Rheometer

A rheometer is a device that measures the way a viscous fluid flows in response to applied forces.

The relation between the strain and stress tensors is an important information when solving the Navier-Stokes flow momentum conservation equation:

$\rho {\Bigl (}{\partial {\vec {u}} \over \partial t}+{\bigl (}{\vec {u}}.\bigtriangledown {\bigr )}{\vec {u}}{\biggr )}=-\bigtriangledown p+\bigtriangledown .{\overline {\overline {\tau }}}+\rho {\vec {g}}+\rho {\vec {f}}$

where ${\vec {u}}$ is the velocity vector, t is time, $\rho$ is the mass density, $p$ is the pressure, ${\overline {\overline {\tau }}}$is the stress tensor, ${\vec {g}}$ is the gravitational field and ${\vec {f}}$ represents external body force on the fluid. Using the generalized Newtonian fluid convention, the relationship between stress and strain rate tensors is:

${\overline {\overline {\tau }}}=\eta ({\dot {\gamma }},t){\overline {\overline {\dot {\gamma }}}}$

where ${\overline {\overline {\dot {\gamma }}}}$ is the strain rate tensor and $\eta$ is the effective viscosity, which is a function of the shear rate (${\dot {\gamma }}$) and possible time. The shear rate is the second invariant of the strain rate tensor:

${\dot {\gamma }}={\sqrt {{1 \over 2}\textstyle \sum _{ij}\displaystyle {\dot {\gamma }}_{ij}{\dot {\gamma }}_{ji}}}$

where ${\dot {\gamma }}_{ij}$ are the components of the strain rate tensor.

It is the purpose of a rheometer to measure the strain-stress relation of a particular viscous fluid, i.e., $\eta ({\dot {\gamma }},t)$.

Rheometer Types

There are essentially four types of shear geometries:

1. Couette drag plate flow
2. Cylindrical flow
3. Poiseuille flow in a tube
4. Plate-plate flow

Couette Concentric Cylindrical Rheometer

A typical rheometer geometry used in the pretroleum industry is the Couette concentric cylindrical rheometer. This correspnonds to the rheometer type 1 in the previous list. The rheometer is constituted of a concentric cylindrical bob inside a cylindrical cup. There are two possible arrangements for the rotating element:

1. a rotating cylinder, called the rotor, rotates inside a fixed cup around a fixed bob (this is the principle used in the Model 35 rheometer used for the measurment of drilling fluids at the rig site).
2. the bob rotates inside a fixed cup (this is a typical configuration for scientific rheometers)

1) a rotor-bob configuration 2) a rotating bob configuration

In both cases the rheometer measures the torque on the bob. The rotational speed of the rotating element is varied to obtain different shear conditions.

The possible rotational speeds can either be fixed when using a mechanical system or adjustable when using an electronic speed controller. A Model 35 rheometer has typically 6 or 8 speeds. The speeds used by an Anton Paar MCR301 or MCR302 rheometer are user-controllable.

The relation between the rotation speed and the shear rate is:

$\Omega =\int _{r_{b}}^{r_{c}}{{\dot {\gamma }}(r) \over r}dr$

where $\Omega$ is the angular velocity, $r$ is the radial distance, $r_{b}$ and $r_{c}$ are respectively the radius of the bob and the radius of cup or rotor. This can be re-written as :

$\Omega ={1 \over 2}{\int _{\tau _{c}}^{\tau _{b}}{{\dot {\gamma }}(r) \over r}dr}$

where $\tau _{b}$ is the shear stress at the wall of the bob and $\tau _{c}$ is the shear stress at the wall of the cup or rotor .

For a Newtonian fluid, the conversion from the angular velocity to the shear rate at the wall of the bob is constant : ${\dot {\gamma }}_{b}={{2\xi ^{2}} \over {\xi ^{2}-1}}\Omega$ where $\xi ={r_{c} \over r_{b}}$ and ${\dot {\gamma }}_{b}$ is the shear rate at the wall of the bob.

For a Model 35 in the configuration R1B1, $r_{b}=17.245$ mm, $r_{c}=18.415$ mm and ${{2\xi ^{2}} \over {\xi ^{2}-1}}{2\pi \over 60}=1.702295$ when converting angular velocity to RPM. It should be noted that the ISO-3219 convention is to use the average shear rate in the gap. For a Newtonian fluid, ${\dot {\gamma }}_{a}={{1+\xi ^{2}} \over {\xi ^{2}-1}}\Omega$ where ${\dot {\gamma }}_{a}$ is the average shear rate in the gap. This is the convention used by certain scientific rheometers like those from Anton Paar. For non-Newtonian fluids, the conversion between the angular velocity and the shear rate at the wall depends on the rheological behavior. Skadsem and Saasen provides a method to calculate the shear rate at the wall for a fluid that follows a Herschel-Bulkley rheological behavior, i.e., $\tau =\tau _{\gamma }+K{\dot {\gamma }}^{n}$ where $\tau _{\gamma }$ is the yield stress, $K$ is the consistency index and $n$ is the flow behavior index.

The torque on the bob is the integral of the shear stress at the wall: $C_{b}=\iint \limits _{S}\tau dA$ where $C_{b}$ is the torque on the bob, $S$ is the surface of the bob. Considering that the bob is cylindrical, the torque can be be related to the surface of a cylinder $S=2\pi r_{b}^{2}h$ where $h$ is the height of the bob. However, the shear stress at the wall also applies on the top and bottom of the bob, and to account for these end effects, it is often considered that the height of the bob is slightly longer than its true value:

$\tau _{b}={{C_{b}} \over {2\pi r_{b}^{2}h(1+\varphi )}}$

where $\tau _{b}$ is the torque on the bob and $\varphi$ is a correction factor for the end effects.

For Newtonian fluids, $\varphi$ is constant and is equal to $\approx$ 0.064 for a Model 35 and 0.1 for an Anton Paar MCR301/302. However, an Anton Paar rheometer provides the average shear stress in the gap between the bob and the cup, as according to the ISO-3219 convention. The relationship between the two shear stresses is:

$\tau _{a}={{1+\xi ^{2}} \over {2\xi ^{2}}}\tau _{b}$

where $\tau _{a}$ is the average shear stress in the gap. For non-Newtonian fluids, $\varphi$ depends on the rheological behavior. Lac and Parry have made computational fluid dynamics (CFD) simulations for Herschel-Bulkley fluids with different configurations of the Model 35 rheometer (R1B1, R1B2 and R1B5) and provided formulas to estimate the end-effect correction factor for that type of rheometer.

Both the shear rate correction from Skadsem and Saasen and the end-effects corrections from Lac and Parry are implemented in C# and made available as open source by the SPE sub-committee Open Source Drilling Community. The code is available on github here. A simple web application running the open-source code is available here.

Pipe Rheometer

In a pipe rheometer, the fluid flows into a cylindrical pipe and the pressure loss along the pipe is measured. This corresponds to the second type of rheometer in the above classification. The flowrate is varied and/or the fluid passes through multiple pipe sizes to create different shear conditions.