A general discussion on rheology

Overview

Before going into the details of rheology and what it is, let’s consider the effects on external force on a purely solid body such as a block of metal and a fluid body like water. The normal intuition or anyone with a high school level understanding of physics would know that the force required to move any object is directly proportional to its mass. So that means, heavier the object the larger the force required to move

The same principle applies to fluids as well. Since fluids move more freely than solids, fluids are generally considered as a continuum and the force required to move a fluid body is directly proportional to its mass.

Now what happens when the fluid contains suspensions of minute solids particles? Is the force required to move the fluid suspended with solids particles proportional to its mass? The answer is yes of course, but it is no longer directly dependent. Fluids with suspended particles in it are called Particulate Fluids. The force required to move such fluids alters depending on the current velocity of the fluid. The required external force to increase fluid velocity either increases or decreases depending on the composition of the fluid.

Due to these behaviors, the fluids are categorized in to two types. Fluids that have the direct proportional relationship between force and mass are called Newtonian Fluids while those that don’t are called Non-Newtonian Fluids.

Rheological Models

The resistance of the fluid that has been discussed for so long is termed the Viscosity of the fluid. For evaluating viscosity, multiple mathematical models have been developed over the years which accurately determines the fluid resistance. The commonly used models in the field are:

• Power Law
• Hershel Bulkley

The viscosity profile for Power Law and Hershel Bulkley fluids are the same. The difference is that Hershel Bulkley fluids have another characteristic property called the Yield Point. The Yield Point is basically the threshold force required to make the fluid move from rest. Consider it like inertia for a liquid body. Not all fluids have this property (like water) but there are many lab studies conducted that show the presence of a Yield Point in Non-Newtonian fluids.

Mathematically, the equations for Power Law and Hershel Bulkley have the following form,

Power Law : τ = Kγⁿ

Hershel Bulkley : τ = τ_y + Kγ^m

• τ - Shear Stress
• τ_y - Yield Point
• K - Consistency Index
• γ - Shear Rate
• n, m - Flow Behavior Index

This clearly looks pretty complex for those not familiar with mathematical models. To make it simple, the shear stress is equivalent to the external force and the shear rate is equivalent to fluid velocity. The ‘K’ is termed Consistency Index which is the value of fluid resistance or Viscosity. The ‘n’ or ‘m’ basically tells us the extent of Non-Newtonian behaviour. If they are less than 1, then the fluid is Shear Thinning and if they are greater than 1, the fluid has Shear Thickening behaviour. If it is equal to 1, then the fluid behaves in a Newtonian manner i.e., the viscosity does not change with increasing fluid velocity.

There are many other rheology models beyond these two models but are more complicated with multiple parameters in it. The details of these models are not discussed here. There are excellent sources available for a detailed explanation of these models.

Model Fitting

So now we have various models defined. How do we choose the right model? Normally, with lack of enough information drilling engineers generally choose the model which they are comfortable with. The commonly chosen models are either Bingham Plastic (not discussed cause its outdated) or Power Law. The engineer simply decides the PV and YP values he/she desires and then calculates the chosen rheology parameters.

In the field however, we have rheological data obtained from the rheogram readings to determine the rheology that fits drilling fluid that has been prepared. The workings of the equipment are not discussed here but in a simplistic manner, the equipment determines the force needed to move the fluid at various speeds. So, these values are converted to a chart as shown,

RPM is the shear rate or the speed and the Dial is the shear stress or the force required to move the fluid.

As you would have observed, the data are just points and not a continuous line. With this data a curve fitting algorithm is performed which would identify the model that best fits the data. The end result would be something that looks like the picture below,

For the example above the best fit rheology model was Power Law and the parameters are:

• a. K – 5.87
• b. n - 0.39

Now with these parameters, we can determine the viscosity at any fluid velocity which is what we need.

Final Remarks

To conclude, all drilling fluids are generally Shear Thinning in nature. It is preferable to have a rheology that has a yield point so that the fluid has suspension characteristics to suspend any cuttings and other solids binding the fluid when the fluid becomes stationary. With the rheological model determined, the pressure required to circulate the fluid can be determined. The critical area to focus on is the annulus (the section where the fluid returns to the surface). If the pressure required in the annulus is too high, then the formation may fracture which leads to many other complications such as loss circulation and well control. Evaluating such requirements can provide the engineer with the limitations of the system during the design stage which allows him/her to put forth contingencies and alternative design options.