The use of statistical methods can improve the selection of a rheological model and the subsequent calculations for critical pump rate and pressure drop for cementing operations.

The accompanying interactive Basic computer program allows the user to analyze fluid rheology to help determine the best data for use in predicting cementing pump rates. An accurate critical pump rate and pressure drop can then be calculated based on the correctly calculated rheological parameters.

The first step in calculating the critical (minimum) pump rate for turbulent flow, and consequently the frictional pressure drop of a slurry at that rate, is the analysis of the rheological data and the selection of the mathematical model that best fits the actual behavior of the slurry.

For cementing operations, the important methods of calculating the critical pump rate are the Hedstrom analysis, based on the Bingham plastic rheological model, and the Nletzner and Reed analysis, based on the power law rheological model.

The equivalent diameter and hydraulic area must be calculated first to determine the critical pump rate and pressure drop (Box 1).1

Some cementing manuals perform the Hedstrom analysis and the Metzner and Reed analysis, as shown in Boxes 1 and 2, respectively. Alternatively, the Bingham plastic parameters can be converted for use in the power law model. The calculations using the concrete power law model are the compared to calculations using the proper power law model (Box 4).23 In this procedure, the consistency index K', is converted into a corrected consistency index K. This K value replaces the K' values in the Box 3 calculations. Tables 1 and 2 list the critical Babbitt and Caldwell and the critical Metzner and Reed modfdiedy nolds numbers, respectively.

Many cementing companies and operators use different empirical equations and definitions of boundary turbulence. Thus, their analyses may produce varying results for the same input data. In one case, a 25% deviation was found for the critical pump rate calculations.4 All of the methods, however, give the same result at the limiting case ofNewtonian In a Newtonian mode, the yield stress, Ty of the Bingham plastic mode equals zero, and the flow behavior index n', of the power law model equals one.

An index of determination can indicate the preference of one mathematical model over another for regression of the rheological data model. The closer the index is to one, the better the fit of the selected rheological model in describing the actual behavior of the slurry.

A statistical analysis can improve the index by determination by discarding highly scattered points from the regression. These analyses for the Bingham plastic model were discussed in a previous article (OGJ, Oct. 25, 1993, pp. 82 88).5 The same principles are applicable to the power law model which is a linear regression of the logarithms of the shear rate and the shear stress. The calculated confidence interval is therefore valid for the logarithms (that is, the linear model).6

The larger the confidence interval, the more scattered the points from the selected model. The same indication is obtained by the coefficient of determination. Removing the widely scattered points from the regression gives a narrower confidence interval and a higher coefficient of determination.

### PROGRAM

The accompanying Basic computer program calculates a statistical analysis for the Bingham plastic and power law models. The program allows the user to add or remove points from the regression and recalculate the regression without having to restart the program This interactive feature allows easy calculation of the rheological parameters.

In certain rare cases and because of rounding errors, the residual variance acquires a very small negative value. Because the root of this value is used in calculating the confidence interval and other statistics, an error message is prompted in these cases. This problem is overcome in the program by replacing any negative value for the residual variance with zero.

The program reports the plastic viscosity, the yield stress, the confidence interval for both values at a 90% confidence level, and the standard error of estimate for the Bingham plastic model. For the power law model, the program outputs the flow behavior index the consistency, index the confidence interval of the intercept (100 x log K') and the slope (n') for the linear model, and the standard error of estimate.

Because the index of determination is rounded to five decimals, some calculations can produce a value of one yet report some nonzero values for the standard error of estimate and the confidence intervals. These cases are quite obvious, and no harm occurs. The user should remember that in most cases a value of one for the index of determination is just an approximation.

### RESULTS

In the accompanying example, the equations in Boxes 1 4 and the computer program are used to determine the critical pump rate and pressure drop for cementing a 7 in. casing in an 8 1/2 in. hole. The pressure drop calculated for turbulent flow is 184 psi/1,000 ft at a flow rate of 7.37 bbl/min, based on the Bingham plastic model and including the 600 rpm Fann viscometer reading in the regression. The same calculations for the power law model yield 150 psi/1,000 ft pressure drop.

The pressure drop value with the highest degree of confidence is obtained from the Bingham plastic model with the 600 rpm viscometer reading excluded from the regression. This pressure drop is 162 psi/1,000 ft at a flow rate of 7.06 bbl/min. This value has an index of determination of one. The power law model, in this case, gave a pressure drop of 130 psi/1,000 ft a the critical pump rate and had an index of determination of 0.99974.

### REFERENCES

- Savins, J.G., "Generalized Newtonian Fluid Flow in Stationary Pipes and Annuli," Society of Petroleum Engineers transactions, Vol. 213, 1958.
- American Petroleum Institute Task Group on Cement Rheology,, Exxon Production Research Co., Houston, April 1981.
- McCabe, W., Smith, J., and Harriott, P., Unit Operations of Chemical Engineering, McGrawHill Book Co., 1987.
- Suman, G.O. Jr., and Ellis, R.C. "Cementing Oil and Gas Wells,', World Oil, 1971.
- Moftah, K.R., "Analysis improves selection of rheological model for slurries," OGJ, Oct. 25, 1993, pp. 82 88.
- Hawkins, C.A., and Weber, J.E., Statistical Analysis, Harper & Row Publishers Inc., 1980.

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