Brian F. Towler

University of Wyoming

Laramie, Wyo.

George G. Chakmakian

Consultant

Laramie, Wyo.

The three parameters needed for hyperbolic-decline-curve analysis can be determined by a new method developed for a computer spreadsheet optimizer routine.

The method can replace graphical or type-curve methods that are tedious and give subjective results.

Production decline-curve analysis is accepted for prediction of future performance of oil and gas wells and fields. One type of cline-curve analysis involves the hyperbolic-decline-curve equation (Equation 1 in the equation box).

The challenge with this equation is to determine simultaneously three parameters.

The new method creates a computer spreadsheet containing a set of production data. Then, the specific arrangement of cells and columns containing data and/or formulas snows for statistical fits of the data. Finally, setting-up the nonlinear solver allows for optimization of the three parameters by maximization of the square of the regression coefficient.

Eight field cases demonstrated the repeatable and rapid curve-fitting of the method.

### DECLINE CURVES

The popularity of decline-curve analysis stems from both its simplicity and its success. There are different types of decline curves, as well as different application techniques. Many articles discuss the general aspects of the analysis.

Decline-curve analysis had its beginnings informally in the year 1908.1 Arps2 formalized the analysis for three types of declines: exponential, hyperbolic, and harmonic. In these equations, the problem was to determine the three parameters: n, Di, and qi.

Slider3 offered a means of finding the three parameters using type-curves. Fetkovich4 introduced a solution by graphical means.

These methods have two problems: They are tedious and lack repeatable results. To overcome these problems, Towler and Bansal1 proposed two methods based on linear regression.

But, potentially linear regression on logs of variables imposes more weight on low production rates. Another problem is that the two methods did not produce equivalent results.

The need existed for a method that was not tedious, could produce repeatable results, and would equally weight the production rates during curve-fitting.

A computer spreadsheet equipped with a solver me these requirements.

The solver works by optimizing the square of the regression coefficient using nonlinear equation solvers while varying only the three parameters in the equation.

Also, because many production data bases exist in spreadsheet format, the method provides an interface between the data base and spreadsheet analysis. Further, the technique can be generalized to nonlinear curve-fitting applications.

In this article, the method is not presented as a spreadsheet tutorial. The reader should refer to the user's manual to answer specific questions on the spreadsheet software. The building of the spreadsheet is developed in a stepwise fashion, however, with specific reference to initialization of variables and specific formulas in cells and columns.

### METHOD

The method was developed for Quattro Pro for Windows, Version 1.0. 5 But the 12 steps that follow ar generic enough to adapt to other spreadsheet programs.

For clarification of the steps, refer to Table 1 that illustrates the spreadsheet setup for Well 5, which was selected because it has the smallest set of production data but is representative of the other cases.

- Create a spreadsheet with columns for oil-production rate, qt, (Cells B8 to B31) and for production time (Cells A8 to A31).
- Create a new column for curve fitting qt that begins at a high point from which the remaining oil-production rates steadily decline (Cells D10 to D31). It is sometimes helpful to plot the data in Step 1 for determining this high point, especially for a large data set. All of the next steps depend on the values of qt from this new column.
- To count qt, place a column of numbers, t, next to qt, beginning at zero (Cans C10 to C31).
- Set up cells for the hyperbolic exponent, n, the initial decline rate, Di, and the initial oil-production rate, qi.
It is recommended to fill these cells with initial guesses of 0.5 for n and Di, and the highest value from qt for qi. Initially, cells A5, B5, and C5 contained estimated values for n, Di, and qi of 0.5, 0.5, and 1,520, respectively.

The values shown in the cells are the results after optimization.

- Set up a cell (E5) to calculate the average oil-production rate, q.,g, from (Cells D10 to D31).
- Set up a column (Cells E10 to E31) containing the hyperbolic-decline-curve equation to fit the production, qt' (Cells C10 to C31) based on the values in Cells A5, B5, and C5.
- Set up a column (Cells F10 to F31) to calculate the errors squared with the formula (qt - qt,)2. Values in cells F10 to F31 are the differences in errors squared from the values in the corresponding cells of columns D and E.
- Set up a cell (Cell G5) that contains the sum of the errors squared (SSE) from Step 7.
- Set up a column (Cells G10 to G31) to calculate the total errors squared from the formula (q, - qavg)2. Values in cells G10 to G31 are the differences in the total errors squared from the values in the corresponding cans of columns D and Cell E5.
- Set up a cell (Cell H5) that contains the sum of the total errors squared (SST) from Step 9.
- Set up a cell (Cell D5) to calculate the square of the regression coefficient (r 2) from the formula: r2 (SSE/SST).
- Initialize the solver to optimize the contents of the cells which contain n, Di, and qi, by maximizing the cell that contains r2

It is recommended to run the solver more than once to ensure that r2 has been maximized. The solver is called and initiated to optimize the contents in cells A5, B5, and C5, by maximizing the contents of cell D5. In Quattro Pro this is under Tools/Optimizer. In Excel this is under Formula/Solver.

The solver is run until the value in Cell D5 is maximized and does not change between runs.

An alterative to maximizing r2 is to minimize the sum of the squares of the residuals (SSR). Both give the same result.

The 12 steps in no way are hard and fast. The reader is encouraged to experiment with the arrangement of cells and columns to suit specific applications.

### TESTING THE METHOD

The method was tested with eight different sets of production data. Results are in Table 2. Each case is identified as follows:

- Slider, which was a data set used in Reference 6 for illustrating decline curves.
- Borie field in Wyoming, total monthly production.
- Wells 1-6, from a field in Wyoming's Powder River basin.

Fig. 1 represents curve-fitting of the actual data for each case.

The two cases selected for comparison were Slider and Borie because results, showing the square of the regression coefficient, were published in the study by Towler and Bansal.1 .

The square of the regression coefficient is used as the measure of comparison between the method of this article and the two methods (Method 1 and Method 2) of Towler and Bansal.

Slider's data formed the base case for testing the two methods using linear regression compared to the spreadsheet method. A comparison of the square of the regression coefficients shows:

- Spreadsheet: 0.999842
- Method 1: 0.9997916
- Method 2: 0.9998759

The comparison reveals that Method 2 has the closest fit to 1.

The regression coefficients of the three methods were also compared with data from the Borie field, as follows:

- Spreadsheet: 0.984216
- Method 1: 0.9664694
- Method 2: 0.9641359

The spreadsheet method is closest to 1.

Six anonymous wells from the Powder River basin were also selected to test the spreadsheet method. The regression coefficients for each of well Nos. 1-6, respectively, are: 0.948098, 0.968656, 0,975136, 0.956547, 0.969890, and 0.973983.

ln Table 2 it should be noted that for Wells 1-6, n is greater than 1. An explanation for this is offered in Reference 1.

Other researchers have postulated that n usually fags between 0 and 1, and some have explained why n cannot be greater than 1, or why n cannot be less than 0. Our research has shown numerous field cases for which these reasons are inaccurate.

Reference 1 suggests that n greater than I can sometimes be attributed to fractured or heterogenous reservoirs. In the case for n less than 0, the suggested reason is possible mechanical problems.

This spreadsheet nonlinear regression technique may not always be the most accurate method of fitting data; its accuracy depends somewhat on the source of the errors that cause scattered data.

If the errors are relative errors (relative to the value of the production rate), then the linear regression technique that regresses on the logs of the data (Towler and Bansal's Method 1) may be more accurate.

If the errors are more absolute (that is, independent of the value of production rate), then the nonlinear regression used in this spreadsheet technique may be more accurate.

In certain circumstances, a weighted regression may be more appropriate. This aspect of the problem requires more research.

### ACKNOWLEDGMENTS

Support for this research was provided by the Wyoming Mining and Mineral Resources and Research Institute under the direction of Dr. David O. Cooney. The technique discussed in this article was based on a method used on a different problem by Dr. Melissa Merrill of the University of Wyoming.

### REFERENCES

- Towler, B., and Bansal, S., "Hyperbolic decline-curve analysis using linear regression," Journal of Petroleum Science and Engineering, Vol. 8, 1993, pp.257-68.
- Arps, J.J., "Analysis of decline curves," Transactions AIME, Vol. 160, 1945, pp. "8-47.
- Slider, H.C., "A simplified method of hyperbolic decline curve analysis," JPT, Vol. 20, No. 1968, pp. 235-36.
- Fetkovich, M.J., "Decline curve analysis using type curves," JPT, Vol. 32, No. 6, 1980, 1065-77.
- Quattro Pro for Windows, Version 1.0. Borland Software.
- Slider, H.C., Worldwide Practical Petroleum Reservoir Engineering Methods, PennWell Publishing Co., Tulsa, 1983, p. 542.

### BIBLIOGRAPHY

Arnold, R., and Anderson, R., Preliminary report on Coalinga oil district, U.S.G.S. Bulletin, 1908, pp. 357-79.

Arps, J.J., "Estimation of primary oil reserves," Transactions AIME, Vol. 207, 1956, pp. 182-91.

Fetkovich, M.J., Vienot, M.E., Bradley, M.D. and Kiesow, U.G., "Decline curve analysis using type curves-case histories," SPE Formation Evaluation, Vol. 2, No. 4, 1987, pp. 637-56.

*Copyright 1994 Oil & Gas Journal. All Rights Reserved.*