Q921 de2 lec5 v1

Drilling Engineering 2 Course (1st Ed.)

1. drilling hydraulics:
A. types & criteria of fluid flow
B. fluid Rheology and models
a.

Bingham plastic & Power-law models

1.
2.
3.
4.
5.
6.

Laminar Flow in Pipes and Annuli
Turbulent Flow in Pipes and Annuli
Pressure Drop Across Surface Connections
Pressure Drop Across Bit
Optimization of Bit Hydraulics
Particle Slip Velocity

laminar flowing pattern application
For drilling operations
the fluid flow of mud and
cement slurries
are most important.

When laminar flowing pattern occurs,
the following set of equations can be applied
to calculate
the friction pressure drop [psi] Δp,
the shear rate at the pipe wall 𝛾 𝑤 and
the circulation bottom hole pressure
for the different flow models:
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Laminar:
Newtonian Fluid model
Flow through pipe:
Flow through annulus:
When comparing
the mean velocity υ with
the so called “critical velocity”,
denoted by υc (υcan, υcp),

the fluid flow pattern can also be determined.
This classification is given by:
υ < υc ... laminar flow
υ > υc ... turbulent flow
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Laminar:
Bingham Plastic Fluid Model
Flow through pipe
υcp in [ft/sec]

Flow through annulus
υcan in [ft/sec]:

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Laminar:
Power-law Fluid Model
Flow through pipe

υcp in [ft/min]:

Flow through annulus
υcan in [ft/min]:

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turbulent fluid flow behavior
description
To describe the flow behaviour,
friction pressure loss and
shear rate at the pipe wall for laminar flow,
analytic equations are applied.
For turbulent fluid flow behavior,
analytic models to calculate these parameters
are extremely difficult to derive.
Therefore, various concepts that
describe their behavior are used in the industry.
The concept based on the dimensionless quantity
called “Friction factor”
is the most widely applied correlation technique.
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friction factor determination for fully
developed turbulent flow pattern
𝜖 [in] ... absolute roughness of pipe,
see from following table
(Absolute pipe roughness for several types of circular pipes)
𝜖
𝑑

 [1] ... relative roughness of pipe
To solve this equation for f, iteration techniques have to
be applied.

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Friction factor for turbulent flow
The friction
factor can also
be obtained
from the
figure.

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Friction factor estimation
In drilling operations,
the relative roughness is oft assumed to be
insignificant (usually less than 0.0004) which
reduces the friction factor equation
to the following equation for smooth pipes:

For smooth pipes and turbulent flow
𝜖
( = 0 and 2,100 <= NRe <= 100,000),
𝑑
the friction factor can be estimated by:
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The pressure drop calculation at
turbulent flow pattern
The pressure drop at turbulent flow pattern is then
computed for the different flow models
when replacing di
with the equivalent diameter de = 0.816 (d2 − d1).
When the friction factor is computed,
the pressure drops
for the individual flow models can be calculated.
For Newtonian Fluid Model:

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Power-law Fluid Model:
For fluids that behave according to the power-law
fluid model, an empirical friction factor correlation
based on the flow behaviour index n is used.
This correlation gives for:
Flow through pipe:

Flow through annulus:

μa [cp] ... apparent Newtonian viscosity
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Friction factor for Power-Law
Reynolds
number is then
compared with
the critical
Reynolds
number,
which is
depended on
the flow
behaviour
index n and
should be
obtained from
the figure
Friction factor for Power-Law fluid model
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pressure drop for power law
Instead of using the figure,
following equation can be applied
to determine the friction factor iteratively:

When the friction factor f is calculated, the
corresponding pressure drop can be calculated with
the Newtonian fluid model equation:

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the total pressure loss
at the surface equipment
The pressure drop in surface connections comprise
the pressure drops along the standpipe,
the rotary hose, swivel and kelly.
Since different rigs do use different equipment, the
total pressure loss at the surface equipment can
only be estimated.
(Δpf )se [psi] ... pressure loss through total surface
equipment, q [gpm] ... flow rate, E [1] ... constant
depending on the type of surface equipment used
Groups of surface equipment

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Equivalent drillpipe lengths
for surface equipment
Another approach is
to determine the equivalent length of drillpipe for each
surface equipment and
then use the relevant equations
to determine the surface pressure loss.

The Figure gives the equivalent lengths of the
different equipment parts.

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pressure drop across the bit
The pressure drop across the bit is mainly due to
the change of fluid velocities in the nozzles.
To increase the penetration rate,
when the mud flows through the nozzles
its speed is increased drastically which causes
a high impact force when the mud hits the bottom
of the hole.
This high fluid speed on the other hand
causes a relative high pressure loss.
This pressure loss is very sensitive to the nozzle seize.

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Calculation of pressure drop
across the bit
The bit pressure drop
itself can be calculated
with:

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AT [in2] ...
total nozzle area
dn [1/32] ...
jet nozzle seize
 𝜐 𝑛 [ft/sec] ...
mean nozzle velocity
q [gpm] ... flow rate
ρm [ppg] ... mud density
Cd [1] ... discharge
coefficient, depending on
the nozzle type and size
(commonly Cd = 0.95)


24

Initiating Circulation
All the equations to calculate the individual
pressure drops presented above assume a
nonthixotropic behavior of the mud.
In reality, an additional pressure drop is observed
when circulation is started due to the thixotropic
structures which have to be broken down.
This initial phase of addition pressure drop may last
for one full circulation cycle.

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Initiating Circulation
pressure drop calculation
The additional pressure drop can be estimated
applying the gel strength τg of the drilling mud as:
For flow through pipes:

For flow through annuli:

τg [lbf/100 ft2] ... gel strength of the drilling mud

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hydraulic program design
The penetration rate in many formations is
roughly proportional to
the hydraulic horsepower expended at the bit.

To drill most efficiently hydraulic programs are
designed
for maximum bottom hole cleaning
(how much bottom hole cleaning is necessary
to reach maximum penetration rate)
combined with maximum bottom hole cleaning based
on the surface hydraulic horsepower availability.

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drilling optimization parameters
For this reason,
mud rheology,
hydraulics (individual pressure drops) and
bit nozzle selection
are the parameters to consider for drilling optimization.

To optimize drilling hydraulics,
different approaches can be made.
The hydraulics can be designed to either
optimize the nozzle velocity,
the bit hydraulic horsepower or
to optimize the jet impact force.
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The total pressure drop
at the circulation system
The total pressure drop at the circulation system
is the summation of
the pressure drop at the bit and
the pressure drop through
the rest of the circulation system.

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Optimum pressure drop across the bit
The pressure drop across the bit can be written as:
Hydraulic horsepower:

Jet impact force:

m [1] slope of the parasitic pressure loss (Δpf )d vs. flow rate
Theoretically m = 1.75
but in general it is better to determine m
from field data than assuming this value.
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optimum nozzle area
When plotting flow rate vs. pressure on a log-log
plot, the optimum design is found at the
intersection between the path of optimum
hydraulics and the (Δpf )d line for either of the
criteria mentioned above.
Having determined the optimum design,
the optimum pump flow rate,
optimum nozzle area and
corresponding pressure losses can be calculated:

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Optimum hydraulic horsepower and
jet impact force
Optimum hydraulic horsepower and
jet impact force are given with:

The optimum nozzle area leads to the respective
nozzle selection.
Nozzles for drilling bits are given 1/32 [in] seizes thus the
calculated nozzle area has to be converted into n/32 [in].
Knowing n (has to be an integer and is commonly
rounded down to ensure the nozzle velocity) and
the amount of nozzles to use,
the individual seizes are found.
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specific hydraulic horsepower
The so called “specific hydraulic horsepower”
is defined as
hydraulic horsepower per unit borehole cross-section.

The optimization as discussed above
is performed for regular intervals (e.g. 1,000 [ft]) and
is included in the drilling program.

In practice, computer programs are available
in the industry that perform
these hydraulic optimization calculations.
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The annular flow of the drilling fluid
The annular flow of the drilling fluid
(carrying drilling cuttings and
a certain amount of gas to the surface,)
is disturbed by frictional and centrifugal forces
caused by the rotation of the drillstring.

In practice, when it is noticed that
inefficient hole cleaning is present,
either the mud flow rate is increased or
the effective viscosity of the mud is increased or
both adjustments are performed.

To estimate the slip velocity of the cuttings,
following correlation methods were developed empirically
and are widely accepted and used in the industry:
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Estimation of the slip velocity;
Moore’s Correlation
Moore’s Correlation:

for NRp > 300:
for NRp < 3:
for 3 NRp < 300:

Fall 13 H. AlamiNia

μa [cp] apparent
Newtonian viscosity
ds [in]
drilling cuttings diameter
NRp [1]
particle Reynolds number
 𝜐 𝑠𝑙 [ft/sec] particle slip
velocity
ρs [ppg] cuttings density
τg [lbf/100 ft2] gel
strength required to
suspend a particle of
diameter ds


37

Estimation of the slip velocity;
Chien’s Correlation
Chien’s Correlation:
The correlation equations determined by Chien are
similar to the ones defined by Moore.
For clay-water muds,
he recommends the usage of the apparent viscosity.

The correlation is performed as:
for NRp < 100:
for NRp > 100:

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transportation velocity
The so called “transportation velocity” 𝜐 𝑇
is defined as
the difference between the mean annular velocity 𝜐 𝑎𝑛 and
the slip velocity 𝜐 𝑠𝑙 . The “transportation ratio” FT given by:

determines whether the cuttings are
transported to the surface (FT is positive) or not and
provides a relative measure of
the carrying capability of the drilling mud.

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minimum mean annular velocity
To have proper hole cleaning and
with the knowledge of the transport velocity,
a minimum mean annular velocity can be determined.
This minimum mean annular velocity has to be calculated
at the annulus with the maximum cross-section area and
in this way determines the minimum pump rate.
As a rule of thumb,
• a minimum mean annular velocity of 3 [ft/sec] is often applied.

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

Q921 de2 lec5 v1

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