```New Comprehensive Equation
Shu Luo
The University of Tulsa
Foam Flow Meeting, Jul.9th, 2013
1
Outline
• Introduction
• Background and Approach
• Model Formulation
• Model Validation
• Program Demonstration
• Summary
Foam Flow Meeting, Jul.9th, 2013
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• Minimum pressure drop in the tubing is
reached
• The liquid drops cannot be entrained by
the gas phase (Turner et al.)
• The liquid film cannot be entrained by
the gas phase (Zhang et al., Barnea)
are not the same
•
Foam Flow Meeting, Jul.9th, 2013
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IPR
Stable
OPR
Unstable
Transition
Point
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• As gas flow rate increases

(∆ )
( )
and
(∆ )
( )
• At low velocities
increase in
(∆ )
(∆ )
( )
decreases faster than
( )
• When two gradients are equal, minimum
occurs
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Definition based on
Mechanisms
• Two potential mechanisms of
transition from annular to slug
flow
 Droplet reversal
 Film Reversal
• Models are either based on
droplet reversal (Turner) or
film reversal (Barnea)
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Air-Water Flow
• Anton Skopich conducted experiments
in 2” and 4” pipes
• The results observed are different
based on film reversal and minimum
pressure drop
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Calculation Procedure
•
•
•
•
Total pressure drop is measured and gradient is
calculated
Holdup is measured and gravitational gradient is
calculated
Subtracting gravitational pressure gradient from total
By dividing the incremental pressure gradient by
incremental gas velocity, changes in gravitational and
frictional gradients with respect to gas velocity are
calculated.
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Magnitude of Gravitational vs. Frictional Gradient
with respect to Gas Velocity
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dPG vs. dPF
Air-Water, 2 inch, vsl=0.01 m/s
Minimum
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Total dp/dz
Air-Water, 2 inch, vsl=0.01 m/s
Film Reversal
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dP/dz)G vs. dP/dz)F
Air-Water, 2 inch, vsl=0.01 m/s
dp/dz)F is
zero
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dP/dz)G vs. dP/dz)F
Data from Netherlands (2 inch)
dp/dz)F is
zero
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dP/dz)G vs. dP/dz)F
TUFFP (3 inch, vsl=0.01 m/s)
dp/dz)F is
zero
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dP/dz)G vs. dP/dz)F
TUFFP (3 inch, vsl=0.1 m/s)
dp/dz)F is
zero
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dPG vs. dPF
Air-Water, 4 inch, vsl=0.01 m/s
Minimum
Foam Flow Meeting, Jul.9th, 2013
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Total dp/dz
Air-Water, 4 inch, vsl=0.01 m/s
Film Reversal
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dP/dz)G vs. dP/dz)F
Air-Water, 4 inch, vsl=0.01 m/s
dp/dz)F is zero
Film reversal
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•
•
occurs
We adopt the model of film reversal to predict
The reason for this adoption, as we will show
later, is because we are able to better predict
methodology.
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Outline
• Introduction
• Background and Approach
• Model Formulation
• Model Validation
• Program Demonstration
• Summary
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Background
Turner’s Equation
•
•
to the minimum gas velocity to lift the
largest liquid droplet in the gas stream.
Turner et al.’s Equation:
,
•
−
= 6.558
2
0.25
This equation is adjusted upward by
approximately 20 percent from his
original equation in order to match his
data.
Foam Flow Meeting, Jul.9th, 2013
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Background
Drawbacks with Turner’s equation
•
Turner’s equation is not applicable to all field data. Coleman
et al. proposed equation (without 20% adjustment )
,
•
•
•
−
= 5.465
2
0.25
Veeken found out that Turner’s results underestimate critical
gas velocity by an average 40% for large well bores.
Droplet size assumed in Turner’s equation is unrealistic
based on the observations from lab experiments.
Turner’s equation is independent of inclination angle which is
Foam Flow Meeting, Jul.9th, 2013
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Approach
Film Model
•
Two film models are investigated to predict liquid
 Zhang et al.’s model(2003) is developed based on slug
dynamics.
 Barnea’s model(1986) predicts the transition from annular to
slug flow by analyzing interfacial shear stress change in the
liquid film.
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Approach
Zhang et al.’s Model
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Approach
Zhang et al.’s Model
•
•
Momentum equation for annular flow:
With other equations and closure relationships, we
can solve this momentum equation and calculate
critical gas velocity
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Approach
Barnea’s Model
•
•

•
Constructing force balance for annular
flow and predict the transition from
annular to slug flow by analyzing
interfacial shear stress changes.
The combined momentum equation:
1
1

+
−
−  −   sin  = 0

Interfacial shear stress from Wallis
correlation:
2
1

=
2
(1 − 2)4
Foam Flow Meeting, Jul.9th, 2013
Schematic of Annular Flow
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Approach
Barnea’s Model
•
•
•
•
Solid curves represent
Interfacial shear stress from
combined momentum equation
Broken curves represent
Interfacial shear stress from
Wallis correlation
Intersection of solid and broken
solution of film thickness and
gas velocity at transition
boundary
Another transition mechanism
is liquid blocking of the gas
core.
Foam Flow Meeting, Jul.9th, 2013
Transition
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Outline
• Introduction
• Background and Approach
• Model Formulation
• Model Validation
• Program Demonstration
• Summary
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Three Main Modifications
• Accounted for variable liquid film thickness
• Changed the equation for liquid film friction
•
factor
Accounted for presence of liquid in the form
of droplet
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Model Formulation
•
In inclined wells, the film thickness is expected to vary
Vertical Well
Foam Flow Meeting, Jul.9th, 2013
Inclined Well
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Original Barnea’s Model
at Different Inclination Angles
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Non-uniform Film Thickness Model
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Non-uniform Film Thickness Model
•
•
Let A1=A2, we can find this relationship.
1
= [ 0,  +  ,  ]
2
If film thickness reaches maximum at 30 degree
inclination angle
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Non-uniform Film Thickness Model
•
We will use the following film thickness equation in
the new model:
≤  ≤
,  =

−  +

>
,  =   −  +
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Non-uniform Film Thickness Model
•
•
Only maximum film thickness will be used in the
model because thickest film will be the first to fall
Find critical film thickness δT by differentiating
momentum equation. δT equals to maximum film
thickness δ(π,30).
1
1
= [0 +  , 30 ] =
2
2
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Non-uniform Film Thickness Model
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Interfacial Friction Factor
•
•
Critical gas velocity calculated by Barnea’s model is
conservative compared to other methods. Fore et al.
showed that Wallis correlation is reasonable for small
values of film thickness and is not suitable for larger
film thickness liquid film.
A new correlation is used in the new model :
= 0.005 1 + 300
Foam Flow Meeting, Jul.9th, 2013
1+
17500 ℎ
− 0.0015

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Outline
• Introduction
• Background and Approach
• Model Formulation
• Model Validation
• Program Demonstration
• Summary
Foam Flow Meeting, Jul.9th, 2013
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Turner’s Data
•
•
•
106 gas wells are reported in his paper, all of the gas
wells are vertical wells.
wells are reported questionable in the paper.
are reported.
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Turner’s Model Results
Turner’s Data
Vg < Vg,c
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Vg > Vg,c
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Zhang et al.’s Model Results
Turner’s Data
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Barnea’s Model Results
Turner’s Data
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New Model Results
Turner’s Data
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Coleman’s Data
•
•
•
56 gas wells are reported, all of the wells are also
vertical wells.
These wells produce at low reservoir pressure and at
well head pressures below 500 psi.
Coleman reported gas velocity after they observed
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Turner’s Model Results
Coleman’s Data
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Zhang et al.’s Model Results
Coleman’s Data
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Barnea’s Model Results
Coleman’s Data
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New Model Results
Coleman’s Data
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Veeken’s Data
•
•
•
•
Veeken reported offshore wells with larger tubing
size.
67 wells, which include both vertical and inclined
wells, are presented.
Similar to Coleman’s data, critical gas rate was
reported.
Liquid rate were not reported in the paper. We
assumed a water rate of 5 STB/MMSCF.
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Turner’s Model Results
Veeken’s Data
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Zhang et al.’s Model Results
Veeken’s Data
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Barnea’s Model Results
Veeken’s Data
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New Model Results
Veeken’s Data
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Chevron Data
• Production data:
 Monthly gas production rate
 Monthly water and oil production rate
• 82 wells have enough information to analyze
•
•
Two tubing sizes: 1.995 and 2.441 inch
Get average gas and liquid production rate when
cap string is installed from service history.
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Production Data
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Turner’s Model Results
Chevron Data
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Zhang et al.’s Model Results
Chevron Data
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New Model Results
Chevron Data
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ConocoPhillips Data
• Daily production data and casing and tubing
•
•
•
pressure data are available
Select 62 wells including 7 off-shore wells
Two tubing size: 1.995 and 2.441 inch
pressure divergence.
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ConocoPhillips Field Data
Pc and Pt diverge
400 MCFD
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Turner’s Model Results
ConocoPhillips Data
Foam Flow Meeting, Jul.9th, 2013
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Zhang et al.’s Model Results
ConocoPhillips Data
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New Model Results
ConocoPhillips Data
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Outline
• Introduction
• Background and Approach
• Model Formulation
• Model Validation
• Program Demonstration
• Summary
Foam Flow Meeting, Jul.9th, 2013
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Program
• This program is developed in .net framework
•
using c sharp.
It consists two pages: single well calculation
and multiple well calculation.
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Summary
•
•
•
•
•
concluded that definition based on liquid film reversal is
most appropriate.
using liquid film reversal method.
The new model is applicable for both vertical and inclined
wells.
The new model is able to better predict the inception of