working-interest owners of the individual leases form a committee. This committee appoints members to a technical subcommittee. The technical subcommittee performs an engineering study to determine the values of parameters for each lease that are deemed relevant to equity parameters for each lease that are deemed relevant to equity determination of the proposed unit. For example, the subcommittee might determine the reservoir volume under each lease or the expected ultimate primary recovery from each lease. Many other parameters are possible, such as current producing rate, cumulative oil production, and the number of wells. All these parameters are determined for each lease involved in the proposed unit and must be based on sound engineering and geological science. Table 1 is an example of a simple parameter table.

At this point, the parameter table contains values in many different measurement units, such as barrels, acre-feet, and well count. The technical subcommittee also prepares the parameter table expressed in percentages by prepares the parameter table expressed in percentages by totaling each parameter for all leases, then dividing each parameter value by this total, and multiplying by 100 to parameter value by this total, and multiplying by 100 to convert to a percentage. In this way, all the parameters become unitless, and the total for each parameter over all leases is exactly 100%. Table 2 illustrates the result of this calculation. With each owner's working interest in the leases constituting the proposed unit, the technical subcommittee also computes a parameter table listed by owner. Tables 3 and 4 depict this computation.

After the completion of its task, the technical subcommittee presents the study and the parameter tables, along with any recommendations, to the working-interest owners' committee. The owners' committee votes to accept or to reject the engineering study and the prepared parameter tables. If rejected, the technical subcommittee parameter tables. If rejected, the technical subcommittee reconvenes to revise the study and the parameter tables on the basis of recommendations from the owners' committee. When the technical study and parameter tables are accepted by the owners' committee, the owners begin the process of unit negotiation on the basis of the prepared parameter tables. parameter tables. Each owner is given the opportunity to propose a unitization formula. This formula is simply a list of weight factors where each weight is applied to a corresponding parameter. The weight factors are expressed as decimal parameter. The weight factors are expressed as decimal fractions between zero and one and must sum to exactly one.

Every owner's unit participation or working interest can be calculated by the multiplication of each percentage parameter by its corresponding weight. Table 5 is a parameter by its corresponding weight. Table 5 is a summary of the calculations performed for a hypothetical formula.

After a proposal has been offered and every owner's interest has been calculated, each owner votes his calculated interest for or against the proposed formula. Any proposal that receives votes exceeding some specified minimum percentage level (usually set by a governmental body) becomes the official unit equity formula. In this paper, the specified minimum percentage level is referred paper, the specified minimum percentage level is referred to as the statutory majority. Variations of this process exist, but the previous discussion generally describes the unit equity negotiation process.

Before voting for various proposals, each working interest owner usually has decided what minimum unit working interest he will probably accept. The sum of these desired minimum interests, however, will undoubtedly exceed 100%.

JPT

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