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API REPORT 81-39 Document Information:
Title
Rheological Evaluation of Cement Slurries
American Petroleum Institute
Publication Date:
May 1, 1982
Scope:
The occurrence of slip, associated with the flow of cement slurries in small diameter tubes, has
been reported by Bannister (1980). Mannheimer (1972) has also shown that high-internal-phase-ratio
emulsions also exhibit slip in tubes. This type of rheological anomaly is characterized by a
decrease in the apparent yiscosity with decreasing tube diameter, and can be readily explained in
terms of a thin film of a low viscosity liquid that effectively lubricates the wails of the tube.
It is important to realize that even a very thin film, only a few micrometers (1 micrometer = 1
× 10−4 cm) thick, can have a profound effect on the apparent viscosity of a highly
viscous or solid-like material.
In order to determine if slip is important in the measurement of the rheological properties of
cement slurries with a rotational viscometer, it is necessary to vary the annular gap (H = Rc-Rb,
where Rc is the inner radius of the cup, and Rb is the outer radius of the bob). However, we must
first show that these changes in the gap size will not significantly change the shear rate. In
order to be able to do this, we will need to consider how the basic rheological variables of shear
stress (τ) and shear rate (γ) are related to the measured torque (M) and the relative
angular velocity of the bob and cup (Ω).
Assuming end-effects are negligible, the shear stress at any radial position τ (Rb ≤ τ
≤ Rc) can be obtained from a simple torque balance:
where the quantities M and L are the measured torque and length of the bob respectively. It is
important to point out that the shear stress does not depend on the rheological properties of the
material in the gap. On the other hand, the shear rate at any radial position in the annular gap is
defined in terms of a velocity gradient:
where ω is the angular velocity of a material element in the θ direction.
Because of this more complex relationship, the shear rate, which cannot be measured directly, will
generally be a function of the rheological properties of the material. As an illustration, consider
the case of a power law fluid for which the shear rate is related to the shear stress by
where K and n are constants. It is important to note that Equation (3) predicts a linear
relationship between log (τ) and log (γ), so that the constants n and K can readily be
determined from the slope and intercept of a log-log plot. Equations (1), (2), and (3) can be
combined (along with the no-slip assumption) and solved to yield, Krieger and Maron (1951):
where Ω is the rotational velocity of the bob relative to the cup,
k = Rb/Rc is the radius ratio of the bob to the cup,
and τb = M/2πRb²L is the shear stress evaluated at the bob.
By comparing the form of Equations (3) and (4), it is evident that the term on the left of Equation
(4) is the shear rate of a power law fluid evaluated at the bob:
For the case of a Newtonian fluid (n = 1) so that Equation (5) becomes
From Equation (6), it is evident that the shear rate of a Newtonian liquid is independent of
viscosity (µ). However, Equation (5) shows that the shear rate of a non-Newtonian liquid
exhibits a complex interaction between the power law constant n and the radius ratio k. This result
suggests that it would be impossible to vary the radius ratio k without affecting the shear rate.
However, we will show that for limited values of n and k the mean shear rate, defined as:
where Rm = (Rb + Rc)/2, differs only slightly from the shear rate of a power law fluid evaluated at
Rm, i.e.:
Equation (8) was obtained from Equation (4) by using the fact that M = 2πRb²τb =
2πRm²τm; and then writing the resulting equation in the form:
In Figure 1 we see the errors that would be involved if either Equation (6) or Equation (7) were
used to estimate the shear rate of a power law fluid. It is evident that Equation(7) estimates the
shear rate better than Equation(6). For example, with n = 0.2, and k = Rb/Rc = 0.9, the error using
Equation (7) is approximately 5%, while the error in Equation (6) is close to 45%. Even less error
would be involved for larger values of k (i.e., less than 1% for k = 0.96); however, in order to
vary the gap size and keep k constant, one would have to simultaneously change the size of both the
bob and the cup. While this procedure might be slightly more accurate, the experimental
repeatability with cement slurries is no better than this 5% error; consequently, the simpler
approach of holding the cup size constant and varying the size of the bob has been used in this
work.
This analysis has shown that relatively small (<5%) error will be introduced by using values of
k such that 0.9 < k < l, provided the power law parameter n is in the range 0.2 ≤ n ≤
1. While these results were derived for a particular rheological model (e.g., the power law model),
it would not be too difficult to show that they should also hold even when n is not a constant.
Therefore, it can be concluded that if the dimensions of the viscometer is such that 0.9 ≤ k
< 1, and if the measured torque-rotational speed data are expressed in terms of log (τm) Vs
log (V/H), then the error in using V/H for the shear rate will be less than 5% provided that the
slope of the curve at any point is greater than 0.2 and that slip is negligible.
We will now consider the effect of varying the gap size of a rotational viscometer when slip is
significant. The Hershel-Bulkley model will be used for these calculations:
This model provides for a yield value (τo) and a shear dependent viscosity described by the
power law constants n and K. This three-parameter model is extremely versatile in that it reduces
to the power law model for τo = O, to the Bingham model for n = 1, and to the Newtonian model
for τo = O and n = 1. If it is further assumed that the quantity V/H is an adequate
approximation of the shear rate when-slip is negligible, then Equations (10a) and (10b) can be
written as:
In Equations (11a) and (11b), we have introduced the concept of a slip velocity Vs. The true shear
rate corrected for slip is given by the quantity H) even when the shear stress is less than
the yield value. Furthermore, the effect of slip on the shear rate is tied directly to the gap size
(H). In order to calculate the slip velocity as a function of the shear stress, we will consider a
model described by Oka (1960). In this case, slip takes place in a thin film adjacent to the wall.
If the film thickness (E) is much smaller than the gap (H), then the slip velocity is related to
the shear stress by:
where µs the viscosity of film. For the purpose of illustration, we will assume that E = 2.5
× 10−4 cm and that the viscosity of the film is essentially that of water, i.e.,
µs = 1 × 10−² dynes - s/cm = 1 m Pas = 1 cP. The results of all these
assumptions are summarized in Figure 2 in which we have plotted log (τm) as a function of log
(V/H). The characteristics shear
behavior of a material with a yield value without slip (i.e., VS = O) is illustrated by the
darkened squares in Figure 2. As one approaches the yield value (τo = 70 dynes and log 70 =
1.845) from slightly higher shear stresses, the slope of the no-slip curve approaches zero as an
asymptote. It is in this range that the error due to the use of V/H to approximate the shear rate
becomes significant. From a practical point of view, rheological measurements are not easily
obtained in this region (i.e., rotational viscometers operate at finite shear rates). Instead, the
yield value is generally obtained by extrapolating the shear stress to zero shear rate on a linear
plot.
The effect of slip in Figure 2 has been calculated for the different gap sizes that were actually
used in this work. These results show that the apparent viscosity is a function of the gap size,
and that the lowest apparent viscosity occurs in the smallest size gap (0.714 mm). At shear
stresses well above the yield value, slip produces what appears to be a family of straight lines
thatarecharacteristic of power law fluids. It is important to note that this is the same result
that was observed experimentally for the flow of cement slurries in tubes by Bannister (1980).
Furthermore, the slopes of these lines differ only slightly. Consequently, the primary effect of
slip appears to be in the intercept if these lines are extrapolated to log V/H = O. Based on these
experimental observations, Bannister has proposed a method for determining the slip velocity from
the extrapolated values of K; however, these theoretical results illustrate the potential danger of
extrapolating rheological measurements.
It is important to note that the effect of slip in Figure 2 is most apparent at low shear stresses.
In particular, at shear stresses below the yield values, results for different size gap become a
series of parallel lines with unit slope. This is due to the particular slip model that was chosen
for this illustration (i.e., that both the viscosity and thickness of the slip film are independent
of the shear stress). Of particular significance is the distinctly different shape of the flow
curves in the low stress regime from the predicted shape of the flow curve when no slip film is
present. To summarize, it is expected on theoretical grounds that even a very thin film
of low viscosity liquid can have a profound effect on rheological measurements. Furthermore, it is
predicted that the slip effect will be most prominent at low shear stresses, and that a distinct
change in the shape of the flow curve should be evident at shear stresses below the yield value
when slip is important.
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