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8/20/2019 Multiphase Flow in Pipes, 2006, Critical Velocity, Presentacion
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Multiphase Flow in Pipes
© Copyright 2006 iPoint LLC. Prepared for iPoint Clients only. All rights reserved. This work contains proprietary presentation of iPoint LLC and may not be copied or stored in an informational
retrieval system, transferred, used, distributed, translated or retransmitted in any form or by any means, electronic or mechanical, in whole or part, without the express written permission of the
copyright owner.
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Outline
1. Components of pressure loss for
multiphase flow in pipes.
2. Liquid holdup.
3. Shape of the tubing curve.
4. Correlation for oil and gas wells
5. Critical rate to unload a well
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Single-phase Flow
q L
q L
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Pressure Loss Components
dL
dv
g
v
d g
v f
g
g
dL
dP
ccctot
ρ ρ θ ρ ++=
2
sin
2
Elevation
Friction
Acceleration
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Single Phase Flow
Fluid occupies 100% cross section of the pipe:
q = Phase rate in bpd, cubic meters per day
A = Area of cross section of pipe, ft2 or m2
v = q/A, Velocity in ft/sec or m/sec
f = Friction Factor = f (NRe)
ρ = Density of fluid (lbm/ft3)
µ = Viscosity of fluid, cp
σ = Surface Tension, dynes/cm
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Moody Friction Factor Diagram
Laminar Critical Zone Transition Zone
Complete Turbulence, Rough Pipes
Pipe Rel.
Roughness
Smooth Pipe
FrictionFactor
Reynolds Number
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Reynolds Number
Where,
v = q/A, Velocity in ft/sec
d = Pipe diameter, ft
ρ = Density of fluid (lbm/ft3)µ = Viscosity of fluid, cp
µ ρ vd N 488,1Re =
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Friction Factor
For Laminar Flow, NRe
< 2000 and
f = 64/NRe
For Turbulent Flow , 3000
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Pipe Roughness
Normally inside wall of a pipe is not smooth In non corrosive environment oil or gas wells tubing may
behave like smooth pipe
Absolute Roughness ε, is the mean protruding height of piperoughness
• Measured with mean protruding height of uniformly distributed, sized,tightly packed sand grains giving same pressure gradient behavior as theactual pipe.
Absolute Roughness = ε in.
Relative Roughness = ε / d Diameter of pipe = d in.
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Friction Factor Rough Pipe:
“ In turbulent flow the effect of wall roughness on pressureloss in pipes depends on both the relative roughness andthe Reynolds number”
If a thick laminar sublayer of l iquid exists in the boudary layeradhering to the pipe wall, the pipe behaves as a Smooth
pipe.
−=
d f ε 2log274.11
Based on his Sand Grain Experiments, Nikuradse Suggested,
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Friction Factor in Transition Region
Transition Region, where friction factor varies
both with relative roughness and Reynolds
number
Colebrook (1938) proposed (Iterative),
+−=
f N d f Re
7.182log274.1
1 ε
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Friction Factor in Transition Region
A simpler equation explicit in friction factor‘f’ was
proposed by Jain (1976) - reproduces Colebrook
equation over the entire range of relative
roughness and Reynolds Number and is
presented as follows:
+−=
9.0
Re
25.21log214.11 N d f
ε
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Single Phase Pressure
Loss
dL
dv
g
v
d g
v f
g
g
dL
dP
ccctot
ρ ρ θ ρ ++=
2
sin
2
Elevation
Friction
Acceleration
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Single Phase Calculations
Calculate Velocity from rate
Calculate friction factor from Reynolds
Number
Calculate pressure losses in smallsegments assuming average fluid physical
properties in case of compressible flow
Use single phase pressure loss equations
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Single Phase Gas
pT TZ p B Bqq
ZRT pM Where
Sc
ScggSc === ;;, ρ
d
v f g
dL
dP gggg
gas 2sin
2 ρ θ ρ +=
dpC g
ZT
p
ZT
p
dL R
M wf
tf
p
p
L
+
= ∫∫ θ sin2
0
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Single-Phase gas Flow
dp
F
ZT
p
ZT
p
I where
+
=
2
2
sin001.
,
θ
,75.18 dp I L
wf
tf
p
p
g ∫=γ
ft andLind MMscfDq RT psia p
d
fq
F and
sc
o
sc
======.;;;;
,
667.0
, 5
22
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Cullender and Smith Ex.
Calculate the flowing bottom hole pressure
in a gas well (γg=0.75),Well Depth,L = 10,000 ft
BH Temp.,T = 245 oF
Wellhead Pressure, ptf = 2,000 psia
Wellhead Temp., Ts = 110oF
ε = 0.00007 ft
d = 2.441 in.
q sc = 4.915 MMscfd
Assume f =0.015 and the first pressure estimate,
p(est) = p(known)(1+2.5x10-5xL/2 Sinθ)
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Multiphase Flow
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Pressure Loss Components
dL
dv
g
v
d g
v f
g
g
dL
dP m
c
mm
c
mmmm
ctot
ρ ρ θ ρ ++=
2sin
2
Elevation
Friction
Acceleration
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Multiphase Flow
q L ,q g
q L ,q g
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Multi phase Flow
Characteristic
More than one phases flow through every-cross section of the pipe
Cross section occupied by a fluid phase
continuously change in the direction offlow due to slippage between phases
Ratio of this cross section for any phase,
over the whole pipe cross section isdefined as the Holdup (HL) for the liquidphase
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Holdup??
q L ,q g
q L ,q g
G
LHL = 0.5
HL = 0.25
HG = 1 - HL
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Definition of Variables
In multi-phase flow calculations,
Single-phase flow equations are modified• To account for the presence of second or third phase
• Involves mixture expressions for velocity, fluid properties
with weighting factors» Based on in-situ volume or mass fraction - holdup
• Weighting factors are flow pattern dependent
• Example: For Liquid-Gas flow, if HL is the weighting factor
for liquid,
( ) LG L Lm H H −+= 1 ρ ρ ρ
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No-Slip Holdup
In oil-water flow,
watercut f w is defined
as,
Where, f 0
= 1- f w
Under no-slip
condition, volume
fraction of liquid, λL Where, λL = 1- λG G L
L
L qq
q
+=λ
ow
ww
q f
+=
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Note!!
For Holdup: HL + Hg = 1
No-slip Holdup: λL + λg = 1Watercut: f O + f w = 1
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Velocities
Superficial Velocity (vSL): Assumes a given phaseoccupies the entire pipe area, Ap
Mixture Velocity (vm): Sum of phase superficial
velocities
p
g
Sg A
qv =
p
LSL
A
qv =
SgSL p
g L
m
vv A
qqv +=
+=
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Actual Phase Velocity
No-Slip flow: Gas and liquid flows at the mixture velocity
Because of the slippage between phases, liquid velocity
will slow down compared to gas in uphill flow and the vice
versa in downhill flow.
Actual velocities vL,vg and slip velocity vs are,,
L
SL L
H
vv =
g
Sg
L
Sg
g H
v
H
vv =
−=
1
Lgs vvv −=
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wwoo L f f ρ ρ ρ +=
wwoo L f f σ σ σ +=
wwoo L f f µ µ µ +=
Mixture Properties
Oil/water mixture flow No Slip condition
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Mixture Properties
Gas/Liquid mixture flow Slip or No-Slip condition
Numerous weighting rules used by different
authors, eg. For mixture viscosity,
)1( Lg L Ls H H −+= µ µ µ
)1(
, L L H
g
H
Lsor
−
×= µ µ µ
)1(, Lg L Lnor λ µ λ µ µ −+=
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Mixture Properties
Gas/Liquid mixture flow
Slip or No-Slip condition
Numerous weighting rules used by different
authors, eg. For mixture density,)1( Lg L Ls H H −+= ρ ρ ρ
)1(, Lg L Ln
or λ ρ λ ρ ρ −+=
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Multiphase Flow
Mixture Properties:
• Holdup Weighting or Dependence
•
Flow Regime Actual phase velocities Affected
by Slippage between phases
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Flow RegimesDuns and Ros (Vertical Uphill)
BubbleBubble Slug Annular Mist MistTaylor BubblePlug
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Stratified Flow - Downhill
q l , q gq l , q g
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Flow Regimes in Two Phase Flow
Bubble flow: (can be present in both upflow or
downflow)
• Slug flow: (can be present in both upflow ordownflow)
• Annular/mist flow: (can be present in both
upflow or downflow)• Stratified flow: (only possible in downflow or
Horizontal well)
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Pressure Traverse- Segmentation
Pressure drops arecalculated for each
calculation increments
(i=1,-----,m) in each
segments (j=1,----,n).
Uses pressure
gradient equation for
each increments and
segments.
Iterative calculation
Segments
1
2
3
4
Calc.increments
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Computing Algorithm
Marching Algorithm
Known Wellhead
pressure, pi
Calculate pi+1 in thecalculation increment
iteratively
A complete traverse is
calculated bysequentially marching
through the traverse.
dLdL
dp p
L
∫
=∆
0
ji
ji
m
i
n
j
L
dL
pd p ,
,11
∆
=∆ ∑∑
==
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Marching Algorithm
1. Assume a rate and calculate the pwf from IPR2. Start from the bottom segment L(more accurate ?) – Why?? And
estimate the end of segment pressure
3. Estimate avg. p and T in the segment
4. Calculate Fluid props in the segment at this avg. p & T
5. Calculate the end of segment pressure, if i t is not the same as theassumed one in step 2
1. Continue the iteration using standard methods such as Newton-
Raphson or Wagstein’s till it converges w ithin acceptable
tollerance
6. Now assume the second segment end pressure and repeat steps 2-5
7. When the surface terminal segment is reached, the calculated
pressure must match this given terminal pressure.
8. If not, either fol low the steps 1-7 till a match is obtained or
graphically solve like in the with systems approach
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Tubing Curve
0
500
1000
1500
2000
2500
3000
3500
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Production rate, STB/D
F l o w
i n g b o t t o m h o l e
p r e s s u r e , p s i
Tubing Curve
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Dimensionless Numbers
4Lv N Number,VelocityLiquid
L
LSL
gv
σ
ρ =
4gv N Number,VelocityGas
L
LSg
gv
σ ρ =
L
Lgd σ
ρ = N Number,DiameterPipe d
43L
N Number,ViscosityLiquid L L
L
g
σ ρ µ =
T Ph Fl R i M
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Two-Phase Flow Regime Map
- Duns and Ros
.
1 10 102 103
1
10
102
10-1
BUBBLE FLOW
PLUG FLOWSLUG FLOW
MIST FLOW
REGION IIIREGION II
REGION I
GAS VELOCITY Number, Ngv
L I Q U I
D V E L O C I T Y N u m b
e r , N L v
Vertical upflow
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Multiphase Flow
Determine Flow Regime• Phase Velocities
• Phase physical properties
• Pipe inclination• Production and injection
Determine Holdup - Dependence on
• Pipe inclination
• Flow Regime
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Multiphase Flow Calculation
Superficial Velocities
Flow Regime Maps
Holdup Slippage Velocity
Two Phase Flow pressure gradients
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Pressure Gradient Prediction
Vertical Upflow• Duns and Ros (1963)
• Hagedorn and Brown (1965)
• Orkiszewski (1967)
• Mechanistic Models: Ansari et al. (1994) Inclined Flow
• Beggs and Brill (1973)
• Mukherjee and Brill (1980)
Horizontal Flow• Dukler (1964)
Important Dimensionless
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Important Dimensionless
Variables
In multiphase flow calculations differentempirical equations for flow regimes and
liquid holdup are correlated with
dimensionless variables first proposed byDuns and Ros.
Knowing Phase rates and pipe inclination,
calculate Flow regime and liquid Holdup
Calculate pressure gradient (Ref. Note)
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Two-Phase Friction Gradient
Two phase friction factoris defined differently by
different authors as it is
no more analytically
predictable as in singlephase flow.
d g
v f
dL
dP
c
SL L L
Bubbleflow 2
2 ρ =
d g
v f
dL
dP
c
Sggg
mistflow 2
2 ρ =
;2
2
d g
v f
dL
dP
c
m f tp
twophase
ρ
=
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Two-Phase Hydrostatic Gradient
Two phase hydrostatic gradient is defined as,
θ ρ sinscc Hydrostati g
g
dL
dP
=
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Tubing gradients
9,810 ft at top perf.
0
2000
4000
6000
8000
10000
12000
0 500 1000 1500 2000 2500 3000 3500
Press ure (psig)
< - - - - -
D e p t h
( f t )
Ansari
Aziz
BB
HB
Muk BR
ORK
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Gradient Curves
0 1000 2000 3000 40000
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Pressure, psig
D e
p t h ,
f t
Gradient (A) Case 2 (B)
Case 3 (C) Case 4 (D)
Case 5 (E) Not Used
ABCDE
Inflow
Outflow
Inflow Outflow
Gas/Liq Ratio, scf/bbl
Gas/Liq Ratio, scf/bbl
(1) 100.0 (A) 100.0(2) 200.0 (B) 200.0(3) 400.0 (C) 400.0(4) 1500.0 (D) 1500.0(5) 3000.0 (E) 3000.0
Reg: Authorized User - Dowell Schlumberger
WHP= 200 psi
Rate = 2000 bpd 27/8”;350 API; 2000 F
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Hagedorn and Brown
Published in 1963
Widely accepted throughout industry
Based on data from 1500’ test “well”
Tubing size: 1”, 1 1/4”, and 1 1/2” nominal
Different liquids: water, oil: 10 - 110 cp
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Beggs and Brill
Published in 1973
Based on experimental data from inclined
90’ long acrylic pipe
Pipe size: 1” and 1 1/2”
Gas flow rate: 0-300 Mscf/D
Liquid flow rate: 30-1000 bbl/D
Inclination: ±90, 85, 75, 55, 35, 20, 15, 10,5, 0°
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Mukherjee and Brill
Published in 1983
Based on data from 1 1/2” ID inclined pipe
Developed three separate correlations
• Uphill and horizontal flow
• Downhill stratified flow
• Other downhill flow regimes
Wellbore Correlations
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Wellbore Correlations
High GLR Gas Wells
Cullender and Smith (1956)• Dry gas only. GLR > 100,000 scf/bbl
Fundamental Flow
• Dry gas only. GLR > 50,000 scf/bbl. Shallow depth,low pressure
Fundamental Flow adj
• Adjusts gas density for GLR > 50,000
Wellbore Correlations
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Wellbore Correlations
Low GLR Gas Wells
Gray (1974)• Wet gases, gas condensates
Ros and Gray (1961)
Oil well correlations may also be useful• Duns and Ros (1963)
• Hagedorn and Brown (1963)
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Pressure Balance
)()()()()()( q pq pq pq pq p pq p acc f cht flhsepwf ∆+∆+∆+∆+∆+=
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Liquid Holdup
Vg
VL
g L
L L
V V
V H
+≡
( ) g L L Lm H H ρ ρ ρ −+= 1
Determination Of Liquid
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Determination Of Liquid
Holdup
Oil/Water Flow
Gradiomanometer
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ρtool
Water
Holdup
ρwater ρoil
0%
100%100%
water
point
In this example
Hw = 40%
100%
oil
point Gradio
DensityError In
Measurement
Error In Expected
Downhole Oil Density
Uncertainty in
Water Origin and
Salinity
ρ= ρo Ho + ρw Hw
1 = Ho + Hw
ow
o H
w
ρ ρ
ρ ρ
−
−=
ow
w
o H ρ ρ ρ ρ
−−=
Hold-Up Determination
C f
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Critical Rate To Lift Liquid
Most gas wells produce some liquids
Liquids may be
• Vaporized in reservoir gas
• Free liquid in reservoir
Liquids will accumulate if not lifted to surface
Accumulated liquids will reduce productivity
For a given set of conditions, there is a minimumflow rate to lift liquids
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Models for Liquid Transport
Continuous film model Entrained drop model
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Critical Velocity
( )
−
=21
4141
912.1
g
g Lt v
ρ
ρ ρ σ
vt = terminal velocity of liquid droplet, ft/sec
ρL = liquid density, lbm/ft3
ρg = gas density, lbm/ft3
σ = interfacial tension, dynes/cm
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Critical Rate
Tz
A pvq t c
3060=
A = area open to flow, ft2
p = flowing pressure, psia
qc = critical rate, Mscf/DT = flowing temperature, ºR
vt = terminal velocity of liquid droplet, ft/sec
z = real gas deviation factor, dimensionless
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Examples : Perform
Recommended