q921 de2 lec4 v1

Drilling Engineering 2 Course (1st Ed.)

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1. Mud Weight Planning

2. drilling hydraulics: A. the hydrostatic pressure

1. drilling hydraulics: A. types & criteria of fluid flow

B. fluid Rheology and modelsa. Bingham plastic & Power-law models

Parameters influence rheological properties of drilling fluidSince multiple aspects of drilling and completion

operations require the understanding of how fluid moves through pipes, fittings and annulus, the knowledge of basic fluid flow patterns is essential.Generally, fluid movement can be described as laminar,

turbulent or in transition between laminar and turbulent.It should be understood that rotation and vibrations

influence the rheological properties of drilling fluids.Also the pulsing of the mud pumps cause variations in

the flow rates as well as the mean flow rates. Furthermore changing solid content influences the actual mud density and its plastic viscosity.

Fall 13 H. AlamiNia Drilling Engineering 2 Course (1st Ed.) 5

Laminar vs. turbulent flow

Fluid movement, when laminar flow is present, can be described as in layers or “laminae”. Here at all times the direction of fluid particle movement

is parallel to each other and along the direction of flow.In this way no mixture or interchange of fluid particles

from one layer to another takes place.

At turbulent flow behavior, which develops at higher average flow velocities,

secondary irregularities such as vortices and eddys are imposed to the flow. This causes a chaotic particle movement and

thus no orderly shear between fluid layers is present.


Ideal laminar flow (animation)

Ideal laminar flow in a tube

(note that the particles to the center of the tube move faster, as affected to a lesser extent dissipative effect of the walls)


Laminar, transitional, and turbulent flow

Laminar vs. turbulent flow Laminar, transitional, and turbulent flow from a faucetFall 13 H. AlamiNia Drilling Engineering 2 Course (1st Ed.) 8

Laminar vs. turbulent flow

Laminar and turbulent water flow Laminar vs. turbulent flow of smokeFall 13 H. AlamiNia Drilling Engineering 2 Course (1st Ed.) 9

Reynolds number

The so called “Reynolds number” is often used to distinguish the different flow patterns. After defining the current flow pattern,

different equations are applied to calculate the respective pressure drops.

For the flow through pipes, the Reynolds number is determined with:

and for the flow through annuli:


Reynolds number range

The different flow patterns are then characterized considering the Reynolds number. Normally the Reynolds number 2,320 distinguishes

the laminar and turbulent flow behavior,

for drilling purposes a value of 2,000 is applied instead.

Furthermore it is assumed that turbulent flow is fully developed at Reynolds numbers of 4,000 and above,thus the range of 2,000 to 4,000 is named transition flow:


Rheological Classification of Fluids

All fluids encountered in drilling and production operations can be characterized as either “Newtonian” fluids or “Non-Newtonian” ones.Newtonian fluids,

like water, gases and thin oils (high API gravity) show a direct proportional relationship between the shear stress and the shear rate, assuming pressure and temperature are kept constant. They are mathematically defined by:

• 𝜏 [dyne/cm2] ... shear stress

• 𝛾[1/sec] ... shear rate for laminar flow within circular pipe

• μ [p] ... absolute viscosity [poise]


Newtonian flow model

A plot of

𝜏 vs. −𝑑𝑣𝑟

𝑑𝑟produces a straight line that passes through the origin and has a slop of μ.


Non-Newtonian fluids

Most fluids encountered at drilling operations like drilling muds, cement slurries, heavy oil and gelled

fracturing fluids do not show this direct relationship between shear stress and shear rate.

They are characterized as Non-Newtonian fluids.

To describe the behavior of Non-Newtonian fluids, various models like “Time-independent fluid model” including

the “Bingham plastic fluid model”,

the “Power law fluid model” and

“Time-dependent fluid models” were developed


Non-Newtonian fluidstime depended modelsThe time dependence mentioned here concerns

the change of viscosity by the duration of shear.

It is common to subdivide the time depended models into “Thixotropic fluid models” and The “Rheopectic fluid models”.

It shall be understood that all the models mentioned above are based on different assumptions that are hardly valid for all drilling operations, thus they are valid to a certain extend only.


Bingham plastic fluid model


𝜏y[lbf/100 ft2] yield point

μp [cp] plastic viscosity

Sketch of Bingham fluid modelFall 13 H. AlamiNia Drilling Engineering 2 Course (1st Ed.) 19

Bingham fluids

In contrary to Newtonian fluids, Bingham fluids do have a yield point 𝜏𝑦 and

it takes a defined shear stress (𝜏𝑡) to initiate flow.

Above 𝜏𝑦, 𝜏 and 𝛾 are proportional defined by the viscosity, re-named to plastic viscosity μp


Power-law fluid model


n [1] flow behavior index

K [1] consistency index

Sketch of Power-law fluid modelFall 13 H. AlamiNia Drilling Engineering 2 Course (1st Ed.) 21


When the characteristics of the Power-law fluid model is done on a log-log scale, the results is in a straight line. Here the slope determines the flow behavior index n and

the intercept with the vertical, the value of the consistency index (logK).

The flow behavior index (n), that ranges from 0 to 1.0 declares the degree of Non-Newtonian behavior,

where n = 1.0 indicates a Newtonian fluid.

The consistency index K on the other hand gives the thickness (viscosity) of the fluid where,

the larger K, the thicker (more viscous) the fluid is.


rheological properties determination

To determine the rheological properties of a particular fluid, a rotational viscometer with six standard speeds and variable speed settings is used commonly.

In field applications, out of these speeds just two are normally used(300 and 600 [rpm])

since they are sufficient to determine the required properties.

rotational ViscometerFall 13 H. AlamiNia Drilling Engineering 2 Course (1st Ed.) 23

individual fluid parameters determinationNewtonian fluid model



r2 [in] rotor radius, r1 [in] bob radius, r [in] any radius between r1 and r2, θN [1] dial reading of the viscometer at speed N, N [rpm] speed of rotation of the outer cylinder


1. Dipl.-Ing. Wolfgang F. Prassl. “Drilling Engineering.” Master of Petroleum Engineering. Curtin University of Technology, 2001. Chapter 4