310Chapter5a

Rheology of Drilling Fluids

It is the science of deformation and flow of matter. VISCOSITY is the most commonly used rheological term and it defines the internal resistance of a substance to flow.

Oil field termsFunnel Viscosity

Apparent Viscosity

Effective Viscosity

Yield point

Low-Shear Viscosity

Low-Shear Rate Viscosity

Gel strength

PNGE 310 1

Functions of Mud

Marsh Funnel Viscosity measures a relative condition of a sample.

All viscosity terms are described in terms of the ratio of shear stress ( ) to shear rate ( ).

This description applies to all fluids.

PNGE 310 2

W J

Functions of Mud

Shear Stress Shear Rate Relationship:

PNGE 310 3

dy

dvF

Surface area = A

J

dy

dv J

A

F W

PNGE 310 4

Functions of Mud

Shear rate for all fluids (drilling muds) is measured using a viscometer.

It is equal to the dial reading (N) in rpm times 1.703

This constant is based on the geometry of the viscometer.

Shear stress is measured using viscometer.

It is equal to the 1.0678 times the dial reading ( )

It is expressed in lb/100 sq. ft.

N703.1 J

TW 0678.1 T

PNGE 310 5

Functions of Mud

Effective Viscosity ( e ): The viscosity under specific conditions such as shear rate,

temperature, and pressure

Apparent Viscosity ( a ): In general it is the viscosity measured at 300 rpm in the laboratory

Plastic Viscosity ( p ): Viscometer data is used to calculate this parameter

N

AVa

TP 300)(

300600)( TTP PV

p

pP

aP

eP

PNGE 310 6

Functions of Mud

Plastic Viscosity ( p ): It is important because it increases with:

Increase in volume of solids

Decrease in size of solids

Change in the shape of particles

Lower the plastic viscosity lower the pressure losses in the system and higher the pressure loss at the bit.

Ideally, the plastic viscosity value should be kept below the two times the mud density value expressed in lbs/gal.

Units: Centipoise (cp)

1 poise = 1 dyne-sec/cm2

1 poise = 1 gm/cm-sec

1 poise = 100 cp

1 cp = 0.01 poise

pP

dmudp

UP *2

PNGE 310 7

Functions of Mud

Yield Point ( o or y or YP): It measures the attractive forces in a fluid.

It is calculated from the measured Fann Viscometer data at 600 rpm ( ) and 300 rpm ( ):

Yield point is the second important parameter and it can indicate several problems such as:

Intrusion of soluble chemicals (e.g. salt, anhydrite or gypsum, and cement)

Mistreatment of mud with chemicals

PVp

YPy

PTTTW 300600300

*2)(

W

300T600T

PNGE 310 8

Functions of Mud

Low Shear Viscosity and Low-Shear Viscosity Muds:

They are more effective in deviated wells.

Thixotropy

It is the property of the mud to form gel when flow is stopped and then becomes fluid as the flow starts.

The gel strength is measured at end of 10-second and 10-minute interval.

In some cases a measurement at 30-min is also desirable.

The gel strength measures the static attractive forces while yield point measured the attractive forces in a fluid system under dynamic conditions.

PNGE 310 9

Functions of Mud

Effect of P&T on Viscosity:

If data exists, the change in viscosity due to P&T change can be estimated.

With data corresponding to two different T and P values the effective viscosity is:

Where

The pressure constant () must be determined for each fluid. The temperature constant () must be determine at each shear rate for each

fluid

> @

2112

)1

()2

(

TT

TT

Te

Te

EPP > @ > @12)

1()

2(

PP

Pe

Pe

DPP

PNGE 310 10

Flow Models

Rheological Models:

They describe the relationship between shear stress and shear rate.

Many models are available, but the most common ones are:

Newtonian

Non-Newtonian

Bingham Plastic

Power Law

Modified Power Law

Shear Stress

Shear Rate

Yield Pseudoplastic

Yield Dilatant

Bingham Plastic

Pseudoplastic

Newtonian

Dilatant

n > 1

n< 1

W

J

PNGE 310 11

Flow Models

Flow Models:

Newtonian

Bingham Plastic

Power Law

Modified Power Law

Where

n is the Power Law Index

K is the Fluid Consistency Index

JPW

ofor

opWWWJPW !

> @nK JW

> @p

nK WJW

PNGE 310 12

Flow Models

Determination of variables:

Newtonian

Bingham Plastic

Power Law and Modified Power Law

NNTP 300

poand

pPTWTTP

300300600

1

2log

1

2log

N

N

N

N

n

TT

nNNK

703.1

510T nK 511300

510T

12

Tank

Mud Pump

Return Line

Casing

Hole

Fluid Circulating System

PNGE 310 13 PNGE 310 14


Conservation of Energy:

Energy balance between points 1 and 2 gives:

Where

Frictional Pressure loss

Work done by the pump

g

P

g

vhWF

g

P

g

vh UU

222

2

12

121

2

11

g

fP

F U'

g

pP

W U'

PNGE 310 15


Also,

h1= h2 v1= v2 P1= P2

And - F + W = 0

W = F

Then

and

g

fP

g

pP

UU'

'

fp PP ''

12

Tank

Mud Pump

Return Line

Casing

Hole

PNGE 310 16


The total frictional loss in the system can be expressed as:

CSGDPHoleDPHoleDCBitDCDPSCf PPPPPPPP ''''''''

PNGE 310 17

Fluid Circulating SystemSurface Connection Losses

Pressure loss through surface connections is given as:

psiQEP pSC 2.08.08.0 PU 'Surface

Equipment

Type

Stand Pipe

Length

ft.

Stand Pipe

ID

in.

Hose

Length

ft.

Hose

ID

in.

Swivel

Length

ft.

Swivel

ID

in.

Kelly

Length

ft.

Kelly

ID

in.

E

1 40 3 45 2 4 2 40 2.25 2.5x10-4

2 40 3.5 55 2.5 5 2.5 40 3.25 9.6x10-5

3 45 4 55 3 5 2.5 40 3.25 5.3x10-5

4 45 4 55 3 6 3 40 4 4.2x10-5

Fluid Circulating SystemFlow through Jet Bits

PNGE 310 18

P1

P2vn

vo

g

P

g

vhWF

g

P

g

vh UU

222

2

12

121

2

11

g

P

g

nv

g

P

UU2

2

2

11

Conservation of Energy can be written for nozzle flow:

Assuming

suming

hhh1

ng minghhh1~ h~ h~ h2

h

F

~ h~ h~ h~ h2hhh1~ h

FFFFF = 0 (Negligible friction)

FF = 0 (Negligible friction)= 0

W = 0 (No work done)

W =

vvvn

W = 0 W =

vvn=

0 (NoW = 0

= = = = vvvo

(No wo

or vvv2

work don wo

vv2> v> v> v1

PNGE 310

19


g

P

g

nv

g

P

UU2

2

2

11

Nozzle flow and pressure drop:

Also, the pressure loss across a bit is defined as:

Then the previous pressure drop equation can be written as:

Where

21PP

bitP '

UbitP

nv

' 2

PNGE 310 20


The velocity through a nozzle is given as:

In field units with h ftftft/sec and nd ppgg values:

UbitP

nv

' 1238

U410074.8 ' x

bitP

nv

UbitP

nv

' 2

PNGE 310 21


Since the friction due to flow through jets is neglected, the actual Since the friction due to flow throughSivelocity is always less than predicted.

A correction factor called Discharge Coefficient is used to A correction factor called DischargA modify the velocity equation as:

UbitP

dC

nv

' 1238

U410074.8 ' x

bitP

dC

nv

PNGE 310 22


The pressure drop p PPbitbitbit must be same for all nozzles regardless The pressure drop Th p PPPbitbitbitbit mumuof the number of nozzles.

of the number of nozzles.

Also, the velocity must be same through nozzles.

Also, thAl

Then

Additionally, the total flow rate, Q through bit is given as:

321qqqQ

3`

3

2

2

1

1A

q

A

q

A

q

nv

321A

nvA

nvA

nvQ

321AAA

nvQ

tA

nvQ

PNGE 310 23


Then:

The nozzle velocity in field units ( (ftftft/sec, ec, gpmm, sq. in.) is given as:

iA

iq

A

q

A

q

A

q

tA

Q 3

3

2

2

1

1

tA

Qnv

117.3

PNGE 310 24


The velocity equation is substituted to the nozzle equation and The velocity equation is substituted to the nozzThpressure drop across the bit is determined as:

With C Cd C Cd=0.95

22

2510311.8

tA

dC

Qxbit

PU '

210858

2

tA

Qbit

PU '

PNGE 310 25


The bit nozzle diameters are often expressed in 32nds of and The bit nozzle diameters are often expreThinch and the total area is computed as:

Also, the energy equation in the form of hydraulic Also, the energy equation in the form of hydraulhorsepower developed by the bit is given as:

2

322

212324

dddtA

S

1714

Qbit

P

bitHP

'

PNGE 310 26

Fluid Circulating SystemPipe Flow

Flow Regimes:

Most Common Regimes

Laminar

Turbulent

Transition

Turbulent Criteria

Reynolds Number

Intersection of laminar and turbulent dP versus Q plot

Q

Laminar

dP

Turbulent

PNGE 310 27


Laminar Flow in Pipes and Annuli:

Assumptions

Drillstring is placed concentrically in the CSG

Drillstring is not rotated

Section of open hole are circular in shape and known diameter

Drilling fluid is incompressible

Flow is isothermal

Newtons Law of Motion is considered

for a shell of fluid at radius r and

fluid is flowing at a constant velocity (thus the sum of forces acting on it must be zero)

PNGE 310 28


Flow in Pipes and Annuli:

r

vF2

F1

F3 'L

r

F2

F1

F3F4

'rF4

'r

PNGE 310 29


The forces F F1 F F1, FF2FF2, FF33 and FF44 are given as:

rrPF ' S211 rrPF ' S222 LrF r ' SW 23

LrrF rr '' ' SW 24

vF2

F1

F3 'LF4

'r

04321 FFFF

PNGE 310 30


r

C

dL

dPr f 12

W

Summing forces, expanding and dividing by 2 r r L , taking limit as r goes to 0 and integrating with respect to r yields:

Where CC1 is the constant of integration. This equation can Where CC1 is the constant of integration. This equation caisbe combined with the shear rate equation to yield the be combined with the shear rate equation to yield the be combibibined with the shear rate equation to yield the binenebineproper pressure drop formula for the fluid type in question.

'S ''

PNGE 310 31


Newtonian Fluids Reynolds Number

Laminar Pressure Loss Equation: (N(NRe < 2100)

PU dv

N928

Re

Ld

vPL ' ' 21500

P

PNGE 310 32


Newtonian Fluids Turbulent Pressure Loss Equation: (N(NRe > 2100)

Friction factor (f) can be approximated by Blasius Equation given as:

Thus, the Turbulent Pressure Loss Equation is:

Ld

vfPT ' '

8.25

2U

25.0

Re

0791.0

Nf

Ld

vPT ' ' 25.1

25.075.175.0

1800

PU

PNGE 310 33

Fluid Circulating SystemAnnular Pipe Flow

Newtonian Fluids Reynolds Number


P

U12

Re

757 ddvN

Lddv

PL ' ' 212

1000

P

PNGE 310 34


Newtonian Fluids Turbulent Pressure Loss Equation: ((NNRe > 2100)



Lddvf

PT ' '12

2

1.21

U

25.0

Re

0791.0

Nf

Lddv

PT ' ' 25.112

25.075.175.0

1396

PU

PNGE 310 35


Bingham Plastic Fluids

ng

Reynolds Number


PU dv

N928

Re

Ldd

vP ypL ' '

2251500 2WP

PNGE 310 36



ng

Turbulent Pressure Loss Equation: (N(NRe > 2100)



Ld

vfPT ' '

8.25

2U

25.0

Re

0791.0

Nf

Ld

vPT ' ' 25.1

25.075.175.0

1800

PU

PNGE 310 37



ng

Reynolds Number


P

U12

Re

757 ddvN

Lddddv

P ypL ' '

12

2

122001000

WP

PNGE 310 38



ng

Turbulent Pressure Loss Equation: (N(NRe > 2100)



Lddvf

PT ' '12

2

1.21

U

25.0

Re

0791.0

Nf

Lddv

PT ' ' 25.112

25.075.175.0

1396

PU

PNGE 310 39


Power Law Fluids Reynolds Number


nn

n

d

K

vN

/13

0416.089100 2

Re

U

Ld

vKP

n

n

L

n

n

' '

1144000

0416.0

/13

PNGE 310 40


Power Law Fluids Turbulent Pressure Loss Equation: (N(NRe > 2100)

Please make a note that all pressure loss equations are the same for turbulent Pleaseflow

Also, the friction factor (f) can be calculated from the following empirical equation Also, the friction factor (f) can be calculated from the followideveloped by Dodge and Metzner for smooth pipes only:

Ld

vfPT ' '

8.25

2U

2.1

2/1

Re75.0

395.0log

0.41

nfN

nf

n

PNGE 310 41


Power Law Fluids Reynolds Number


nnn

dd

K

vN

/12

0208.010900012

2

Re

U

LddvK

Pn

n

L

n

n

' '

1

12144000

0208.0

/12

PNGE 310 42


Power Law Fluids Turbulent Pressure Loss Equation: ((NNRe > 2100)

Please make a note that all pressure loss equations are the same for turbulent Pleaseflow

Lddvf

PT ' '12

2

1.21

U

PNGE 310 43

Fluid Circulating SystemSummary

Equations are given for pressure calculations in pipes for

laminar and turbulent flow conditions

different fluid flow models

Newtonian

NonNon-n-Newtonian:

Bingham Plastic

Power Law

See Table 4.6, page 155, SPE Textbook #2

310Chapter5a

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