TECHNOLOGY Spreadsheets simplify oil field unitization calculations

Aug. 28, 1995
Scott B. Cline Hefner Corp. Oklahoma City, Okla. Optimization Equations (25027 bytes) Linear programming, now available with most spreadsheet programs, allows companies involved in unitizing oil producing properties to determine their minimum and maximum interest. With spreadsheets, even the smallest companies can do linear programming or optimization techniques without having to know much about the process involved.

Scott B. Cline
Hefner Corp.
Oklahoma City, Okla.

Optimization Equations (25027 bytes)

Linear programming, now available with most spreadsheet programs, allows companies involved in unitizing oil producing properties to determine their minimum and maximum interest.

With spreadsheets, even the smallest companies can do linear programming or optimization techniques without having to know much about the process involved.

If each company can easily determine its and the competition's optimum parameter weight, then reasonable compromise and unitization are more likely. When suspicion and lack of knowledge prevail, parties are less likely to cooperate and form production units.

Unit participation formulas vary greatly, even if units have similar reservoir characteristics.1

Weighted parameters

Many different parameters can be combined in a unit formula. Some common parameters include original oil in place, usable well bores, and cumulative production.

The companies usually first settle on the parameters to include and the reservoir engineering or geologically determined values of these parameter. For instance, each company will negotiate over the acre-ft of pay underlying a unit or the cumulative production from each well.

Companies often disagree about the parameters to include in unit formulas but more frequently about what weight to assign to each parameter.

For instance, if the parameters are acre-ft, usable well bores and cumulative production, the question is then what fraction should be assigned to each parameter. One solution might be 40% to acre-ft, 10% to usable well bores, and 50% to cumulative production.

Agreement on parameter weights is difficult because, unlike the engineering or geologically determined value of each parameter, the weight of each parameter is very subjective. The relative weight of each parameter greatly impacts each company's total unit interest and ultimate economic return. There is an optimum weight formula that maximizes or minimizes unit working interests for each company.

There are many ways to approach the problem of optimization, but optimization problems have certain common characteristics. Each problem is concerned with maximizing or minimizing some linear objective function given a set of restrictions or constraints that limit the degree to which the objective can be pursued.2 Non-negativity of constraint is also a general feature of linear programming.

The most common approach is to optimally allocate each company's parameter weight factor once the parameter list is approved and each party's parameter value is calculated. Each company obviously wants to maximize its share of a unit or at least know the possible ranges that it could have.

Example

As an example, after preliminary negotiations, four companies decided that cumulative production, usable well bores, and reservoir acre-ft are the three most reasonable parameters for a proposed unit. The companies also agreed on the values for these parameters, such as the total number of usable wells bores, cumulative production, and acre-ft of reservoir underlying the proposed unit tracts.

If the acre-ft of reservoir became a matter of discussion, each party would theoretically want to determine the relative parameter or decision variable weight that would maximize its interest in the unit. Often trial and error or plain subjective opinions are used at this stage in a typical small unit.

Fig. 1 (25027 bytes) shows the outline of the proposed unit. Within the unit are eight spacing tracts, 800 acres of surface, and seven usable well bores. The wells have a total cumulative production of 370,580 bbl of oil.

The eight spacing tracts have four different companies with working interests within the unit. The four companies within the aggregate unit have agreed that the unit contains 6,474 acre-ft of reservoir.

Each of the four companies owns various working interests in each of the eight tracts. Based on the various companies' working interests within the eight tracts, Table 1 (15384 bytes) shows the aggregate division of interest within the entire 800 acre unit for the four companies involved.

The equation box shows the generalized problem objective and constraints for maximizing the interest of Company 1 given the parameters of acre-ft, cumulative production, and usable well bores. The problem is specified as the optimization formula:

W1 3 0.2294 + W2 3 0.2857 + W3 3 0.139

Additionally, it was decided that more emphasis was to be given to acre-ft and cumulative production than to usable well bores, as shown by the weight constraints. These limits can, in general, be any range between zero and one.

This problem is then easily set up and solved in a spreadsheet with an optimization subroutine. Most spreadsheet programs have optimization subroutines such as Solver in Excel. The basic data are input to the spreadsheet as shown in Table 2 (15384 bytes). The user asks the program to maximize the total participation factor (21.47 % for Company 1) subject to the constraints previously specified. In this case, the program calculates weights as 0.35 for cumulative production, 0.3 for usable well bores, and 0.35 for acre-ft for maximizing Company 1's interest and then computes each unit interest in Column L. Part B of Table 2 (15384 bytes) shows the individual spacing tract participation factors that maximize Company 1's unit interest.

Table 3 (18073 bytes) and Fig. 2 (68386 bytes) show the results of maximizing each company's interest and the relative weights needed. The relative variance in the weighted parameters resulted in a variation of as much as 6% of the total unit interest for Company 4. This could be a substantial economic difference.

References

1.Cline, S.B., Stanley, B.J., "Unitization formula's need scrutiny," OGJ, Sept. 13, 1993, pp. 73-4.

2.Anderson, David, Sweebet, Dennis, Williams, Thomas, An Introduction to Management Science, West Publishing Co., St. Paul, Minn., 1982.

OPTIMIZATION EQUATIONS

Maximize Company 1's interest

Maximize: W1 3 CP1 + W2 3 UW1 + W3 3 AF1

Subject to: W1 + W2 + W3 = 1.0

LL1 W1 UL1

LL2 W2 UL2

LL3 W1 UL3

Where: CP1 = Company 1's cumulative production, % of total

W1 = Cumulative production weight

UW1 = Company 1's usable well bores, % of total

W2 = Usable well bores weight

AF1 = Company 1's acre-ft, % of total

W3 = Reservoir acre-ft weight

LLi = Lower limit of weights (zero min., unity max.)

ULi = Limit of weights (zero min., unity max.)

Optimization formulation

Maximize: W1 3 0.2294 + W2 3 0.2857 + W3 3 0.139

Subject to weight constraints: W1 + W2 + W3 = 1.0

0.35 W1 0.61

0.10 W2 0.50

0.35 W3 0.61

Table 1

PARAMETER VALUES FOR UNIT

Cumulative Acre-ft 5 ft

production, % Usable % underlying %

Company bbl of unit wells of unit unit of unit

===========================================================================================

1 85,000 22.94 2 28.57 900 13.90

2 90,102 24.31 3 42.86 1,500 23.17

3 75,000 20.24 1 14.29 750 11.58

4 120,478 32.51 1 14.29 3,324 51.34

Totals 370,580 100.00 7 100.00 6,474 100.00

The Author

Scott B. Cline is vice-president and manager of exploration for Hefner Corp. in Oklahoma City. He previously worked for Slawson Oil Co., Union Texas Petroleum Co., and Gulf Oil Corp. Cline has a BS in geological science from Pennsylvania State University, a masters in business administration from Oklahoma City University, and an MS in petroleum engineering from the University of Oklahoma.

Copyright 1995 Oil & Gas Journal. All Rights Reserved.