fluid mechanics for drilling

Fluid Mechanics for Drilling

Fluid Rheology& Flow

RegimesDr Reza Ettehadi Osgouei

Petroleum Engineering

Department

PAB4333 - Advanced Drilling Engineering

The concept that a fluid cannot maintain a rigid shape is a basic,

but important characteristic, which means that fluids cannot

sustain a shear stress (a tangential force applied to the surface).

Rheology is the study of the deformation of fluids.

The fluid flow behavior is described by an applied shear stress

and the resulting shear rate within that fluid.

In general

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shear stressTwo parallel fluid layers are separated by a distance dy. An applied force, F, acting over an area, A, causes the layers to slide past one another. The resistance to this sliding movement, the frictional drag, is called shear stress.

Shear stress, , is defined as the force per unit area required to sustain a constant rate of fluid movement.

Or, an applied force, F, acting along a unit surface area, A, tending to deform the fluid element.

Mathematically:

F

A

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shear rate: Consider that a fluid is placed between two parallel

plates that are 1.0 cm apart, the upper plate moving at a velocity of 1.0cm/sec and the lower plate fixed. The fluid layer at the lower plate isnot moving and the layer nearest the top plate is moving at 1.0 cm/sec.Halfway between the plate, a layer is moving at 0.5 cm/sec. The velocitygradient is the rate of change of velocity with distance from the plates.This simple case shows the uniform velocity gradient with shear rate (v1- v2)/h = shear rate = (cm/sec)/(cm/1) = 1/sec.

Shear rate, , is the velocity gradient, i.e., is the rate of change of velocity at which one layer of fluid passes over an adjacent layer.

Mathematically:

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dv

dy

It has been shown experimentally that the force per unit area (shear stress) applied to a fluid is proportional to the velocity change of two fluid layers (shear rate) a unit distance apart:

where n (power index) depends on the type of fluid.

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Time independent

Fluid

Newtonian Fluids

Non Newtonian

Fluids

Pseudoplastic

Power lawHerschel-Bulkley

Bingham plastic

Dilatant

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Newtonian fluid Newtonian fluids are those whose flowbehavior call be fully described by asingle term called the Newtonianviscosity, .For these fluids. examples of whichinclude water and light oil.The shear stress is related to shear ratelinearly with the proportionalityconstant being the Newtonianconstant viscosity, .

In engineering units, is in dyne/cm2 = 4.79 lbs/100 ft2

is in s-1

is in poise= dyne x s/cm2

The field unit of viscosity is the centipoise (1 poise = 100 centipoise). The field unit of shear stress-is lbs/100 ft2.

non-Newtonian fluid

•A fluid whose viscosity is notconstant at all shear rates anddoes not behave like a Newtonianfluid.

•Most successful drilling fluidsare non-Newtonian.

•Within that group are severalgeneral types and rheologicalmathematical models to describethem.

•Pseudoplastic is a general type ofshear-thinning, non-Newtonianbehavior that is desirable fordrilling fluids.

Source / Illustrations: glossary.oilfield.slb.com

PseudoplasticPseudoplastic is a general type ofshear-thinning, (i.e., the apparentviscosity decreases as the shear rate

increases) non-Newtonian behaviorthat is desirable for drilling fluids.

Pseudoplastic rheology: low viscosityat high shear rates and high viscosityat low shear rates, benefits severalaspects of drilling-higher drilling rateand improved cuttings lifting.

Bingham plastic and power-lawmodels describe a psuedoplasticbehavior using only twomeasurements (two parameters).

The Herschel-Bulkley model is athree-parameter rheological model


nK

y p

n

y K


Bingham plastic model. Fluids thatconform to the Bingham plastic modeldo not have a constant viscosity andrequire a certain minimum stress toinitiate flow. The yield point, orthreshold stress, is the y intercept.Bingham Plastics include thickened

hydrocarbon greases, certain asphaltsand bitumen, some emulsions

PV should be as low as possible forfast drilling and is best achieved byminimizing colloidal solids.

YP must be high enough to carrycuttings out of the hole, but not solarge as to create excessive pumppressure when starting mud flow.

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y p

Power-law fluid

A fluid described by the two-parameter rheological model of a pseudo plastic fluid, or a fluid whose viscosity decreases as shear rate increases

In this equation, K is the consistency index and n is the flow behavior index. The flow behavior index is readily determined as the slope of a plot of vs on logarithmic coordinates. The value of n is less than unity for Power Law .

Example: Water-base polymer muds, especially those made with XC polymer

n

K

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Herschel-Bulkley fluid (Yield Power Law)

A fluid described by a three-parameter rheological model. A Herschel-Bulkley fluid can be described mathematically as follows:

Many clay-water behave as Herschel-Bulkley fluid

n

y K

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Some Important Definitions

Viscosity : Viscosity is the internal resistance of a fluid to flow. It is equal to the ratio of shearing stress to the rate of shearing strain.

Plastic Viscosity : The plastic viscosity is the part of the flow resistance of the fluid caused by mechanical friction within the fluid.

This mechanical friction is due to

1. the interaction of individual solid particles,

2. solid and liquid particles

3. the deformation (shearing) of the liquid particles under shear stress.

The amount of solid particles, their size, distribution and their shape have direct effect on the plastic viscosity.

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Some Important Definitions

Yield Stress : The yield stress is the part of the flow resistance of the fluid caused by electrochemical forces within the fluid.

These electrochemical forces are due to

1. the electrical charges on the surface of reactive particles,

2. the electrical charges on the sub-micron particles

3. the presence of the electrolytes in the case of water-base muds.

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Effective Circulating Viscosity

Most drilling muds commonly used in the field exhibit viscousproperties which are shear rate dependent. During normal drilling, themud being circulated experiences different velocities in the varioussections of the circulating system ranging from practically 0 ft/s in thepits to over 300 ft/s across the jets of the drill bit. These wide rangingvelocities give rise to mud shear rates of less than 5 sec-1 in mud pits toover 100,000 sec-1 across the bit jets.

The effective circulating viscosity: therefore, can be defined as theviscosity of the mud at a given shear rate in a particular section of thecirculating system under given conditions of pressure and temperature.

where e is in cp, K is in lbf/ft2(sec-1)n, annular velocity, va, is in ft/s andwellbore diameter, Dw, and pipe diameter, Dp, are in inches.

1

47913.6

n

ae

w p

vK

D D

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1. Place a freshly stirred sample in acontainer and immerse the rotor-sleeveexactly to the scribed line.

2. Start the motor by placing the switch inthe high-speed position with the gearshift all the way down. Wait for a steadyindicator dial value, and record the 600RPM reading. Change gears only whenmotor is running.

3. Change switch to the 300-RPM speed.Wait for a steady value and record 300-RPM reading.

4. Plastic viscosity in centipoise = 600reading minus 300 reading (see Figure).

5. Yield Point in lb/100 ft2 = 300 readingminus plastic viscosity in centipoise.

6. Apparent viscosity in centipoise = 600reading divided by 2.

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Field measurements of viscosity:

Newtonian Fluid :

where is the Newtonian viscosity in cp, N is the viscometer dial reading at rotational speed, N.

Bingham Plastic Fluid :

For N=300 and 600 rpm:

where p is the plastic viscosity in cp, y is the yield point in lbf/100 ft2

300 N

N

y p 300yN

pN N

600 300p 300 6002y

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Power Law Fluid and Herschel-Bulkley fluid (Yield Power Law):

In filed units, ( Power Law )

where n is the Power Law index, K is the consistency index in eq.cp and o is the zero gel in lbf/100 ft2. The shear rate, (sec-1), can be expressed in terms of N as

n

K

23.32log N o

N o

n

N o

n

N

K

510 N

n

N

K

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1.704 N

Example - You are given the following data

Mud, powerlaw

• θ600= 65: θ300 = 40;

• 10s/10min; gel = 15/25 lbs/l00 ft2

• Drill pipe

• OD = 4.5 in.

• ID=4in.

• Air weight = 14 lbs/ft

• Drill collar

• OD = 7.5 in.

• ID=4in.

• Air weight = 110 lbs/ft

• LDC = 1,000 ft

• flow rate of 400 gpm.

•Hole•Last intermediate casing set •8.5 in. ID set at 16.000 ft TVD open hole washed out to 8.5 in. using a 7.5 in. bit

-Determine •Plastic viscosity•Yield Point•Apparent viscosity•Effective Circulating •n is the Power Law index,• K is the consistency index•Viscosity in each section

Flow Regimes: A range of stream flows having similar bed forms, flow resistance, and means of transporting .

Laminar flow is a streamline flow where allfluid particles move along lines parallel to theaxis of the conduit, and adjacent fluid layersslip past each other with no mixing ofparticles. In steady state conditions, insidecircular conduits, laminar flow can bevisualized as a series of concentric cylindersslipping pass one another as shown in figure.

http://www.youtube.com/view_play_list?p=C20E2B3106BCA23D

As the fluid velocity becomes higher, Theparticle travel in irregular paths with noobservable pattern and no definite layer.

Turbulent flow is characterized by theirregular movement of particles of the fluid.

http://en.wikipedia.org/wiki/File:Airplane_vortex_edit.jpg

Laminar Flow is usually found inthe annulus during drillingoperations.

This type of flow is generally desired inthe annulus since it does not lead to hole erosiondoes not produce excessive pressuredrops.

Smoke rising from a cigarette isturbulent flow. For the first fewcentimeters, the flow is certainlylaminar. Then smoke becomesturbulent as its Reynolds numberincreases

http://upload.wikimedia.org/wikipedia/commons/c/cb/Humphrey_Bogart_by_Karsh_(Library_and_Archives_Canada).jpg

Turbulent Flow is the type offlow regime found inside thedrill string during drillingoperations.Since high mud velocities arerequired to achieve turbulent flow,this results in high pressure drops.This type of flow is generally notdesired in the annulus due to itstendency to cause excessive holeerosion and high “equivalentcirculating densities”.However, turbulent flow canmove the mud like a plug, causingthe mud to move at approximatelythe same rate. This provides forbetter hole cleaning and issometimes required on high angleholes.

The Reynolds number Re is a dimensionless number that gives a measure of theratio of inertial forces to viscous forces. It is generally used to determine whether aflow regime is laminar or turbulent. For Newtonian straight pipe flow, it has beenestablished experimentally that the critical Reynolds number, i.e., the number abovewhich flow is no more laminar, has a value approximately equal to 2100.Mathematically, Reynolds number is given by

where is the fluid density, v is the average fluid velocity, D is the pipe diameter and is the fluid viscosity. In field units, NRe can be expressed as

where is in ppg, v is in ft/s, D is in in and is in cp.

Re

v DN

Re

928 v DN

Example

An oil well is expected to produce at a rate of 480SBPD (standard barrel per day). If the viscosity of theoil is 5 cp and its specific gravity is 0.8, determine theminimum size of production tubing to be installed inthe well such that laminar flow is maintained. Assumefluid is Newtonian.

Introduction The calculations of friction pressure losses in the

rotary rig circulating system are important and essential because of their direct bearing on

Drill bit hydraulic program design

ECD during tripping in and out of wellbore

ECD during drilling and well control operations

Fluid Flow and Associated Pressure

Drill bit hydraulic program design To ensure optimum drilling conditions, necessary calculations

Pump operating requirements, optimum flow rates, corresponding optimum drill bit nozzle sizes

In drilling, pump pressure (Pp ) required in order to drill to a certain depth for a given set of operating conditions is equal to the total friction pressure (Pf) losses in the circulating system plus the dynamic pressure changes (generally those across jet bits PB )-that is,

where Pp is the required pump delivery pressure Pf is the circulating system friction pressure loss PB is the dynamic pressure across bit nozzles


Pp=Pf + PB

Pressure changes during tripping and casing operations (Swab/Surge Pressures)

What is swab pressure? Swab Pressure: If a drill string , casing

string or logging tool is being pulled out ofhole too fast, due to bigger diameteralmost same hole size, BHA/ bit, casing orlogging tool will possibly swab mud out ofhole, like pulling small a piston of syringe.For this reason, hydrostatic pressure ofbottom hole will be reduced. Pressurereduction created by this situation is called“Swab Pressure”. If swab pressure is toomuch, kick (wellbore influx) may be intothe hole and well control must beconducted in order to secure well.


Mathematically

where

Paei is the equivalent mud pressure at some well depth Di in the annulus

Pahi is the mud hydrostatic pressure at Di

∆Pa swb is the swab pressure gradient in the annulus (in psi\ft)

or in terms of mud weight,

where

ρme is referred to as the equivalent circulating mud weight

ρmh is the actual mud weight while not moving

∆ ρ a swb is the change in mud weight due to the swab pressure

• for safe drilling, ρme > ρff where ρff is the formation pressure gradient

Paei=Pahi-Di ∆Pa swb

ρme= ρmh- ∆ρa swb


Pressure changes during tripping and casing operations (Swab/Surge Pressures)

What is surge pressure?Surge Pressure: When pipe moves downward with mud circulation through drill string, additional bottom hole pressure called “Surge Pressure” is created. If surge pressure is too much, many problems will occur as formation brake down, partial mud loss and lost circulation.


And... where

Paei is the equivalent mud pressure at some well depth Di in the annulus

Pahi is the mud hydrostatic pressure at Di

∆Pa swb is the surg pressure gradient in the annulus (in psi\ft)

or in terms of mud weight,

where

ρme is referred to as the equivalent circulating mud weight

ρmh is the actual mud weight while not moving

∆ ρ a swb is the change in mud weight due to the surg pressure

• for safe drilling, ρme < ρfrac where ρfrac is the formation fracture gradient

Paei=Pahi+Di ∆Pa surg

ρme= ρmh+ ∆ρa surg


Pressure changes during drilling During drilling, the friction pressure losses in the

annulus will effectively increase the mud weight, resulting in an ECD that ,as in the case of surge pressures, may cause fracturing 0f formation:

where ∆ρa f is the change in mud weight in annulus owing to the friction pressure loss gradient in the annulus.

For safe drilling, ρme < ρfrac . A similar situation may be encountered during well control operations.

ρme= ρmh+ ∆ρa f


Mechanical Energy and Pressure BalanceThe study of fluid dynamics is based on three physical laws:

• Conservation of energy

• Conservation of momentum

• Conservation of mass

These laws, when combined with

•the fluid rheological models (Newtonian, Bingham, power law, and

yield power law)

• the fluid state (compressible or incompressible)

•fluid-flow regime (Laminar or turbulent)

• the conduit type (pipe, annular or slot flow)

constitute all the conditions required in order to formulate a fluid dynamic

problem.


Mechanical Energy and Pressure BalanceThe mechanical energy balance equation

for an incompressible fluid entering a

physical system at point i and leaving it at

some point j can be written as

where

D is the elevation

ρ is the fluid weight density

V is the average fluid velocity

P is the pressure

Wp is the work done by pump

Wf is the friction energy loss

ρ(Dj-Di)+(ρ\2g)(V2j-V

2i)+Pj -Pi=

Wp + Wf


Mechanical Energy and Pressure BalanceThe mechanical energy balance equation for an incompressible fluid entering a

physical system at point i and leaving it at some point j can be written as

where

Ph is the hydrostatic head

Pd is the dynamic pressure

Pp is the pump pressure

Pf is the pressure loss due to friction In drilling, the circulation system can De visualized as a U-tube

Ph + Pd + Pj -Pi = Pp-Pf


Mechanical Energy and Pressure BalanceIn drilling, the circulation system can be visualized as a U-tube

Pp=Pf + PB


Mechanical Energy and Pressure Balance The available pump surface pressure or what is normally

referred to as the circulating stand-pipe pressure is expanded throughout the circulating system in the following manner:

where Pp is the operating pressure, Ps is the frictional losses at the surface connections, Pdp is the frictional losses inside the drillpipes, Pdc is the frictional losses inside the drill collars, Padc is the frictional losses in the annular space of the wellbore and the drill collars, Padp is the frictional losses in the annular space of the wellbore and the drillpipes, and Pb is the frictional losses at the bit.


p s dp dc adc adp bP P P P P P P

Pressure Drop across the Bit Nozzles (Jets)

Ph + Pd + Pj -Pi = Pp-Pf

Pi -Pj= (ρ\2g)(V2j-V

2i)

Pp=0 Pf , Ph negligible



Pi -Pj= Pb=(ρ\2g)(V2j)= (ρ\2g)(V2

exit)

Vi negligible

The actual exit velocity of the jets is always smaller thanwill be predicted by this equation owing to the assumptionmade regarding the frictionless state of the jets. Tocompensate for this difference, a modifying factor calledthe nozzle (jet) discharge coefficient. Cd, which will beimplemented in equation

The value of Cd, is assumed to be 0.95 unless otherwise specified.



when the following field units are used:Q is in gpmρ is in Ibs\galAt is in in.2

Cd is a nondimensional termPb is in psi

Since


References

Slides presented were prepared with illustrations from the following referenced documents:

-Drilling Engineering: Amazon.ca: J. J. Azar, G. Robello Samuel: Books-http://www.glossary.oilfield.slb.com

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