ANALYSIS IMPROVES SELECTION OF RHEOLOGICAL MODEL FOR SLURRIES

Khaled Moftah
ECG Engineering Consultants Group SA
Cairo

The use of a statistical index of determination can help select a fluid model to describe the rheology of oil well cement slurries. The closer the index is to unity, the better the particular model will describe the actual fluid behavior.

The Theological behavior of a cement slurry (and most muds) typically follows one of the fundamental models shown in Fig. 1.1 Cement slurries for oil wells are normally treated as Bingham plastic or power law fluids. Many engineers have a preference for one model or the other and tend to label the fluid accordingly.

The Bingham plastic model is a linear regression of the shear stress as the dependant variable and the shear rate as the independent variable. The power law model is the linear regression of the logarithms of the shear stress and the shear rate on the same axes.

In the Bingham plastic model, the yield stress is the intercept of the regression line with the ordinate (shear stress). This model has highest shear stress yielding a zero shear rate. The slope of this line is the fluid's plastic viscosity. One drawback inherent in this model is that some regression analyses can produce a negative yield stress, and a negative yield stress is meaningless.

The minimum yield stress of zero is associated with true Newtonian fluid. Only time-dependant models show shear rate while there is no stress during the relaxation period.

NEGATIVE YIELD STRESS

Why do some slurries produce negative yield stresses? The negative yield stress either may be related to the slurry itself or may be a statistical phenomenon.

The regression line in the Bingham plastic model is the best linear fit according to the least mean square method.

This method minimizes the sum of the squares of the errors between the data points and the corresponding points on the regression line. If the apparent viscosity (shear stress divided by the shear rate) increases at higher shear rates, the final slope of the line will be high, and the line will probably intersect the ordinate at a negative value.

Cement slurries typically face shear thickening phenomena during turbulent conditions (Fig. 2). The dashed portions of the curves in Fig. 2 lie in the turbulent region. Successful Bingham plastic modeling should exclude from the regression those points measured at turbulent conditions .2

With the Farm 35 VG viscometer speed set at 600 rpm and sometimes at 300 rpm, fluid turbulence may occur. In these cases, excluding the 600-rpm reading from the regression can improve the analysis, but the 300 rpm reading cannot be excluded because of the limited number of measured points.

The field versions of the Fann viscometer have 600 and 300-rpm readings only. Thus, the results in the field may be questionable as a basis for the calculations of the Theological parameters and the subsequent calculations of the critical pump rate and the pressure drop.

CONFIDENCE INTERVAL

The other source of a negative yield stress is chance error. Chance error may convert slightly positive yield stress into negative values.

A large chance error widens the scatter of the data points. This scattering is best illustrated by the level of confidence. For a 90% level of confidence, the probability that a certain value (the yield stress, for example) lies within the calculated range of confidence (the confidence interval) is 90%. If the data points fit exactly on the regression line, the range around the optimum regressed value for any level of confidence is zero That is, it is certain that the points he exactly on the regression line.

For a normal distribution, 68.0% of the points are expected to he within Se (the standard error of the estimate), 90.0% within 2Se, and 99.7% within 3Se. After regression, any point having an absolute error (the vertical distance between the measured point and the regressed line) greater than 3S, may be omitted from the calculation. The regression is then recalculated to obtain a more probable line.

These points are omitted because it is highly improbable that actual points lie farther than 3Se from the regression line. The probability of that occurrence is 0.3%. Fig. 3 shows the scattering of data around a regression line. 3

The points measured in the turbulent region are widely scattered from the linear portion and can therefore be thought of as resulting from chance error. This assumption in turn leads to a wide confidence interval for a selected confidence level.

Applying the statistical rules can thus help in deciding whether to eliminate the 600-rpm reading from the regression analysis. These same statistical rules can be applied to any measured point (that is widely scattered from other points) to eliminate it from the linear regression.

An examination of the equation for the residual variance of any point reveals that the farther a point is from the middle of the line, the wider the confidence interval. Thus, unfortunately, the yield stress (the Y intercept) has the widest confidence interval. That is, the exact value is more uncertain (Fig. 4).4

PROGRAM

Table 1 lists a computer program written in Quick Basic to calculate Theological parameters and an index of determination for the Bingham plastic and power law models. The points used for the calculation of the rheological parameters can be selected from the data set. The skipped points can then be introduced and the calculations continued, not restarted, to obtain the parameters for the full set of data. The two sets of results are then compared for the decision to include or exclude the added points in the regression. Table 2 is a printout of an example run.

The program also calculates the apparent viscosity to help determine where turbulence or high gross error occurred.

In addition, the program calculates the confidence interval of the Theological parameters for a 90% level of confidence. The selected level of confidence may be changed, and for this change, the t value should be changed accordingly. The statistical equations are listed in the accompanying boy and Table 3 lists the t values. 3 5

The accompanying example box shows the calculation of Theological parameters for two Bingham plastic fluids.

RESULTS

• Excluding the Farm viscometer 600-rpm reading from the regression analysis may convert slightly negative yield stress values into positive values, lowering the plastic viscosity. Normally, this method improves the correlation factor (index of correlation). The resulting improved Theological parameters can be used in calculating critical pump rate and the pressure drop.

• If a confidence interval of the yield stress contains negative and positive values, it must therefore include zero. A line stemming from the graph origin represents a true Newtonian fluid model (in which the critical Reynolds number is 3,000 and the corresponding Hedstrom number is 100). The calculations based on that model are not statistically different from those based on a line selected within the confidence interval of a Bingham plastic correlation.

• One procedure, which is sometimes used by field engineers, is to assume a small positive fractional value of the yield stress and use it instead of the negative value, but without changing the plastic viscosity. This method is considered conservative because it uses the higher values for both the plastic viscosity and the yield stress. In many cases in the field one decimal fraction used for the yield stress produces a Reynolds number 3,300 and Hedstrom number 1,000.

• If more than one point is widely, scattered from a regressed line, the Theological test should be run again.

• Other models (power law, for example) may fit the data points well while the Bingham plastic produces a bad fit (lower index of correlation). It is advisable to treat each cement slurry individually for the best model to fit and not to assume general preference of any particular model.

• Based on the assumption of normal distribution, for a fluid model to be significant or fit well, there is a minimum for the index of determination associated with the number of data points. A value of the index of determination less than this minimum means that the selected fluid model is insignificant, and another model should be tried. This minimum value of the index of determination is 0.9877 for three points and a 90% level of confidence. The corresponding value for 4 points is 0.8100.

ACKNOWLEDGMENT

The author would like to thank Professor M.A. Shalaby for reviewing this article.

REFERENCES

1. Douglas, J.F., Gasiorek, J.M., and Swaffield, J.A., Fluid Mechanics, second edition, English Language Book Society.

2. American Petroleum Institute Task Group on Cement Rheology, Exxon Production Research Co., Houston, April 1981.

3. Levin, R.I., and Rubin, D.S., Applied Elementary Statistics, Prentice-Hall Inc.

4. Hawkins, C.A., and Weber, J.E., Statistical Analysis, Harper & Row Publishers Inc., 1980.

5. Perry, R.H., and Chilton, C.H., Chemical Engineers' Handbook, fifth edition, McGraw-Hill Kogakusha Ltd.

Copyright 1993 Oil & Gas Journal. All Rights Reserved.