API sedimentation tests
June 8, 2020Compressive strength development at high temperature
December 10, 2020Errors in rheological measurements
Maximum recommended rotation speed
Ramp up and ramp down measurements

Reading 

Rotational Speed (rpm) 
Rampup 
Rampdown 
Ratio 
Average 
3 
8 
5 
1.6 
6.5 
6 
9 
7 
1.29 
8 
30 
18 
16 
1.13 
17 
60 
27 
26 
1.04 
26.5 
100 
39 
39 
1 
39 
200 
69 
70 
0.99 
69.5 
300 
97 
 
 
97 
Rheometer configuration 
R1B1F1 

Conditioning 
Atmospheric pressure, 30 minutes at 50°C 

Initial slurry temperature 
50°C 

Final slurry temperature 
48°C 
Model fitting  Bingham Model
Even though curve fitting is easy to do with spreadsheets there is still a tendency for people to use just two data points to fit the data, which was the case with the peer reviewed journal article (using 600 and 300 rpm data points). API RP 10B2 clearly states that 2point calculations should not be used.
As an example, I will use the average data points from the above table  I have converted the rpm and dial reading to shear rate (s^{1}) and shear stress (Pa) for the following analysis. Data fitting for the Bingham model has been done using the built in linear regression function in Excel.
The results for the fit are shown in Figure 1.
The curve fit using just the two data points at the highest shear rates does not match the data over the entire range and gives a yield stress value of 7.6 Pa. The curve fit using all the data points give a much better fit of the data over the entire range. The yield stress value is 3.7 Pa. The reading at 5.1 s^{1} (3 rpm) is 3.3 Pa.
In most cases fitting the data using all the data points will give a higher plastic viscosity and lower yield stress than using a twopoint data fit at high shear rates. Exceptions will be fluids exhibiting dilatancy (shear thickening).
The differences between the fits and the actual data can be seen more clearly on a loglog (Figure 2). The fit using all 7 data points overestimates the low shear rate viscosity but is much closer to the data than the twopoint fit.
Plotting the data on a loglog plot also shows that there are no signs of slippage at the wall, as the data points follow a smooth curve.
Model fitting – HerschelBulkley model
The HerschelBulkley model is a threeparameter model that combines power law and Bingham plastic behaviour.
τ = τ_{y }+ kγ̇^{n}
Where k is the consistency index, and n is the power law index.
The HerschelBulkley model simplifies to the Bingham model (if n=1) and to the power law model with τ_{y}=0.
Microsoft Excel can also be used to fit the HerschelBulkley model to a set of data as follows:
 Convert the rpm values and average dial readings to shear rate (s^{1}) and shear stress (Pa) respectively.
 Estimate a yield stress value, this can be from 0 to the shear stress determined at 3 rpm (5.1 s^{1}).
 Use this estimated yield stress to perform a linear regression on log((shear stress – estimated yield stress)/Pa) v log(shear rate/s^{1}) to give n and k.
 Calculate the sum of the squares of the differences between the data points and the calculated values.
 The Excel solver function can then be used to determine the value of the yield stress, and the associated n and k values, that minimises the sum of the squares.
The best fit to the data above has:
τ_{y}=2.6 Pa, n=0.897, k=0.175 Pa.s^{n}
The fit is plotted in Figure 3:
There is a more detailed discussion of fitting rheological models to data of drilling and cementing fluids in:
Hans Joakim Skadsem and Arild Saasen, “Concentric cylinder viscometer flows of HerschelBulkley fluids”, Appl. Rheol. 2019; 29 (1):173–181.
This is an open access paper available at: https://doi.org/10.1515/arh20190001