forfatter fornavn etternavn institusjon viscosity viscous fluids examples dependency of viscosity on...

Forfatter Fornavn

EtternavnInstitusjon

ViscosityViscous Fluids

Examples

Dependency of Viscosity on Temperature

Laboratory exercise

Non-Newtonian

Fluids

Fluid Flow Characteristi

cs

Viscometer

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-Viscous fluids

-Fluid flow characteristics

-Dependency of viscosity on temperature

-Non-Newtonian fluids-Examples

-Laboratory exercise

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Section 1: Viscous Fluids

Horizontal flow of viscous

fluids

Continuity equation for viscous flow

Viscous flow in a cylindrical

tube

Viscous flow through a porous medium

made up of a bundle of identical tubes

Exercise: capillary tube

viscosity measurement

Select one

subject

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Horizontal flow of Viscous Fluids

t < 0Y

V

t = 0

V

tyvx ,t is small

t is large

V

yvx

y

x

Two large parallel plates of

area A

Y – a very

small distance

Lower plate is set in motion with constant

velocity

The fluid start to move due to motion of the

plate

After a while the fluid enter a steady state

velocity profile

To maintain this steady state

motion, a constant force F

is required

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The force may be expressed:

The force F per unit area A is proportional to the velocity V in the distance Y; the constant of proportionally is called the viscosity of the fluid.

VA

YF

Y

V

A

F

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Also used is the so called Newton`s equation of viscosity: dy

dvx

The shear stress

The fluid viscosity

The fluid velocity in

x- direction

The dimensions:

sms

m

dy

dv

m

N 1,

2

2/ m

sN

dydv

And

By definition: .10)(1011 32

cppoisepm

sNsPa

Explicit: smPacp 11

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Typical values of viscosity of some fluids:

2180,2840,17100

2090,3570,3080

2000,4690,8060

1900,6562,3140

1811,0059,8620

1711,79253,000

Viscosity Air, Micro Poise[ p]

Viscosity Water, centiPoise[cp]

Viscosity Castor Oil, Poise[p]

Temperature Celsius

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y

X

y

yVx

The flow will have the biggest velocity

at the top (the surface )

The flow velocity will approximately be zero at the bottom, due to retardation when liquid molecules colliding with

the non-moving bottom

A graphic presentation of this

phenomena is shown here

A liquid is flowing in a open channel

Fluid flows in an open channel

Continuity equation for viscous flow

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y

vj x

p

Transferred momentum pr. time

pr. area

The change in fluid flow velocity pr. distance

between the two layersViscosity is here defined as s

proportionality constant, similar to what was done in the case of defining absolute permeability

Jp – is often characterised as the

momentum intensity or as the shear stress,

The unit : 2/ mN

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y

z

xS

S` xv

pj`

pj

The change of momentum intensity the box:

Sdyy

jSjjSjSj p

pppp

Momentum transfer between layers S and S` in Newtonian viscous flow

dy is the width of the box and S = S` is the cross-

section

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Sdyt

P

y

j

t

P p

fy

j

t

P p

f

y

v

t

v xx

2

2

is the sum of all forces acting on the box. Let f symbolise a force of more general origin.

tP /

The momentum density inside the box is defined by .

The change of momentum pr. time inside the box is :

PCombining with the equation from last page gives us:

Substituting:

xvP

y

vj x

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s`s

rr + dr

R

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Viscous flow in a cylindrical tube

Viscous flow in a tube is characterised by a radial decreasing flow

velocity, because of the boundary effect of the

tube wall

To develop the continuity

equation for this example, dr (a thin layer), of

liquid is considered at a

radius r.

Cross-section of viscous flow through a cylindrical tube

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The momentum flux through the cylindrical volume:

lrdrr

j

r

jldrrjlrjSjSj pp

pppp

222

Here the flux area S is varying and . SS

Defining the change in the momentum density inside the cylindrical volume, as , the general equation for laminar flow is:

lrdrt

P

2

fr

j

r

j

t

P pp

The momentum intensity

jp is redefined according to the geometric conditions in this example:

r

vj x

p

The minus sign show that the flow velocity vx is decreasing when

the radius r is increasing

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The continuity equation for viscous flow in a cylindrical tube is:

f

r

v

rr

v

t

v

1

2

2

This is under stationary conditions.

At a constant pressure drop along the tube, no velocity variation is observed .

p0/ tv

f

rr

vr

r

f

r

v

rr

v

01

2

2

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212 ln

4

1CrCr

fv

f

rRv 22

4

1

The general solution of the previous equation, found by integrating twice, is:

The general constants are found by

considering the boundary conditions. For this example the flow is directed along

the x-axis1. Since the maximum flow velocity, vx(r=0) in the centre of the tube is less then infinity; C1=0

2. Since the flow velocity is zero along the tube wall; vx(r=R)=0 and 2

2 )/(4/1 RfC

The solution is then:

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R

-R

XV

Velocity profile in cylindrical tube flow

This flow velocity profile in the tube is given by the

formula above.

l

prRv

22

4

1

If the outer force, driving the liquid through the tube is the pressure drop along a tube length then and:

pl lpf /

Defines a the laminar viscous flow pattern

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Viscous flow through a porous medium made up of a bundle of identical tubes

drl

prR

rdrrrvrdq

22

4

22)()(

l

pRdrrR

l

pdr

dr

dqq

RR

82

4

0

22

0

l

pRAq

8

2

The incremental flow rate through a fraction of the cross-section of a capillary tube can be expressed:

The total flow rate can be found by integration:

For the sake of convenience we may present the last equation in the following form:

This is also known as the Poiseuille`s equation, where A is the cross-section

of the capillary tube

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l

pkA

l

pRAqq

i i

iiip

8

2

832

22 R

Rk

Considering a porous medium as a bundle of identical capillary tubes, the total flow qp through the medium is:

Where is the capillary tube cross-section and A is the cross-section of the porous medium. From the equation above the permeability of the medium where capillary tubes are packed together is found:

2ii RA

mn

Where the porosity of the bundle of capillary tubes is given by:

4/

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Exercise: Capillary tube viscosity measurement

L

ph ~

V2R

Fluid viscosity may also be estimated by the measuring the volume of fluid flowing through a capillary tube pr. time (as the figure below). Rewriting the Poiseuille`s equation an expression for the dynamic fluid viscosity is written:

V

t

l

pR

8

4

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Which means that if the relative accuracy in the tube-radius is +/- 2-3 %, then the relative accuracy in the viscosity is about +/- 10 %. A small variation in capillary tube fabrication induces large uncertainty in the viscosity measurement.

For horizontal flow through a capillary tube of length and radius R, the time it takes to fill a certain volume is measured. The flow pressure , is fixed during the process, by maintaining a constant fluid level at a certain elevation above the capillary tube.

The accuracy in these measurements is strongly related to the fabrication accuracy of the capillary tubes, as can be seen from the formula above. If the relative uncertainty is considered,

Lt V p

R

R

4

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Section 2: Fluid Flow Characteristics

For laminar flow in a cylindrical tube we have derived Poiseuille`s equation:

l

pR

A

qv

8

2

For turbulent flow in a cylindrical tube, an empirical law, called Fanning`s equation has been found:

l

p

F

Rv

2

F – the Fanning friction factor

This factor is dependent of the tube surface

roughness, but also on the flow regime established in

the tube

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Rv2

Re

The flow regime is correlated to the Reynolds number, which characterises the fluid flow in the tube. The Reynolds number, Re, is defined as a dimensionless number balancing the turbulent and the viscous (laminar) flow [ ]. LT lplp ///

2R is the spatial dimension where the flow occur – the diameter of a capillary

tube or width of an open channel.

v – the average flow velocity – density – the fluid viscosity

From experimental studies an upper limit for laminar flow has been defined at a Reynolds number; Re =2000. Above this number, turbulent flow will dominate. (This limit is not absolute and may therefore change somewhat depending on the experimental conditions.)

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In the case of porous flow, the velocity v in the previous equation should be the pore flow velocity, where and R is the pore radius. The Reynolds number for flow in a porous medium is:

))1(/( orpore SAqv

or

B

S

v

1

2Re

- the bulk velocityBv

residual oil saturation

viscosity

porosity

fluid density

bulk velocity

pore dimension

Bv

orS

R mm 6101010 smscm /01,0/1

33 /1000/0,1 mkgcmg 25,0

1,0%10 smKgcP /1011 3

Typical parameters for laboratory liquid flow experiments:

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2

2Re

l

pRk

Using the previous table we find Reynolds number Re = 1 (ca.), for laboratory core flow, which is far below the limit of turbulent flow.

In case of reservoir flow, the ”normal” reservoir flow velocity is ca. 1 foot/day or , which indicate that turbulent liquid flow under reservoir conditions is not very likely to occur.

sm /5,3

For gas, turbulent flow may occur if the potentials are steep enough. If the formula for Reynolds number and Poiseuille`s law is compared:

The only fluid dependent parameters are the density and the viscosity

Comparing the Reynolds number for typical values of gas and oil :

2500)0,2/(/8,0

)02,0/(/2,0

)/(

)/(

Re

Re23

23

2

2

cPcmg

cPcmg

oil

gas

oil

gas

Demonstrates the possibilities for turbulent when gas is flowing in a porous medium

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Section 3: Dependency of Viscosity on Temperature

Heavy oil

Light oil

Temperature, K

Vis

cosi

ty, c

p

The Viscosity of liquids decreases with increasing temperature. For gas it`s the opposite; viscosity increase with increasing temperature.

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Another commonly equation:n

T

T

00Where n depends on the

type of gas (1 < n < 0,75)

Temperature depending viscosity of gases expressed by the Satterland`s equation:

CT

TK

2/3

Where K and C are constants depending on the type of gas.

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Section 4: Non-Newtonian Fluids

Viscous-Plastic fluids

Bingham (1916) and Shvedov (1889) investigated the rheology of viscous-plastic fluids. These fluids also feature elasticity in addition to viscosity. Equation describing viscous-plastic fluids:

00 ,` whendy

dv •Here is the breaking shear stress• is the so-called structural viscosity

0`

0 when there is no fluid flow

Some oils, drilling mud and cements slurries represent viscous-plastic fluids.

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Pseudo-Plastic fluids

Some fluids do not have breaking shear stress but rather, their apparent viscosity depends on a shear rate:

1,

n

dy

dvk

n

Which means that their apparent viscosity decreases when dv/dy grows:

1

/

n

dy

dvk

dydv

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Section 5: Examples

6.2.1Water viscosity at reservoir

conditions

6.2.4Rotating cylinder


6.2.2Falling sphere


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Water viscosity at reservoir conditions

Water viscosity is primarily a function of temperature. Salinity has also a slight influence on . The pure water viscosity is listed in viscous fluids section. Due to correction foe salinity and reservoir temperature, the normal range of viscosity at reservoir conditions is from 0,2 to 1,0 cP.

w

Correlation for estimation of water viscosity at res. temp. :

493624 1056,110415,11073,407,033,4 TTTTw

Water viscosity is measured in cP and temperature in

Fahrenheit At a res. temp. of , the water viscosity will be:)230(110 FC

cP

cPw

40,0

)2301056,123010415,12301073,423007,033,4( 493624

NB! Even if the temperature is above , water at reservoir condition will still be in a liquid phase since the res. Pressure is quite high as compared to surface condition.

C100

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h

1t

0t

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Falling sphere viscosity measurement

A metal sphere falling in viscous fluid reaches a constant velocity vs

Then the viscous retarding force plus the buoyancy force equals

the weight of the sphere

The force Fs is given by Stoke`s law:

ss rvF 6

r – sphere radiusVs – thermal speed – viscosity

– density of the metal sphere – density of the fluid

sf

The weight of the sphere will balance the viscous force plus the buoyancy

force at the terminal sphere velocity when the

sum of forces acting on the sphere is zero

fss grrvgr 33

3

46

3

4

g – the constant of gravitation

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sfs v

gr1

)(9

2 2

Alternatively; measuring the time : 01 ttt

Viscosity can be estimated by measuring the falling speed of the metal sphere in a cylindrical tube: ( )thvs /

tC fs )(

C – characteristic constant, determined through calibration with a fluid of known viscosity

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Rotating cylinder viscosity measurement

h

M

Here we will also Look at the motion of a fluid between two coaxial

cylinders

Due to the elastic force (in the fluid) a viscosity shear will

exist there

One cylinder is rotating with an angular velocity

The other one is kept constant

The inner cylinder:

– Radius – Constant Angular velocity

innerr

The outer cylinder:

– RadiusHeld stationary by a spring balance which measures the torque M

outerr

1

22

114

outerinner rrhM

h – the fluid height level on the two cylinders

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For a certain viscometer, the viscosity as function of angular momentum and angular velocity is:

M

C

C – the characteristic constant for the viscometer

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Section 6: Laboratory exercise

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Section 6:

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