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
Fluid Circulating System
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
Fluid Circulating System
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
Fluid Circulating System
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
Fluid Circulating SystemFlow through Jet Bits
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
Fluid Circulating SystemFlow through Jet Bits
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
Fluid Circulating SystemFlow through Jet Bits
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
Fluid Circulating SystemFlow through Jet Bits
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
Fluid Circulating SystemFlow through Jet Bits
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
Fluid Circulating SystemFlow through Jet Bits
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
Fluid Circulating SystemFlow through Jet Bits
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
Fluid Circulating SystemPipe Flow
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
Fluid Circulating SystemPipe Flow
Flow in Pipes and Annuli:
r
vF2
F1
F3 'L
r
F2
F1
F3F4
'rF4
'r
PNGE 310 29
Fluid Circulating SystemPipe Flow
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
Fluid Circulating SystemPipe Flow
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
Fluid Circulating SystemPipe Flow
Newtonian Fluids Reynolds Number
Laminar Pressure Loss Equation: (N(NRe < 2100)
PU dv
N928
Re
Ld
vPL ' ' 21500
P
PNGE 310 32
Fluid Circulating SystemPipe Flow
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
Laminar Pressure Loss Equation: (N(NRe < 2100)
P
U12
Re
757 ddvN
Lddv
PL ' ' 212
1000
P
PNGE 310 34
Fluid Circulating SystemAnnular Pipe Flow
Newtonian Fluids Turbulent Pressure Loss Equation: ((NNRe > 2100)
Friction factor (f) can be approximated by Blasius Equation given as:
Thus, the Turbulent Pressure Loss Equation is:
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
Fluid Circulating SystemPipe Flow
Bingham Plastic Fluids
ng
Reynolds Number
Laminar Pressure Loss Equation: (N(NRe < 2100)
PU dv
N928
Re
Ldd
vP ypL ' '
2251500 2WP
PNGE 310 36
Fluid Circulating SystemPipe Flow
Bingham Plastic Fluids
ng
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 37
Fluid Circulating SystemAnnular Pipe Flow
Bingham Plastic Fluids
ng
Reynolds Number
Laminar Pressure Loss Equation: (N(NRe < 2100)
P
U12
Re
757 ddvN
Lddddv
P ypL ' '
12
2
122001000
WP
PNGE 310 38
Fluid Circulating SystemAnnular Pipe Flow
Bingham Plastic Fluids
ng
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:
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
Fluid Circulating SystemPipe Flow
Power Law Fluids Reynolds Number
Laminar Pressure Loss Equation: (N(NRe < 2100)
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
Fluid Circulating SystemPipe Flow
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
Fluid Circulating SystemAnnular Pipe Flow
Power Law Fluids Reynolds Number
Laminar Pressure Loss Equation: (N(NRe < 2100)
nnn
dd
K
vN
/12
0208.010900012
2
Re
U
LddvK
Pn
n
L
n
n
' '
1
12144000
0208.0
/12
PNGE 310 42
Fluid Circulating SystemAnnular Pipe Flow
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