Grupo 4. SPE-26121-PA Settling Velocity of Irregularly Shaped Particles

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Settling

Velocity of rregularly

Shaped

articles

Sze-Foo Chien: SPE Texaco Inc.

Summary

A new correlation has been developed to predict the settling velocity

of

irregularly shaped particles in Newtonian and non-Newtonian

fluids for all types of slip regimes. The correlation was derived from

extensive data on the drag coefficients and particle Reynolds num

bers of irregularly shaped particles. The effective fluid viscosity at

the settling shear rate is used in the correlation. A trial-and-error

or

numerical iteration method is required to predict the settling veloc

ity for non-Newtonian fluids. The correlation predicted and verified

the effects of fluid properties, particle properties, and operation pa

rameters on the settling velocity.

Introduction

The settling process occurs in many petroleum, mining, and process

engineering operations. Applications include lifting

of

drill cut

tings, transportation of fracturing proppants, design of settling and

separating tanks, pipeline transportation of mining and coal par

ticles, and deposition

of

sediments in river channels. In most practi

cal applications, the particles involved are irregularly shaped. The

irregular shape changes the settling behavior compared with

smooth, symmetrical particles. Another practical consideration is

that the fluid medium, such as drilling fluid, polymer fluid, and clay

slurry, through which the particles settle is often non-Newtonian.

Non-Newtonian fluid rheology is more complex than that

of

Newto

nian fluids. The viscosity

of

such fluids is generally shear-rate de

pendent. Some may have time- and history-dependent properties.

Chien

1

presented two empirical correlations for the settling ve

locity of drill cuttings for rotary drilling operations: one for deter

mination

of

the settling velocity

of

cuttings in all slip regimes and

the other a simplified version for the turbulent-slip regime. Since

then, more experimental data on the settling velocity of irregularly

shaped particles have been published, and new models describing

the rheology

of

non-Newtonian fluids have been introduced. These

developments have been incorporated into a new correlation. The

viscosity used in the correlation is an effective viscosity at the set

tling shear rate. With the new correlation, effects of fluid and par

ticle properties and operating parameters on the settling velocity are

presented and compared with experimental observations.

ackground

Settling Velocity, Slip Regime,

and

Settling

Shear

Rate. Rich

ards

2

reported settling-velocity data for galena and quartz particles

in water for a wide range

of

diameters. Quartz particles have a densi

ty comparable with that of drill cuttings and silica sands. Fig. 1

shows the settling velocity

of

quartz particles as a function

of

nomi

nal particle diameter.

In a given fluid, the settling velocity increases with particle dia

meter, but the rate of increase is different for different particle-size

ranges. The logarithmic plot in Fig.

1

shows three distinct regimes

of

settling behavior. For particles < 0.018 cm in diameter, settling

velocity increases approximately proportionally to the square of he

particle diameter. For particles

>

0.13 cm in diameter, the settling

velocity increases proportionally to the square root of he particle di

ameter. The settling behavior of

the small-diameter range is known

as laminar slip, and that

of

the large diameter range as the turbulent

slip. Between these two regimes is the transitional-slip regime. In

·Now

retired.

Copyright 1994 Society of Petroleum Engineers

Original SPE manuscript received for review Nov. 13, 1992. Revised manuscript received

May 23 1994. Paper accepted for publication

Nov.

8 1993. Paper (SPE 26121) presented

at the 1994 SPE Annual Technical Conference and Exhibition held in New Orleans, Sept.

25-28.

SPE

Drilling Completion,

December

1994

the laminar-slip regime, the settling velocity is affected by both the

rheology and the density

of

the fluid, while in the turbulent-slip re

gime, the settling velocity is affected mainly by the density

of

the fluid

and the surface characteristics of the particle.

Because

of

the unique velocity-to-particle-diameter relationship

in each slip regime, the ratio of the settling velocity to the particle

diameter, vs/d also changes with particle size. This ratio is the set

tling shear rate. For non-Newtonian fluids, viscosity depends on the

shear rate and knowledge of the settling shear rate is important for

evaluation

of

the viscous force experienced by the particle. In the

turbulent-slip regime, the fluid viscosity has only a minor effect on

the drag force; therefore, the settling shear rate does not have an im

portant role in turbulent slip. For Newtonian fluids, viscosity is in

dependent

of

the shear rate and the concept

of

a settling shear rate

is not used. Fig. 2 also shows settling shear rate as a function of par

ticle diameter for the data in Fig. I. In the laminar-slip regime, set

tling shear rate increases with particle diameter, while in the turbu

lent-slip regime, shear rate decreases with particle diameter. The

maximum occurs somewhere in the transitional-slip regime. For

comparison, Fig. 2 shows settling shear rates of irregular particles

in several drilling fluids (Fluids

Lj L2,

L3, and

L4

from the work

of Walker and Mayes.

3

The trend of the settling shear rate in the drilling fluids is the same

as that in water. The settling velocity for a given particle decreases

as the fluid becomes more viscous; therefore, the settling shear rate

curve for viscous fluid shifts downward as the fluid viscosity in

creases. For the fluid in the immediate neighborhood of the particle,

the settling shear rate represents the shear rate that the fluid is expe

riencing during the settling process. For fluids that have shear-de

pendent viscosity, the settling shear rate should be used to determine

the effective viscosity of the fluid.

Besides the general trend

of

the settling shear rate with respect to the

particle diameter, one should also note the magnitude

of

the settling

shear rate. The maximum settling shear rate

is

=

120 seconds-

1

in wa

ter. In

drilling fluids, the maximum is in the 20

to

50 seconds-

1

range.

Therefore, the rheological properties

of

the fluid used to predict the set

tling velocity should be measured in the same low-shear-rate range.

Fig. 3 shows settling velocities for one

of

Walker and Mayes 3

test series (settling of disks

ofthe

same diameter but different thick

nesses in the four fluids mentioned earlier) and confirms the impor

tance of the effect

of

rheological properties on settling velocity at

low shear rate. The settling velocity of a disk increases as the effec

tive viscosity of the fluid increases. Fig. 4 shows effective viscosi

ties of these fluids at various shear rates. The decrease in viscosity

among these fluids was only in the low-shear-rate range (

< 50

se

conds - J) rather than in the high-shear-rate range.

Novotny4 and Hannah and Harrington

5

used the concept of set

tling shear rate in their studies of the settling of proppant between

rotating concentric cylinders. In their work, settling shear rate is the

gravitational component

of

shear rate experienced by the fluid.

Drag

Coefficient

and Particle

Reynolds

Number.

Analysis of the

drag force on a particle in a flowing system generally uses a relation

ship between the drag coefficient

CD,

and particle Reynolds number

NRe.

The same treatment can be applied to the settling of particles

in fluids. Basically, the drag coefficient represents the fraction of he

kinetic energy of the settling velocity that is used to overcome the

drag force on the particle, while the Reynolds number is a ratio be

tween the inertial and viscous forces of a fluid. For particles with a

nominal

or

an equivalent diameter d, the drag coefficient and par

ticle Reynolds number in the settling process are defined as

CD = 1308.7d Pp -

PNV Pf ...................... , (1)

281



100 ~ .

10

0.1

t'

f:

/'

l

."./

J - : ~

/ :

:A:

:

.,. -

Laminar ~ a n s i t i o ~

Turbulent

- -S l i p -+ r r - Slip

Slip':.: .------+

t

:

0.01

l..--1-...J-l.-U...I.uJ....---l......J......I...LUJJ.L--1-..L..J...J..WJ.U....----Jc......L...w..

. . . . . .

0.001 0.01

0.1

10

Particle Diameter, em

Fig.1 Settl ing

velocity

of

irregularly

shaped particles in water

(data

from

Ref. 2).

and

N

e

= dV

s

pt/(1O.0f.l.e). .

..

,

. . . . . .

, ,

..

,

. . . . . .

..

(2)

The drag force consists of a viscous drag, which is the result

of

the

fluid viscosity, and a profile drag, which is the resistance of the fluid

against the particle profile. A low NRe (

<

10) implies a relatively high

viscous force, and a major portion of the drag force is used to over

come the viscous resistance of the fluid. At high NRe (> 50), the iner

tial force becomes dominant and the fluid density and the particle pro

file and surface roughness affect the drag force. At NRe 100, the

drag coefficient

of

a given particle approaches a constant value.

To show the range

of

particle Reynolds numbers

of

the laminar

and turbulent-slip regimes, Fig. 5 plots NRe for Richards'2 data vs.

particle size and another set

of

data representing particle settling in

a drilling fluid. Note that the regimes representing laminar and tur

bulent slips can be identified by the slopes of the curves. For most

fluids, the laminar slip prevails when

NRe

<

10. Because the effect

of

the fluid viscosity

on

the settling process is mainly in the laminar-

100 .....----------------------------------

50

30

20

10

5

3

-

FluidLl

. . ·c·· Fluid L2

-

- - ~ -- .

Fluid

L3

--_-.

Fluid L4

.0

2 ~ ~ ~ L ~ ~ ~ ~ ~ ~ ~

__

0.05 0.1 0.2 0.3 0.5 1.0 2.0

Disk Thickness, em

Fig. 3 Settl ing

velocity

of discs in non-Newtonian

fluids

(data

from

Ref. 3).

282

1,000

100

10

0.1

0.001

---

Fluid

L1

....

Fluid L2

- - - - Fluid L3

--- Fluid L4

•

Water

0.001 0.1


10

Fig. 2 Settl ing shear rate vs. particle diameter (data

from

Refs.

2 and 3).

slip regime, attention should be given to the drag coefficient in the

range of

NRe

< 10. The turbulent-slip regime occurs whenNRe > 50.

Fig. 6 shows experimental data, collected from Hottovy and Syl

vester,6 Moore,? Zeidler,8 and Hopkins,9 Richards

2

and Walker and

Mayes,

3

that were used to establish the relationship between the

drag coefficient and

NRe

for irregularly shaped particles. As de

scribed earlier, Richards' data were for settling of quartz particles

in water. Hottovy and Sylvester's data were for settling of 0.88 gI

cm

3

particles in a Newtonian fluid

of

0.506-g/cm

3

density at three

different temperatures. Walker and Mayes' data and some ofZeid

ler's data were for particles settling

in

non-Newtonian fluids.

Moore's data and some of Zeidler's data were for settling of par

ticles in Newtonian fluids. Hopkins used glass and rock particles of

various shapes and sizes settling in water and twelve drilling fluids.

Most of his data were in the transitional- and turbulent-slip regimes.

10r-------------------------------

5

2

0.5

.. " - 0 . ..

0.2

0.1

0.050

0.020

c,

-<1

.......

0

..

"' .. "'000"''' '

a

-

FluidLl

. . ·c··

Fluid L2

- - _. Fluid L3

----. Fluid L4

...... 0... .....

0.010

L-...l...-J...J...L.Ll..1. l.--L...L..l....L. ..JWL--L-L...I...LJ.LJ..U

J.-L...J...J...I.U.J.J

0.1 0.3 3 10 30 100 300 l,OO(

Shear Rate, S-l

Fig.

4 f.l.e

of non-Newtonian

fluids

used

in

Fig. 3.

SPE Drilling & Completion, December 1994



10,000 ~ - - - - - - - - - - - - - - . - - - -

Qj

..0

E

:: l

Z

n

0

0

c:

>

Q)

a::

Q)

u

.€

CIl

a..

1,000

100

10

0.1

0.01

0.001

0.0001

0.003

0.01 0.03

Water

P,

=

1.000

g cm

3

- Drilling Fluid

PI =

1.678 g/cm

Pp =

2.696

g/cm

0.1

0.3

3


Fig. 5-NRe vs. particle diameter.

5

Three guidelines were followed to obtain a relationship between

the drag coefficient and NRe for the settling process.

1.

Settling velocity is usually the parameter of most concern where

settling or sedimentationof he particles can be overcome. In practice.

a settling velocity that is on the high side for a prescribed particle size

and fluid condition will be used to provide a safety factor. In terms of

drag coefficient, this means a lowe r value of

CD

for a given Reynolds

number. In other words. th e correlation is one that fits most data close

to the lower boundary of the spread of the experimental data.

2.

Laminarslipis

likely

to

occur

where NRe

< 10. As faras

fluid

rheology

is

concerned, the main interest

is

the drag coefficien t for

NRe< 10.

3. For NRe > 100, turbulent slip prevails and the surface condition

ofthe particle has a dominant effect on the drag coefficient. Spherici

ty,

'P,

is used to characterize particle surface condition 10 and is de

fined as

'P

= As/Ap

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

A smooth sphere has a 'P value

of

1. Most drill cuttings, sand par

ticles, and other frequently occurring irregular particles have a 'P

C

.91

(.)

0

,@

,0

,0>

,a'

' I ' ' 0.2

_

.•

_ .. _ .. 'f'

..

0.3

------

'f' '

0.4

.• . . . . .

t

' 0.6

-- 1 -0.8

. .

't' • 1.0

1 '

al,;.3--....I..I..Lill,

l-:

2

-'--LLll.w, L:.,...J....W-U.illL, ..J....LLll.1Jl,1:-

..J....J....I..I..L.w, >L.:- -I...J..J..W .I,

o l;;-'...w..w.uI,

Particle Reynolds Number

Fig.

7-Relationships

between CD and NRe'

SPE Drilling

&

Completion, December 1994

C

if '

.,

0

(.)

'

Cl

o

Rlchards

2

a Moore

7

t:. Walker & Mayes

3

: ~ ~ f

& y l v e s t e ~

0

v

•

.

I

'0 ';:--J...w.JjJJJ -:-.LLlWJUL..L.L.l.J.il.llL....w....J..llillL-LLllliUL;-l-.J.JJWJ.JJJ';:--J...LJ.lillll

10.

3

10-

2

10-

1

1

10

10

2

10

3

10

Particle Reynolds Number

Fig.

6-Experimental

data of CD vs. N

Re

of irregularly shaped

particles.

value close to 0.8 (0.7924). With the turbulent-slip data of Hopki

ns

9

and of Walker and Mayes,

3

the drag coefficient for the turbulent-slip

regime is correlated as

CD)t

=

67.289/e5.0301J1

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

(4)

for 0.2 'P 1.0.

Smooth

spheres will have a drag coefficient of

0.44 in the turbulent-slip regime. For frequently occurring irregular

particles, the drag coefficient for turbulent-slip regime is

(cDl, = 1.250 (5)

Fig. 7 shows the relationship

between

the drag coefficient

and

NRe with these guidelines, which is

CD = (30/N

Re

)

+ 67.289/e5.0301Jl), for 0.2

'P 1.0.

(6)

The relationship in Eq. 6 is valid for irregularly sha ped particles in

either Newtonian or non-Newtonian fluids and for NRe from 0.001

n

E

(.)

:>.

-

g

~

50

30

10

5

3

Cl

.£

E

0.5

Q)

CJ)

0.3

0.1

0.05

0.03

Pp = 2.696 g/cm

P, c

1.678

g cm

3

Non-Newtonian Fluids

R3 =

1.007

Rs

=

1.678

R,oo

=

14.2 (H-B only)

Newtonian Fluid

R3 1.007

Rs

= 2.014

Power· Law

. . . Bingham-Plastic

_ . _ .

Casson

- - -

_.

Herschel-Bulkley

- - - - -

_.

Newtonian

0.03 0.05

0.1

0.3 0.5 1

3


5

Fig.

8-Settling

velocity of irregularly shaped particles in fluids

of various models.

283



1.0 -------------------

--- Power-Law

. . . .. Bingham-Plastic

0.5

_._. Casson

- - - _. Herschel-Bulkley

- - - - - _ Newtonian

0.3

0.1

: ~ . : . . . . : . : :

0.05

0.03 L-_ --- - - - .w.. .L. . .W .._- ---- -- - . .J-L.L.I . . . . I- -_- --- -J . . - l

0.1

0.3 0.5 3 5 10

30 50

Shear Rate,

s-l

Fig. 9-Effective

viscosity of fluids

used

in

Fig.

8

to 10,000. The trend of the curves ofEq. 6 (Fig. 7) is identical to that

of

Wasp et al.

and most

ofthe

experimental data fall within

±

25

of the values calculated with Eq. 6.

For frequently occurring irregular particles

\11

=0.8), Eq. 6 can

be simplified:

CD

= (30.0/N

Re

)

+

1.250 (7)

The form

of

Eq. 7 is identical to that proposed by Whittaker.

12

The

laminar portion of the drag coefficient in Eqs. 6 and is slightly higher

than the classic Stoke s law, where CD = 24.0INRe-

Settling

Velocity

orrelations

Derivation

of

Settling Velocity Correlations.

A settling velocity

correlation is obtained by introducing the definitions

of

the drag co

efficient andNRe, Eqs. I and 2, respectively, into the drag coefficient

correlation, Eq. 6, and rearranging:

+ 4.458e

5 3 IJl

Jv

s

-

19.44ge50301J1 - I ) = O

(8)

For frequently occurring irregular particles, Eq. 8 can be simpli

fied to

v;

+ 240.0 ( ~

v -

1046.878d - I) =

O

..... (9)

The value of settling velocity can be solved from Eq. 9 by a qua

dratic formula and choosing the positive root of

Vs:

1 0 7 2 7 d ~ -1

( ~ : )

-1]

(10)

Note that Eq. 10 has the same parameters and form as Chien' sl set

tling velocity equation except for the numerical coefficients. The

difference in the numerical coefficients is because the derivation

of

the earlier equation essentially is based on Richards'2 data and the

parameters involved are in customary units. Another difference is

284

C\l

Q)

..c

(/)

0>

c:

'E

Q)

f)

50

30

10

5

3

1

0.03 0.05

Pp =

2.696

g cm

3

PI =

1.678

g cm

3

Non-New1onian Fluids

R3 =

1.007

R6

=

1.678

R100

=

14.2

(H-B only)

Newtonian Fluid

R3

=

1.007

R6 =

2.014

- Power-Law

... Bingham-Plastic

_ Casson

0.1 0.3 0.5

Herschel-Bulkley

- - - Newtonian

3

Particle Diameter, cm

5

Fig.

10-Settling

shear rate vs. particle size relationship

of

ex

amples

in

Fig.

8

that in the previous correlation, an empirical equation is used to ex

press the effective viscosity as a function of the plastic viscosity of

Bingham-plast ic fluid and annula r velocity. In this work, the effec

tive viscosity will be evaluated at the settling shear rate.

For those interested only in the settling velocity in the turbulent

slip regime, settling velocity can be obtained by substituting the tur

bulent-slip drag coefficient

of

Eq. 4 in Eq. I:

v

s

,

= 4.410 e25151J1

j

d[ Pp/Pj) - I].

(11)

And for frequently occurring irregular particles,

Effective Fluid Viscosities.

In determining the settling velocity

of

drilling particles, Moore

7

and Walker and Mayes

3

proposed use of

an effective viscosity equal to that which resulted in the frictional

pressure loss in the annular flow. Their effective viscosity is at a

shear rate equal to eight times the annular velocity divided by the

hydraulic diameter of the annulus. Chien I suggested use of either

the plastic viscosity or an effective viscosi ty at a shear rate equal to

annular velocity divided by the nominal particle diameter. In this pa

per, an effective viscosity at a shear rate equal to the settling shear

rate is used. This settling shear rate can be calculated according to

Ys

=

vs/d (13)

As discussed earlier, the settling shear rate of the laminar-slip re

gime, where fluid viscosity has a dominant role, ranges from 0.1 to

50 seconds-I and is usually < 25 seconds-I. Therefore, fluid rheo

logical data also should be measured in the same shear-rate range.

The effective viscosity at various shear rates will depend on the

constitutive equation of the fluid, or the relationship between the

shear stress and shear rate

of

the fluid. For Newtonian fluids, viscos

ity is independent of shear rate and the effective viscosity is the same

as the dynamic viscosity; i.e.,

fl.e = fl.N· •• • • ••• • ••• •• (14)

For non-Newtonian fluids, the effective viscosity depends on the

shear rate. Eqs. 15 through 18 are used to determine the effective

viscosities for four popular models of non-Newtonian fluids: Bing

ham-plastic, power-law, Casson, and Herschel-Bulkley.

SPE Drilling Completion, December 1994



E

u

>

-0

o

~

C>

c

t :

Q)

C/)

50

30

10

5

3

0.5

0.3

0 1

Pp = 2.696 g/

cm

3

I

0.155 Pa's

i

0.737

I

I

I

I

I

I

I

I

I

I

I

I

/

I

I

/

I

I

I

I

/

/

/

.-

'

'

/

/

.-

,,

,,

P, = 1.198g/cm

3

P,

= 1.678 g/cm3

P = 2.157 g/

cm

3

0.05 ~ ~ ~ ~ _ _ ~ ~ ~ ~ ~ U

0.03 0.05

0 1

0.3 0.5 3


Fig. 11-Effect

of

fluid density on settling velocity.

Bingham-Plastic Model.

f.le = asd/vs) + f.lB •

Power-Law Model.

f.le

=

l vs/dr

1

•

Casson Model.

15)

16)

f.le

=

[ /0;;/

jvs/d)

+

~ ....................

17)

Herschel-Bulkley Model.

f.le =

atfl/v

s

)

+

[1 vs/d)fH-l .

. . . . . . . . . . . . . . . . . . . . 18)

E

u

::E-

u

o

~

Ol

c

E

Q)

n

50 ~

30

10

5

3

0.5

0.3

0 1

Pp = 2.696 g/cm

3

PI

=

1.678 g/cm

3

i

=

0.737

I

I

I

I

I

I

I

/

/

I

I

I

I

I

I

/

/

I

/

/

I

/

/

/

/

/

=

0.155 Pa's

i

I = 0.309 Pa's

i

I

I

I

- -

I =

0.619

Pa s

i

I

'

,

'

'

0.05

L..-L...L....L....I.A...L.l..-_-- ---- ---- --- --.J.....J...u..J.

- - - - - - - - -

0.03 0.05

0 1

0.3 0.5

3 5


Fig.

12-Effectof Ivalue of

power-law fluid on settling velocity.

SPE

Drilling

&


The first three non-Newtonian fluid models are two-parameter

models that require two sets

of

shear-stress/shear-rate data to deter

mine their rheological parameters. The Herschel-Bulkley model is

a three-parameter model that requires three sets

of

shear-stress

vs

shear-rate data to determine its parameters. The Appendix describes

the use measurements from a Fann viscometer to determine the rhe

ological parameters

of

these four fluids. Similar procedures can be

used with any other appropriate viscometer.

Settling Velocity Solution Methods.The settling velocity for a pre

scribed particle and fluid can be solved by use of the appropriate ef

fective viscosity equation and Eq. 8 or

9

As described previously,

a Newtonian fluid

is

one whose viscosity

is

independent of the shear

rate; therefore, its dynamic viscosity can be used readily as the ef

fective viscosity to solve the settling velocity.

Although the effective viscosity is a function of the settling veloc

ity for Bingham-plastic fluids, the settling velocity correlation can

be grouped to become a quadratic equation of Vs.

For the other three types of non-Newtonian fluids, their effective

viscosity may be such that the settling velocity correlation is no

longer a quadratic equation. Either a trial-and error or a numerical

iterative method, such as Newton-Raphson method,13 can be used.

Such a method can also be used to solve settling velocity for any

type

of

fluid, Newtonian or non-Newtonian. Only the positive root

of

the solutions should be chosen as the practical solution.

Effect

of

Particle,

Fluid,

and

Operating Parameters

on Settling Velocity

Particle-Diameter Effect. Fig. 8 shows examples

of

the effect of

particle diameter on the settling velocity. Particle density is 2.696

g/

cm

3 and fluid density is 1.678 g/cm

3

. All non-Newtonian fluids

in the examples are assumed to have the same Fann viscometer mea

surements at 3 and 6 rev/min R3

=

1.007,

R6 =

1.678). The Hers

chel-Bulkley fluid has an additional measurement at 100 rev/min

(RIOO= 14.213). The Newtonian fluid has a viscosity based on

R3 =

1.007 and

R6 =

2.014. Because all non-Newtonian fluids in the

examples have identical rheological measurements shear stresses)

at

5 11

and 10.22 seconds-I, their effective viscosities close to these

shear rates are approximately equal to one another Fig. 9). Thus,

for a given particle size, the settling velocities are also very close

among various models. Beyond these shearrates, effective viscosity

and therefore settling velocity differ in the laminar-slip regime.

~

E

u

::E-

u

o

Ol

C

~

n

50

r------------------------------.

30

10

5

3

0.5

0.3

0 1

i = 0.537

i = 0.737

i = 0.937

• I

• I

. I

. I

. I

• I

. I

I

I

I

I

I

I

I

/

I

I

/

/

/

/

/

/

/

/

/

/

/

/

/

. ',..

/ '

'

'

Pp = 2.696 g cm

3

PI 1.678

g/cm

3

= 0.155 Pa's

i

0.05

L-L......L....l....L.J...U

____ ----- ---- -.l....I...J.....J...I..l-_-- -

......l....J

0.03 0.05

0 1

0.3 0.5

3 5


Fig. 13-Effect

of ivalue of

power-law fluid on settling velocity.

285



1.0

0.5

0.3

/)

ill

c

:>

-

0

0

u

0 1

/)

:>

U

~

0.05

UJ

0.03

0 01

'

'

'

'

.

..

Pp

= 2.696 g/cm

3

PI =

1.678 g/cm

3

=

0.155 Pa's

i

' '

'

'

'

'

---

i =

0.537

i= 0.737

. . . .

i

=

0.937

'

'

'

'

'

'

'

'

'

0 1 0 3

0.5 1 3 5

10 30

50

Shear Rate,

5.

1

Fig. 14-Effect

of

Ivalue

of

power-law fluid on the effective vis

cosity of fluid examples in Fig. 13.

Fig.

10

shows settling shea r rate as a function

of

particle diameter

for the examples in Fig. 8. All data have a settling shear rate ranging

from 1 to 22 seconds-I. At settling shea r rates

>

11 seconds-I, the

settling starts to become transitional slip where the fluid rheology

has a smalle r effect on settling velocity. Because all the fluids have

the same density, they have almost the identical settling velocity in

the transitional- and turbulent-slip regimes.

Fluid Density Effects. Fig. 11 shows examples

of

the effect of fluid

density on settling velocity. All fluids in the example are power-law

fluids with the same rheological property at 5.11 to 10.22 seconds-

I

shear rate range (i.e.,

R

= 1.007 and

R6

= 1.678) but different densi

ties (1.198, 1.678 and 2.157

g/cm

3

.

An increase in fluid density in

creases the buoyancy force and thus reduces the settling velocity.

Rheological

Property

Effects . Examples illustrated here are for a

power-law and a Bingham-plastic model. Similar analyses can

be

made for other models. Figs. 12 shows that an increase in I at a

constant value of i increases the effective viscosity for a given shear

rate and therefore decreases the settling velocity. Fig. 13 shows the

effect

of

an increase in the

i

value for a given I value. Fig. 14 shows

the effective viscosities of the power-law fluids used in the exam

ples.

For

shear rates

>

1 second-I, the effective viscosity will de

crease with an increase in

i

value, but for shear rates < 1 second-I,

the effective viscosity will increase. All settling velocities in Fig.

3

have a settling shea r rate

>

1

second-I;

therefore, the settling veloc

ity decreases as the value

of

i increases.

Figs. IS and 16 show the effects of yield stress and plastic viscos

ity, respectively, on settling velocity in a Bingham-plastic-model.

The

yield stress of a Bingham-plastic fluid can be considered as the

fluid strength that is capable of supporting a certain particle weight

or

size.

I f

he yield stress is high enough, the settling velocity can be

reduced readily to a very small value. The OR

=

0.688 Pa curve in

Fig. 15 shows that the fluid is capable

of

supporting any particle

< 0.15

cm

in diameter. Rheologically, when yield stress is high, the

effective viscosity could become so large (Fig.

17)

that it could re

duce the settling velocity to a very small value.

Other fluid models that have a yield stress, such as the Casson and

Herschel-Bulkley models, will have the same characteristics of sup-

286

/)

-...

E

u

;6

0

0

Cl

.5

E

Q)

C/)

5 0 ~ = = = = = = = = = = = = ~ ~ ~

30

10

5

3

0.5

0.3

0 1

aBC

0.172Pa

aBc

0.344

Pa

aBc

0.688

Pa

.

.

. /

I

: I

• I

I

I

• I

I

I

I

I

I

I

I

I

I

I

I

I

J

J

J

J

I

7

Pp

= 2.696 g/cm

3

=

0.0671

Pa s

PI = 1.678 g/cm

3

0.05

L....L.....I...J....L...U..l.---L-- --- -- - --..J...J.....LJ...JL...-_- --- ---- --

0.03 0.05

0 1

0.3 0.5 3 5


Fig. 15-Effect of yield stress of Bingham-plastic fluid on set

tling velocity.

porting particles at a high yield stress. Hopki ns'9 experimental data

showed that the settling velocity

of

a given particle will readily

be

reduced to zero when the yield stress increases to a certain value.

This study is able to verify this trend analytically.

Fig. 16 shows that an increase in plastic viscosity of a Bingham

plastic fluid for a given yield stress reduces the settling velocity.

This is similar to the settling in Newtonian fluids where an increase

in fluid viscosity reduces the settling velocity. Not much change oc

curs in the settling velocity in turbulent-slip regime because viscos

ity does not have a major role there.

Fluid

Velocity Effects. Settling velocity refers to the motion

of

a

particle with respect to the fluid through which the particle is set

tling. If the fluid itself is in motion, such as annular velocity in a

drilling operation, the effect of such velocity on the settling process

can be manifested through the change in the settling shear rate. If he

particle is settling in a static fluid, the shear rate experienced by the

fluid is the settling shear rate,

vs/d. I f

he fluid is moving against the

settling direction at a velocity equal to

v

s

the settling shear rate be

comes twice that

of

settling in a static fluid. Thus, by introducing a

multiplier on the shear-rate term in the effective viscosity, one can

evaluate the effect

of

fluid velocity on the settling velocity. For fluid

models, such as Bingham-plastic

or

Casson, which have both shear

dependent and -independent terms in the effective viscosity, the ef

fect

of

fluid velocity on settling velocity also will depend on the rela

tive magnitude

of

the rheological property in these two terms.

Fig.

18

shows a relatively small change in settling velocity as the

fluid velocity is changed in a power-law fluid. Several papers

l4

,15

have presented experimental data in agreement with this prediction.

It is the first time an analytical method is able to verify this effect.

Conclusions

1

New settling velocity correlations for irregularly shaped par

ticles have been derived. These correlations consider size, surface

condition, and density

of

the particle and rheology, density, and ve

locity

of

the fluid; they cover all types

of

fluids and slip regimes and

NRe

from 0.001 to 10,000.

2. An effective viscosity at a shear rate corresponding to the set

tling process is used to predict settling velocity. This shear rate,

which is equal to settling velocity divided by particle size, is defined

SPE Drilling

&




50 r ~

30

10

E 5

o

3

0

o

~

0>

E

Q)

C/)

0.5

0.3

0 1

0.05

0.03

.

,

,

'

, ,

,

,

0.05

,

,

,

,

.,

,

~ ; ; ~

,

,

.,

· 1

, ,

, ,

,

, ,

-

lis

=

0.0336 Pa's

...

Ils

=0 0672 Pa's

- -

_.

lis

=0 1007 Pa's

Pp =

2.696

g/cm

3

Pt = 1.678

g/cm

3

as

=

0.172 Pa

0 1

0.3

0.5 3

5


Fig.

16-Effect of

plastic viscosity

of

Bingham-plastic fluid on

settling velocity.

as settling shear rate. The settling shear rate

of

most settling pro

cesses in drilling and fracturing operations is in the 0.1 to 50 se

conds- I range. To predict a settling velocity, rheological data should

be measured in a similar low-shear-rate range.

3. The mathematical form of the effective viscosity for Bingham

plastic, power-law, Casson, and Herschel-Bulkley models of non

Newtonian fluids are presented.

4.

For non-Newtonian fluids, the effective viscosity depends on

the settling velocity and a trial-and-error or a numerical iteration

method, such as the Newton-Raphson method, can be used to solve

for the settling velocity.

5. For Newtonian fluids, viscosity

is

independent

of

shear rate

and the settling velocity can be solved by a quadratic formula.

Of

course, the trial-and-error and numerical iterative methods may also

be used to solve the settling velocity for Newtonian fluids.

6.

In the turbulent-slip regime, the fluid rheology plays a minor

role and the settling velocity is essentially determined by the fluid

density and particle density and surface characteristics. Settling ve

locity correlations specifically for turbulent slip are proposed.

omenclature

Ap

=

surface area of particle, L2, cm

2

As =

surface area of a sphere

of

the same volume, L2,

cm

2

CD

=

drag coefficient, dimensionless

CDh

=

drag coefficient

of

turbulent-slip regime,

dimensionless

d =

nominal or equivalent particle diameter, L, cm

I =

consistency index

of

power-law model,

mt(i-2)/L2,

Pa

si

IH =

consistency index of Herschel-Bulkley model,

mt(iH-

2

)/L2, Pa slH

i =

power-law index

of

power-law fluid,

dimensionless

iH =

power index of Herschel-Bulkley model,

dimensionless

NRe =

particle Reynolds number, dimensionless

R3,R6,R] ) )

=

Fann viscometer reading at 3, 6, and 100 revlmin

Vs

=

settling velocity, Lit, crnls

SPE Drilling

&


5

I

as=

0.172 Pa

I

0.344 Pa

3

I as=

I

as=

0.688 Pa

I

I

I

I

I

I

cp

I

c

I

0..

I

:tE

I

I

0

I

0

I

0

0.5

I

/)

>

I

I

Q)

\

>

\

·u

0.3

\

2.696 glcm

3

\

Pp =

UJ

\

lis

0.0671 Pa's

\

\

Pf

=

1.678 glcm

3

\

,

,

-

~

...

0 1

....

---

0.05

L . . . J - - - ~ L . . J . . . L _ - - l . _ L - . . . . l - l - L . . J . . . . J . . . . J . . . . I . . . . _ - - l . _ J . . . . . . . . J . . . . J

0.03 0.05 0 1 0.3 0.5 3

5


Fig. 17-Effect of yield stress of Bingham-plastic fluid

on

effec

tive viscosity.

vsh =

settling velocity in the turbulent-slip regime, Lit,

crnls

y=

shear rate, lit, second-I

y

=

settling shear rate, lit, second-I

\II =

sphericity of a particle as defined in Eq.

3,

dimensionless

fl =

plastic viscosity

of

Bingham-plastic model, mILt,

Pa s

flc =

plastic viscosity of Casson model, mILt,

Pa

s

fle

=

effective viscosity of non-Newtonian fluids, mILt,

Pa s

flN

=

viscosity

of

Newtonian fluids, mILt,

Pa

s

Pf Pp =

density

of

fluid and particle, respectively, mlL

3

,

glcm

3

r =

shear stress, mlLt

2

, Pa

OB

=

yield stress of Bingham Plastic model, mlLt

2

, Pa

Oc =

yield stress of Casson model, rnI(Lt

2

, Pa

0H

=

yield stress of Herschel-Bulkley model, mlLt

2

, Pa

References

1.

Chien, S.F.: Annula r Velocity for Rotary Drilling Operat ions,

IntI. J

Rock Meeh. Min. Sci. (1972) 9, 403.

2. Richards, R.H.: Velocity of Galena and Quartz Falling in Water,

Trans. AIME (1908) 38, 210.

3.

Walker, R.E. and Mayes, T.M.: Design of Muds for Carrying Capac

ity,

JPT

(July 1975) 893.

4. Novotny, E.1.: Proppant Transpor t, paper SPE 6813 presented at the

1977 SPE Annual Technical Conference and Exhibition, Denver, Oct.

9-12.

5. Hannah, R.R. and Harrington, L.1.: Measurement

of

Dynamic Prop

pant Fall Rates in Fracturing Gels Using a Concentric Cylinder Tester,

JPT

(May 1981) 909.

287



50

r---------------------------------,

30

10

E 5

u

0

o

C>

c

3

E

Q) 0.5

CJ

0.3

0 1

0.05

Fluid Vel.

Fluid Vel.

- - - Fluid Vel.

o

Pp = 2.696 g/cm

3

Pf

1.678 g/cm

3

I

0.155 Pa' Si

0.737

0.03 0.05

0 1

0.3 0.5


3

Fig. 18-Effect of fluid viscosity

on

settling velocity.

5

6. Hottovy, 1.D. and Sylvester, N.D.: Drag Coefficients for Irregularly

Shaped Particles, Ind. Eng. Chern. Process. Des.

Dev.

(1979)

18,

No.

3,433.

7. Moore, P.L.: Drilling Practices Manual, Penn Well Publishing Co., Tul

sa (1974) 228 .

8. Zeidler, H.U.: Fluid and Drilled Particle Dynamics Related to Drilling

Mud Carrying

Capacity, PhD dissertation, U

of

Tulsa, Tulsa (1974).

9. Hopkins, E.A.: Factors Affecting Cuttings Removal During Rotary

Drilling,

IPT(lune

1967) 807; Trans., AIME, 240.

10. Waddell, H.: The Coefficient

of

Resistance as a Function

of

Reynolds

Number for Solids ofVarious Shapes , 1. Franklin Inst. (1934) 217, 459.

11 Wasp, E.1., Kenny, 1.P., and Gandhi, R.L.: Solid Liquid Flow Slurry

Pipeline Transportation, Gulf Publishing Co., Houston (1979) 39.

12. Whittaker, A.: Theor y and Application of Drilling Fluid Hydraulics

IntI. Human Resources Development Corp., Boston (1985) 122.

13. Hornbeck, R.W.: Numerical Methods, Quantum Publishers Inc., New

YQrk City (1975) 66.

14 Sifferrnan, T.R.

et al.:

Drill-Cutting Transport in Full-Scale Vertical

Annuli, IPT(Nov. 1974) 1295.

15. Sample, K 1 and Bourgoyne, A.T.: An Experimental Evaluation of

Correlations Used for Predicting Cutting Slip Velocity, paper SPE

6645 presented at the 1977 SPE Annual Technical Conference and Ex

hibition, Denver, Oct.

9-12.

Appendix-Use of Fann Viscometer Measurements

for

Calculating

Rheological

Parameters of

Four Types

of

Non-Newtonian Fluids

The settling shear rate

of

a laminar slip is generally

<

50

seconds-I

and in most cases <

25

seconds-I. Therefore, rheological measure

ment of the fluid involved in the settling process should be measured

with a viscometer at low shear rates. The examples given here use

a Fann viscometer (assuming a Fann Model 35A or better with a

standard bob and rotor). For Bingham-plastic, power-law, and Cas

son models, Fann data at 3 and 6 rev/min are used to calculate rheo

logical parameters, and for the Herschel-Bulkley model, an addi

tional measurement at 100 rev/min is needed. The Fann

measurements at 3, 6, and 100 rev/min are represented by R3 R6 and

RIOO, respectively. The settling shear rate can be calculated accord

ing to Eq. 13. Parameters used in the effective viscosity

of

the four

models follow.

288

0.8

0.6

0.4

0.2

o ~ ~ __ __ ~ L ~ __ ~ ~

o

5

10 15 20

25

30

35

Fig. A l iH value of Herschel-Bulkley fluid as a function of Fann

viscometer readings.

Bingham-Plastic Model. The shear-stress vs. shear-rate relation

ship of a Bingham-plastic fluid is

T

= aB

+

flBi . .

•

(A-I)

In a settling process, the effective viscosity at the settling shear rate is

fle = flB + (aBlY,)

,

(A-2)

where flB

=

0.1O R6 - R

3

(A-3)

and a

B

= 0.5.11(2R

3

- R6) (A-4)

Power-Law Model. The shear-stress vs. shear-rate relationship of

a power-law model is

T = 1/.

......................................

(A-5)

Effective viscosity, in pascal seconds, at the settling shear rate is

fle =

Iy:-l,

(A-6)

where

I

= 0.511[R

6

/(l0.2I2)i] (A-7)

and i = 3.32210g(R

6

/R

3

•

••••••••••••• ••••••••• (A-S)

CassonModel. The shear-stress vs shear-rate relationship

of

a Cas

son model is

T

= lac +

; ; ; ;fit

....................... (A-9)

The effective viscosity, in poise, at the settling shear rate is

fle = [ JGc/ ffs + ~ · · · · · · · · · · · · · · · · · · · · · (A-lO)

2

where a

c

= 2.97S([2R;

- fifr, .................. (A-ll)

SPE Drilling Completion, December 1994



and,uc = 0.583 ;R;,-[R;f.

.....................

A-12)

Herschel-Bulkley Model. The shear-stress vs. shear-rate relation

ship of Herschel-Bulkley fluid is

l

=

H

+ IHi/H. .

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

A-13)

The

effective viscosity at the settling shear rate is

where iH can be solved from

or read from Fig. A-I, where it is plotted as a function of

RlOO

- R3)/ R6

-

R3).

IH

and

aH

values can be calculated from

The

value

of

the fluid yielding stress is

Calculation

of

iH

• IH

and

aH

for a given set

of R3 R6

and

RlOO

values can be done in many ways; this Appendix represents only one

of the possibilities.

SPE Drilling Completion. December 1994

SI

etric

onversion Factors

cp x 1.0*

dyne/cm2

x

1.0

ft

3

x 2.831

685

gal x 3.785412

in.

x2.54*

Ibm x 4.535 924

psi

x 6.894 757

·Conversion

factor is exact

E-03

=Pa s

E-Ol

=Pa

E-02 =m

3

E-03 =m

3

E+OO

=cm

E-Ol

=kg

E+OO =kPa

SPEDC

Sze-Foo

hien

recently retired from Texaco Inc. in Houston,

where he was a research consultant in

the E P

Technology

Dept.

He

had

been

with Texaco Research since 1961. His re

search focus was in fluid mechanics and heat transfer

of

multi

phase and non-Newtonian fluids

and

recovery of

unconven

tional energy resources.

He

s an honorary professor at the U. of

Petroleum (Shangdong, China). Chien holds a BS degree from

Natl. Taiwan U. and

MS

and

PhD

degrees from the U. of Minneso

ta all in mechanical engineering.

289

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Grupo 4. SPE-26121-PA Settling Velocity of Irregularly Shaped Particles

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