Gas Kick Behavior During Well Control Operations in ...
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LSU Historical Dissertations and Theses Graduate School
1990
Gas Kick Behavior During Well ControlOperations in Vertical and Slanted Wells.Edson Yoshihito NakagawaLouisiana State University and Agricultural & Mechanical College
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Recommended CitationNakagawa, Edson Yoshihito, "Gas Kick Behavior During Well Control Operations in Vertical and Slanted Wells." (1990). LSUHistorical Dissertations and Theses. 5082.https://digitalcommons.lsu.edu/gradschool_disstheses/5082
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Gas kick behavior during well control operations in vertical and slanted wells
Nakagawa, Edson Yoshihito, Ph.D.
The Louisiana State University and Agricultural and Mechanical Col., 1990
U M I300 N. Zeeb Rd.Ann Arbor, MI 48106
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GAS KICK BEHAVIOR DURING
WELL CONTROL OPERAT8QNS
IN VERTICAL AND SLANTED WELLS
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and
Agricultural and Mechanical College in partial fulfillment of the
requirements for the degree of Doctor in Philosophy
in
The Department of Petroleum Engineering
by
Edson Yoshihito NakagawaB. S., Escola de Engenharia de Piracicaba, Brazil, 1979
M. S., Universidade de Ouro Preto, Brazil, 1986 December, 1990
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ACKNOWLEDGMENT
The author expresses his gratitude to Petroleo Brasileiro S. A.,
PETROBRAS, for having sent him for this doctoral program and for the financial
support during the whole program.
Special thanks to his advisor, Dr. Adam T. Bourgoyne Jr., whose ideas,
suggestions, guidance and encouragement were of great value for the
completion of this work. Acknowledgements are extended also to Mr. O. A. Keliy
and Dr. V. Casariego for their attention, ideas and support during the
experimental phase of this project.
Thanks to Minerals Management Service for funding the experimental
research and to the following companies for donating or lending materials and
equipments for the experiments: Baroid, Rosemount, Teneco Inc. (Chevron)
and Cooper Ind..
The author dedicates this work to his wife Albertina, to his children
Renato and Aline, and to his parents Kichisaburo and Celia.
ii
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TABLE OF CONTENTS
ACKNOWLEDGMENT...................................................................................... ii
TABLE OF CONTENTS....................................................................................... iii
LIST OF TABLES..................................................................................................vii
LIST OF FIGURES.............................................................................................. viii
ABSTRACT............................................................................................................ ix
1. INTRODUCTION........................................................................................... 1
2. LITERATURE REVIEW.................................................................................. 4
2.1. GAS KICKS EXPERIMENTS IN VERTICAL W E LLS ......... 4
2.2. TWO-PHASE VERTICAL FLOW.......................................... 5
2.2.1. BUBBLE F LO W .................................................... 6
2.2.1.1. BUBBLE FORMATION AND STABLE
DIAMETER............................................................. 8
2.2.1.2. VELOCITY OF BUBBLES ..................... 12
2.2.1.2.1. SINGLE BUBBLE VELOCITY . . 13
iii
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2.2.1.2.2. INFLUENCE OF GAS
FRACTION ON BUBBLE VELOCITY 20
2.2.1.2.3. VELOCITY PROFILE
COEFFICIENT.......................................... 23
2.2.1.3. RELATIVE VELOCITY............................ 24
2.2.1.4. CONDITIONS FOR BUBBLE FLOW
EXISTENCE........................................................ 25
2.2.2. SLUG FLO W ........................................................ 26
2.2.2.1. LIMITS FOR SLUG FLOW REGIME . . . 28
2.3. TWO-PHASE INCLINED UPWARD FLOW ..................... 30
2.4. TWO-PHASE FLOW THROUGH ANNULAR
SECTIONS ................................................................................ 32
3. EXPERIMENTAL PROGRAM .................................................................... 35
3.1. EXPERIMENTAL APPARATUS.......................................... 35
3.2. EXPERIMENTAL PROCEDURE........................................ 39
3.2.1. DESIGN OF EXPERIMENTS............................... 39
3.2.2. TEST PROCEDURE............................................. 40
4. EXPERIMENTAL RESULTS ...................................................................... 43
4.1. RESULTS IN VERTICAL POSITION................................. 44
4.2. RESULTS IN INCLINED POSITIONS................................. 52
4.3. RESULTS OF THE PLASTIC PIPE TESTS....................... 54
iv
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5. MODELING OF THE EXPERIMENTAL RESULTS................................... 56
5.1. DESCRIPTION OF THE MODEL FOR VERTICAL
FLO W ......................................................................................... 56
5.2. DESCRIPTION OF THE MODEL FOR INCLINED
FLO W ......................................................................................... 61
5.3. COMMENTS ABOUT THE MODEL AND HOW IT
COMPARES WITH THE EXPERIMENTAL DATA ................... 65
6. SUBROUTINES FOR APPLICATION OF THE MODEL............................ 72
7. CONCLUSIONS ......................................................................................... 75
8. RECOMMENDATIONS............................................................................... 77
NOMENCLATURE........................................................................................... 80
REFERENCES................................................................................................ 85
ADDITIONAL BIBLIOGRAPHY...................................................................... 95
APPENDIX A: EXPERIMENTAL D ATA.......................................................... 99
v
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APPENDIX B: SUBROUTINES FOR IMPLEMENTATION OF THE
BUBBLE MODEL ............................................................ 103
APPENDIX C: BASIC EQUATIONS USED IN THE ANALYSIS OF THE
EXPERIMENTAL D A TA ...................................................116
VITA.................................................................................................................... 123
vi
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LIST OF TABLES
Table 3.1. Variables collected during the experiments. . ......... . ................ 42
Table 5.1. Best fit coefficients for the gas velocity equation............................ 58
Table 5.2. Coefficients for the equations of Ug = f (Um).................................... 64
vii
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LIST OF FIGURES
Figure 2.1. Different types of bubble velocity..................................................... 7
Figure 2.2. Velocity of d r bubbles in water..................................................... 16
Figure 2.3. Bubble Velocity reduction due to swarm effect............................. 21
Figure 2.4. Variation of K factor with mixture Reynolds number.................... 24
Figure 3.1. Flow Loop...................................................................................... 36
Figure 3.2. Detail of the annulus in horizontal position (0 = 90° from vertical).. 37
Figure 3.3. Experimental apparatus................................................................. 38
Figure 4.1. Gas Velocity x Mixture Velocity in Vertical position...................... 46
Figure 4.2. Gas Fraction from Experiments and from Bubble Model............. 48
Figure 4.3. Gas Velocity from Experiments and from Bubble Model.............. 49
Figure 4.4. Gas Fraction x Usg from Experiments and from Bubble Model. . 50
Figure 4.5. Comparison between Slip and Non-slip Gas Fractions............... 51
Figure 4.6. Variation of Gas Fraction at Different Inclinations........................ 53
Figure 5.1. Curves of K factor for gas-water and gas-mud mixtures.............. 60
Figure 5.2. Relation between gas and mixture velocities............................... 62
Figure 5.3. Gas Velocity and Gas Fraction for l= 0° (vertical)......................... 66
Figure 5.4. Gas Velocity and Gas Fraction for l= 10°....................................... 67
Figure 5.5. Gas Velocity and Gas Fraction for l= 20°....................................... 68
Figure 5.6. Gas Velocity and Gas Fraction for l= 40°....................................... 69
Figure 5.7. Gas Velocity and Gas Fraction for l= 60°....................................... 70
Figure 5.8. Gas Velocity and Gas Fraction for l= 80°........................................ 71
Figure 6.1. Structure of the subroutines shown in Appendix B....................... 74
viii
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ABSTRACT
An experimental study for the determination of the gas fraction and gas
velocity in the mud during gas kick control operations was performed. It
consisted of two-phase flow of gas-water and gas-mud mixtures through a 14 m
(46 ft) fully eccentric annular section of an experimental apparatus. The
inclination from the vertical position was varied from 0° to 80°. The results from
these tests allowed the evaluation of a previous model for gas bubbles in
vertical wells. They also provided support for modifications of that model
concerning two aspects: extending its application to slanted wells and
improving the simulation in vertical wells with non-newtonian fluids.
ix
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1. INTRODUCTION
Sometimes, during drilling operations, fluids from the formation being
drilled start to flow into the well. The basic reason for this inflow is the presence
of a higher pore pressure in ihe formation than the pressure exerted by the
drilling fluid inside the well. In general, this situation in not desirable since this
fluid influx from the formation, if not controlled properly, can be the cause of
many problems such as fractures in shallower formations, contamination of the
environment and blowouts. A blowout is an uncontrolled flow of formation fluids
to the surface. It often results in personal injury, loss of life, environment
damage, or loss of drilling equipment.
Immediately after detecting a fluid inflow, called kick, it is necessary to
take steps to handle the situation. The application of methods and techniques to
control the fluid inflow and to remove it from the well in a safe way is designated
as well control operations. Well control operations are an important
consideration when planning a well and when training drilling personnel. For
simulation of these operations, some assumptions are made concerning the
behavior of the kick inside the well. In general, the main assumption is that the
kick flows as a unique slug through the well to the surface. This hypothesis is
not true in most of the cases. However, it can give good results in case of water
or even oil influx. Nevertheless, in case of gas influx that premise is definitely
1
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Introduction 2
not valid. It is known that the gas breaks up into bubbles and spreads along the
well during well control operations.
This gas behavior leads to very different results for wellhead pressures,
gas velocity and time to control the kick when compared to the available results
using the continuous slug assumption. Thus, a better understanding of the gas
behavior inside the well during well control operations is essential. This
knowledge would lead us to more efficient and safer procedures to control gas
kicks.
Presently, a model for gas kicks in vertical wells is available (Bourgoyne
and Casariego, 1988). The model is based on experimental and theoretical
studies done on kick control in the last few years at Louisiana State University.
The present work is a continuation of this project and its goal is to contribute for
the improvement of the model in general and specifically, in considering the
effect of wellbore inclination.
The focus in studying this two-phase flow of gas and drilling fluid inside
the well is centered on the evaluation of the gas concentration profile along the
well and the gas velocity. However, these two factors are inter-related and
dependent on other parameters like bubble size and shape, liquid and gas flow
rates, phase properties and wellbore inclination.
The program of study included three main sections. First, an
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Introduction 3
experimental program was developed to obtain some data on gas-water and
gas-mud flows through an annular section in vertical and slanted positions. An
experimental apparatus was built for this purpose. The second part of this
program was the analysis of the data obtained from the experiments and
comparison with the present model for vertical wells. For the tests simulating
slanted wells, the analysis was focused on the differences, with respect to the
vertical flow, regarding the flow pattern, gas concentration and gas velocity.
Finally, the third section involved some improvements to the present model and
the definition of a method accounting for the effect of well inclination.
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2. LITERATURE REVIEW
The first part of this review includes a summary of previous works that
show the importance of studying the velocity of bubbles and slugs during well
control operations. The second part is focused on the analysis of bubble and
slug flows in vertical sections. Since the primary interest of this study is on
two-phase flow through annular sections in slanted holes, some few works
available in the literature are summarized in the last two sections. Most of the
results from these works are based on concepts and correlations for vertical
pipes.
2.1. GAS KICKS EXPERIMENTS IN VERTICAL WELLS
Until recently, the procedures for gas kick control and the kick simulators
had as a primary assumption that the gas flowed through the annulus to the
surface as a continuous slug. With this concept in mind, Rader et al. (1975)
performed some experiments trying to define the factors contributing for the
velocity of large slugs in annular sections. Therefore, they developed a
correlation «or the velocity of large bubbles and tried to apply it in a large scale
experiment using a 1829 m (6000 ft) research well. However, the correlation did
not work well because the gas kick did not behave as anticipated. It was noticed
the occurrence of bubble fragmentation which caused the spreading of the gas
4
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Literature Review 5
zone, lower gas velocity, and smaller than expected casing pressure during
well control operations.
Later, Mathews (1980) also reported the occurrence of bubble
fragmentation when experimentally evaluating some kick control methods for
handling gas rise in a shut-in well. He observed for the conditions of this study
that the bubble fragmentation rate did not depend on the initial kick volume and
was smaller in more viscous fluids. After this work, it was evident the need for a
better understanding of the bubble fragmentation process and its effects on
bubble rise velocity and concentration.
Motivated by this need, Casariego (1981, 1987) performed experimental
and theoretical studies on gas kick behavior in vertical wells and developed a
bubble fragmentation model that achieved good agreement with data from large
scale experiment in a 1829 m (6000 ft) research well.
2.2. TWO-PHASE VERTICAL FLOW
This review is limited to bubble flow, slug flow, and the transition between
them. These are the flow patterns that one should expect inside the well during
normal kick control operations in vertical wells (Casariego, 1987). Bubble flow
is characterized by the gas phase being distributed as discrete bubbles in a
continuous liquid phase. When the gas fraction in a bubble flow is increased, a
transition to slug flow will occur with the bubbles starting to coalesce and form
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Literature Review 6
larger bubbles. After this transition, the slug flow takes place as a sequence of
large bullet-shaped bubbles occupying most of the flow cross-sectional area
and being separated by each other by liquid bridges containing small bubbles.
For inclined wells, the expected flow patterns are bubble, elcngated
bubble, slug and churn flows. The elongated bubble flow is a type of intermittent
flow where long bubbles flow along the top side of the cross-sectional area
segregated from the liquid phase and separated from other bubbles by liquid
bridges. Churn flow is a chaotic flow where the continuous phase is not clearly
defined and the recirculation of liquid and gas is very pronounced. The
development of annular flow, defined by a continuous gas phase in the center
of the cross sectional area and a continuous liquid film along the walls, would
be expected only if one looses control of the well.
2.2.1. BUBBLE FLOW
The literature is not consistent in the definition of gas bubble velocity
terms. Thus, the definitions of bubble velocity used throughout this work will be
addressed prior to reviewing previous work in this area. The previous work will
then be presented using the defined parameters. A single bubble flowing in an
infinite and static liquid medium has velocity U „ (Fig. 2.1.a) relative to an
observer above the liquid container.
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Literature Reviezv 7
* u
6
V o oo ° . S o
o W oo°§o
O ° o
Single Bubble Swarm of Bubbles in Static Liquid in Static Liquid
(a) (b)
V « i
°o °oi y <3
O '6 U “°
oAV /
o ° ° oo o °9 o o ° oO Q r 1— k
Bubble Velocity with Fluid Injection
(c)
Figure 2.1. Different types of bubble velocity.
A medium is considered infinite if the bubble velocity is not influenced by the
walls of the medium. If the same bubble is in a swarm of bubbles flowing
through a stagnant liquid, now its velocity relative to an observer above the
liquid container will be U0 (Fig. 2.1 .b). A third situation occurs if gas and/or liquid
is being injected into the medium with gas flow rate qg and/or liquid flow rate qi
(Fig. 2.1.c). In this case, the bubble velocity will be Ug. The relative velocity Ur is
defined as the velocity that the bubble has with respect to the average velocity
of the liquid around it.
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Literature Review 8
For this work, the emphasis in studying bubble flow is on the prediction of
bubble velocity and gas concentration expressed as a volume fraction
commonly called the gas hold-up. However, before coming to this point, we
must define how these parameters are inter-related and how other variables
influence them. Essentially, it is the resultant of forces caused by surface
tension, viscosity, inertia, and buoyancy that affects the shape, size, trajectory,
concentration and velocity of the bubbles.
2.2.1.1. BUBBLE FORMATION AND STABLE DIAMETER
Many authors have studied bubble formation processes to come up with
some expression for the equivalent diameter or volume of the bubbles formed.
Bubble size is known to affect its velocity and concentration. Wallis (1969) and
Tsuge (1986) presents a summary of these works for Newtonian liquids while
Rabiger and Vogelpohl (1986) cover this subject in detail for Non-newtonian
fluids. They state that, for Newtonian and pseudo-plastic liquids (whose
viscosity decreases with increasing shear rates), the bubble is deformed
according to the pressure conditions prevailing during its ascension. Therefore,
the bubble’s shape right after the bubble's formation is unimportant for the
prediction of the shape and velocity of the bubble while ascending in the
column. But, for visco-elastic fluids, the bubble's initial shape persists during the
ascension. Thus, for simulation purposes, it should be vital to produce the
bubbles as ciose as possible to the real conditions.
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Literature Review 9
Drilling fluids generally exhibit a pseudo-plastic behavior. Under drilling
conditions, bubbles are not in a stable equilibrium by the time of their formation.
Generally, the flow field and the properties of the fluids will define the maximum
stable size (critical size) of a bubble. Marrucci and Nicodemo (1967) concluded
from their tests with air and water (Newtonian behavior) that coalescence
ocurred near the bubble formation device and defined the bubble size along the
column. This coalescence process depended on the gas flow rate and
concentration.
Taylor (1932, 1934), Karam and Bellinger (1968), Rallison and Acrivos
(1978), and Hinch and Acrivos (1979, 1980) addressed the problem of stable
size for liquid drops in another iiquid. Hinze (1949, 1955) developed
expressions for the evaluation of stable diameters of droplets in air under
different flow conditions. He stated that two dimensionless groups govern the
splitting up of a drop or bubble: a generalized Weber number and a viscosity
group. For turbulent flows, this generalized Weber number is a relation between
external dynamic pressure (inertia) and surface tension forces:
(2 .1)
where
p i = liquid density,
Ur = relative velocity,
Db = bubble diameter,
cs = surface tension.
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Literature Review 10
The viscosity group relates the viscosity to the surface tension forces:
Vpg aD b
where
(2 .2)
\l g = gas viscosity,
p g = gas density.
Hinze stated that the break up of the drop or bubble would occur at a certain
critical Nwe which is related to Nv by:
f (Nv) = function of the viscosity group.
Unfortunately, each type of flow field and fluid properties require different
experimental tests for the definition of this correlation. The reason is that the
mechanism of bubble break-up is not the same in laminar and turbulent flows.
In laminar flows, the interaction between the dispersed and continuous phases
generates viscous forces that lead to the deformation and break-up of the drops
or bubbles. In turbulent flows, the kinetic energy of the continuous phase is the
main factor in the breaking-up of the dispersed phase. Later, Sleicher (1962)
refined Hinze's results and justified the use of a viscosity group different from
the one given by Eq. 2.2:
However, he still used the same form of Eq. 2.3 with adjusted coefficients for the
determination of the stable drop sizes in turbulent flows.
NWe = C [1 + f (Nv)j (2.3)
where
(2.2a)
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Literature Review 11
More recently, Krzeczkowski (1980) also studied this problem and
defined the process as being regulated by the Weber number, Nwe: Strouhal
number, Nstr. Laplace number, Ni_a. and viscosity ratio pi /pg, where:
t = bubble break-up time,
p i = liquid viscosity.
The Strouhal number is the relation between the oscillatory nature of the drop
break-up mechanism and the relative velocity of the phases in the medium. The
Laplace number is the ratio between surface tension and viscosity forces.
Krzeczkowski was interested in the kinematics of drop deformation and in the
duration of disintegration. He concluded that the Weber number had the
strongest influence in the mechanisms of drop deformation and disintegration
as well as in the break-up duration. Larger Weber numbers lead to smaller
break-up times (or Strouhal numbers).
Casariego (1987) proposes a similar way to estimate the stable diameter
of bubbles. The difference compared to Hinze's approach is that Casariego
uses the Reynolds number, a relation between inertial and viscosity forces,
instead of the generalized Weber number (Hinze could have used the
(2.4)
(2.5)
(2 .6)
and
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Literature Review 12
Reynolds group also). He also uses another form of viscosity number, the
Morton number (shown in Eq. 2.15) and defines the relation between both
numbers for which the fragmentation would occur. In this way it is possible to
calculate the stable bubble diameter and the velocity of a single bubble
simultaneously. Although the dimensionless numbers used by Hinze are not the
same as the ones used by Casariego, the numbers used by both correlate the
same basic effects of inertia, viscosity and surface tension.
2.2.1.2. VELOCITY OF BUBBLES
The equation for the average gas velocity Ug (Zuber and Findlay, 1965)
is
Um = average mixture velocity,
K =velocity profile coefficient,
U0 = average bubble swarm velocity in stagnant liquid.
The average values are calculated over the flow cross sectional area. The
velocity of a swarm of bubbles is primarily a function of the single bubble
velocity Uoo and of the gas fraction a. Among several correlations, the most
common one assumes that the bubble velocity decreases exponentially with the
liquid fraction:
Ug = U0 + K Um (2.7)
where
U0 = U„„ (1 - cx)n, (2 .8)
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Literature Review 13
where
n = exponent for the effect of a swarm of bubbles on the bubble velocity.
The mixture velocity Um is defined as
Um = Usl + Usg (2-9)
where
Usi = = superficial liquid velocity, ^ 1 q)
Usg = - = superficial gas velocity, ^ 1 1 )
qi = liquid flow rate,
qg = gas flow rate,
A = cross-sectional area of the medium.
An alternative way to define average gas velocity, using the concept of
superficial gas velocity, is
Ug = = bubble velocity ^2 1 2)
From equations (2.7) to (2.12), the conclusion is that the gas velocity and
the gas fraction are inter-dependent and both can be determined if Uw n and K
are known. The following sections consider each of these factors separately.
2.2.1.2.1. SINGLE BUBBLE VELOCITY
An extensive literature is available on single bubble velocity in infinite
media containing Newtonian fluids (Davies and Taylor, 1950, Peebles and
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Literature Review 14
Garber, 1953, Haberman and Morton, 1954, Saffman, 1956, Harmathy, 1960,
Mendelson, 1967, Maneri and Mendelson, 1968, Grace, 1973, 1976, 1986,
Sangani, 1986) or non-Newtonian fluids (Wasserman and Slattery, 1964,
Astarita and Apuzzo, 1965, Acharya et al., 1977, Buchholz et al, 1978).
The experiments show that the single bubble velocity depends on the
properties of the phases, size (equivalent bubble diameter Db) and shape of the
bubbles (Grace and Harrison, 1967). For the simplest cases in Stokes regime,
some models were developed for prediction of bubble shape and velocity (Cox,
1969, Buckmaster, 1972, Youngren and Acrivos, 1976). However, in most
practical situations this problem becomes very complex and experimental
results are used for the development of correlations. These results can be
presented in different ways. Some authors correlate the drag coefficient Cd
defined as
Pi U„ ti D§ (2.13)
with the bubble Reynolds number
m _ _ _ Db u~ PiNbRs" pi (2.14)
where
gc = conversion factor for unit systems,
F = drag force.
The fluid properties effect is either expressed through the Morton number
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Literature Review 15
Nm = Nfl = - flite- = Pa).N^Npr Pi2 o3 gg (2.15)
or by another similar dimensionless number. The Morton number depends just
on the fluid properties and it represents the relation between the viscosity and
surface tension values. Given the relation Cq = f (Nrs) for each bubble shape
and fluid properties, one can then calculate IL, and Db simultaneously.
Another way to present the results is through correlations between IL
and Db or Eotvos number which relates the buoyancy and the surface tension
effects and is defined as
(di - pn) a D§n E6_ - (216)
The advantage of this type of correlation is that U „ and Db are explicitly defined.
In this work, this approach will be adopted. In Figure 2.2 the relationship
between bubble velocity and Eotvos number is shown for air bubbles in water.
In Region 1, the Stokes expression for solid spheres is
y 9 Dl (Pi - Pg)18 w (2.17)
For ellipsoidal bubbles (region 3 and 4) the velocity is defined by
p | D b U o o / | o e - 7 \ m . . - . 1 4 9— w — =(J-.857) Nm (218)
where
J = 0.94 N'757 for 2< N < 59.3 (2.18a)
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Literature Review 16
or J = 3.42 N'441 for N > 59.3 (2.18b)
N = 4 En Nm’-149 (—El—)"'143 0 M .0009 (2.18c)
for w in N s / m2.
Single Air Bubble Velocity
wE
QO
&G>JQ3m
100
Region £ Slug
Region 3 Region I
Sphericasoidal10
Region 2
1
Region 1 Spherical Transitior
Spherical Ellipsoidal
0.1 * 1—0.0001 0.001 0.01 0.1 1 10 100 1000
Eotvos number
Figure 2.2. Velocity of air bubbles in water.
Region 2, for 0.001 < Eo < 0.1, corresponds to a transition between
spherical and ellipsoidal shapes and can be represented by
Uoo = e<a ln nEo + b) (2.19)
where
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Literature Review 17
a =In Uool'
[U»2j
In Neo1Neo2.
b = In U„o2 - a In Neo2
and
Uo»i = spherical bubble velocity for Egi = 0.001
Uoo2 = ellipsoidal bubble velocity for E02 = 0.1.
In region 5, the velocity of spherical cap bubbles is expressed by
Uoo = .707 J (p l' pg) 9 V pi
and the limit for this region is defined by the slug velocity given by
Usiug = -312 ^g B L ie a (D o + D0'
(2 .19a)
(2 .19b)
(2 .20)
(2 .21)
Abou-EI-Hassan (1986) developed a generalized correlation for bubble
velocity that can be applied to regions 2, 3 and 4 shown in Fig. 2.2. The
advantage of this type of correlation is that it is unnecessary to first determine
the flow regime and bubble shape to be able to apply the approppriate
correlation. His solution relates two dimensionless numbers Nv (velocity
number) and NF (flow number) in the following way:
Nv = 0.75 (log NF)2 (2 .22)
where
Nv =U- D f3 p,2/3
p,173 a173 (2.22a)
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Literature Review 18
N gDfpPCPrPg)F ^,4/3 a1/3 (2.22b)
These two numbers were obtained through an dimensional analysis made with
the following parameters related to the bubble motion: bubble diameter, liquid
kinematic viscosity, gravity, momentum per unit volume (pilL), and the physical
parameter ap2. This correlation was compared successfully with data in the
literature for liquid density between 0.72 and 1.2 g/cm3, liquid viscosity between
2.33 x 10‘4 N.s/m2 (0.233 cp) and 5.9 x 10'2 N.s/m2 (59 cp) and surface tension
between 15 d/cm and 72 d/cm.
To here, all the correlations were from experiments using Newtonian
fluids. For Non-newtonian fluids, some works have been done for prediction of
the bubble volume during its formation (Acharya et al., 1978) and finding
correlations for drag coefficients in Power Law and Bingham fluids (Hirose and
Moo-Young, 1969, Bhavaraju et al., 1978) in the creeping flow region.
Acharya et al. (1977) present equations for bubble velocity in non-
Newtonian media for a wide range of modified Reynolds number. In the low
modified Reynolds number region, the drag factor is
Cn _ 24 F(n')NMRe (2.23)
where
F (n') = a function of n\ bubble velocity and size,
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Literature Review 19
np' i j2 - n* _NMRe=—— rr— — = Modified Reynolds number lo
n' = pseudo-plasticity index
k' = consistency index.
For the transition low-high Reynolds number, the drag factor is given by
Bhavaraju et al (1978):
Cn 16 F-i(n')NMRe (2.24)
where
Fi(n’)= 2n’ - 1 3(n’ - 1)y2 1 - 7.66 n '- 1(2.24a)2
In the region of large Reynolds number, Mendelson's (1967) expression can be
used:
U = .. / -2-CL + 9- Pb. “ V Dbpi 2 (2.25)
Lastly, for completely developed slug flow, the same equation for Newtonian
fluids can be used (Eq. 2.21).
Abou-EI-Hassan (1986) compared his generalized correlation for bubble
velocity with the results from Acharya et al. (1977) for air bubble velocity in
pseudo-plastic fluids. The only modification necessary in Eqs. (2.22a) and
(2.22b) is the substitution of f (n') k' (Uo=/Db)n'1 for the viscosity m, where f (n') is a
correction term, n' is the flow behavior index and k' is the consistency index.
Another aspect to be considered is the wall effect. If the bubble is large
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Literature Review 20
when compared to the vessel, the proximity from the confining walls will reduce
the actual speed of the bubble. In this case, the value for LL, calculated by the
previous equations will reduce by a factor which is a function of the diameter
ratio of the bubble and vessel (Uno and Kintner, 1956; Strom and Kintner, 1958;
Harmathy, 1960; Wallis, 1969). Fluid properties also have influence on the
reduction of the bubble velocity due to wall effects (Uno and Kintner, 1956;
Wallis, 1969; Sangani, 1986).
2.2.1.2.2. INFLUENCE OF GAS FRACTION ON BUBBLE VELOCITY
Several authors have suggested different ways to account for the
reduction on bubble velocity with increasing gas fraction. Some theoretical
investigations are available (Sangani, 1986) but they are restricted to either
small gas fractions or specific ideal geometrical configurations. Normally,
semi-empirical methods are applied in the development of relations for velocity
of swarms of gas. Marrucci (1965) applied the potential flow theory (irrotational
and incompressible flow) for spherical bubbles in the range 1 < Nr0 < 300 and
proposed this relation based on the energy dissipation formulation:
Bhatia (1969) developed another equation applicable to low viscosity, pure
gas-liquid systems, and restricted to large bubbles in large pipes:
(2.27)
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Literature Review 21
However, the most common approach for this problem is to assume the
relation:
Ur = U„ (1 - a)nw (2.28)
or U0 = Uoo (1 - a)nR (2. 28a)
where nw is the exponent suggested by Wallis (1969) and or is the one
suggested by Richardson and Zaki (1954). Wallis (1969) summarizes the
results from several investigators and recommends values of nw between 2 and
0 for regions 3, 4 and 5 in Fig. 2.2. When the bubble size reaches region 5, nw
becomes less than unity due to the entrainment of bubbles in each other’s
wakes. Eventually, the transition to slug flow begins and from this point, nw is
zero. Figure 2.3 shows a comparison of Equations 2.26, 2.27 and 2.28a.
0.90.8
0.7
0.68
Marrucci0.3
0.2
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Gas Fraction
Figure 2.3. Bubble Velocity reduction due to swarm effect.
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Literature Review 22
Richardson and Zaki (1954) use Eq. (2.28a), instead of Eq. (2.28), in
sedimentation and fluidisation experiments of solids in liquids. For these
systems, they prove that nR depends on the particle Reynolds number
M PI Uoo Ds0" Pi (2.29)
and of the diameter ratio (Ds / Dp), where Ds is the particle diameter and Dp is
the pipe diameter. They present equations for nR = f (Npr0, Ds / Dp) in different
ranges of Reynolds number. Notice that nR is the same n defined in Eq. (2.8).
When the diameter ratio approaches zero, the equations for spherical particles
reduce to:
n = 4.65 for Npr6 < 0.2
n = 4.45 NpRe'0 03 for 0.2 < NPRe < 1.0
n = 4.45 NpRe-0.10 for 1.0 < Nprq < 500.
n = 2.39 for 500.0 < NPRe (2.30)
Lockett and Kirkpatrick (1975) compared several correlations for swarm effect
and proposed a correction factor for Richardson and Zaki’s correlation for
regions 3 and 4 and n equal to 2.39 (constant). They concluded that no
correction factor would be necessary for gas fractions less than around 0.3.
Based on Richardson and Zaki's results, Casariego (1987) uses the
following equation for gas bubbles in liquids:
„ _ 3 + 5.805 Nisi!01 -------------2 (2.31)
This equation was recommended for the range 2.39 < n < 4.65. When Eq. (2.31)
gives a value outside this range, n assumes the appropriate limiting values.
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Literature Review 23
It is important to notice that the values of the exponent nw from Wallis are
in a different range when compared with n = nR from Richardson and Zaki. The
reason is that, Ur in Eq. (2.28) and U0 in Eq. (2.28a) are related to each other
according to
U0Ur = 1 „ ~ I VA> (2.32)
Therefore, nw should be n - 1. From Richardson and Zaki's (1954) experimental
results, we notice that the difference is larger than the unity.
2.2.1.2.3. VELOCITY PROFILE COEFFICIENT
The velocity profile of the two-phase mixture contributes to the increase
of gas velocity. This contribution is defined through the coefficient K in the gas
velocity equation (Eq. 2.7). As the velocity profile flattens, when the flow
changes from laminar to turbulent, K decreases (Nicklin, 1962; Wallis, 1969;
Sadatomi and Sato, 1982) as shown in Figure 2.4. The factors affecting this K
coefficient, when no recirculation occurs, are the phase flow rates, geometry
and fluid properties.
The recirculation of the continuous phase can also be included in this
coefficient. Recirculation occurs when the dispersed phase creates a
preferential path through the section and causes back-flow of the continuous
phase. Rietema and Ottengraf (1970) studied this problem for laminar flow of air
bubbles through stagnant liquids in pipes and developed a model using the
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Literature Review 24
principle of minimum energy dissipation. Lockett and Kirkpatrick (1975) also
studied this problem for ideal bubble flow (with K = 1). However, these
1.6■ Circular Pipe
•Annular Section1.4
1.2
1.00 10000 20000 30000
Mixture Reynolds Number
Figure 2.4. Variation of K factor with mixture Reynolds number.
theoretical developments are restricted to conditions much different from
practical cases in well control. For this reason, in this study, the effect of liquid
recirculation will be incorporated to the velocity profile coefficient.
2.2.1.3. RELATIVE VELOCITY
Another important type of velocity is the relative velocity between the
phases. It is defined as
U ^ i ^ l V K U , (233)
where
Ur = gas velocity with respect to the liquid around it,
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Literature Review 25
(2. 33a)
It can be shown that Ur, as presented in Eq. 2.33, results from combination of
Eq. 2.7 and Eq. 2.32, and it is applicable to either case shown in Fig. 2.1.
2.2.1.4. CONDITIONS FOR BUBBLE FLOW EXISTENCE
Taitel et al. (1980) define the conditions under which bubble flow is
possible to occur. In large diameter pipes, they assume the transition to slug
flow beginning at a=0.25 when the dispersion forces resulting from turbulence
are not dominant. Furthermore, they use Harmathy's equation (1960) for bubble
velocity and come up with the equation for the transition:
The limit between small diameter and large diameter pipe is defined according
to:
This equation results from comparison of Harmathy's equation for large bubbles
and slug velocity given by Nicklin (1962):
In small diameter pipes, bubble flow cannot exist because the bubble velocity is
larger than Usiug- Here, the bubbles approach the slugs and coalesce with
them.
(2.34)
(2.35)
Usiug — -35 Vg Dp (2.36)
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Literature Review 26
Another transition exists between bubble flow and dispersed bubble flow.
Dispersed bubble flow occurs under conditions of high turbulence where
dispersion forces break up the larger bubbles and extend the bubble flow for
situations where a>0.25. Using Hinze's study (1955) in the determination of the
maximum stable diameter of the dispersed phase, Taitel's suggestion for the
transition bubble - dispersed bubble is
Usl + Usg — 4 ■D-429P
s i.PI.
'089[g ( p i - p g ) } 446
PI
PI.
■°72 446Pi (2.37)
McQuillan and Whalley (1985) propose the use of Weisman relation
which is similar to Eq. (2.37) but presenting smaller effect of pipe diameter:
U 6 -8 Dp1 1 2 [ g g ( p i - p g)] 278
Pf112 Pf444 (2.38)
This relation comes from data in horizontal experiments. However, McQuillan
states that it should not be any effect of inclination on this transition because of
the turbulence caused by high liquid velocities.
2.2.2. SLUG FLOW
Similarly to the bubble flow case, the emphasis of this review is on slug
velocity and concentration. Fernandes et al. (1983) proposed a very detailed
model to predict 17 parameters related to slug flow and obtained good
agreement with experimental results. However, this model was developed just
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Literature Review 27
for vertical pipe flows. Wallis (1969) reviews several works related to the
determination of slug velocity and presents this equation for Usiug:
usl«g=kyiH|E£arwhere
k = .345 (1 - et--01 Nt/-345]j |1 . @[(3.37-Neb)An]}
M [^ / gPp(Pl-Pfl) PIV Lipnr
N e. _ (PI - Pg) g D P a
for
Nf = inverse viscosity number,
Neo = Pipe Eotvos number,
and m being a function of Nf:
Nf < 18 m = 25
18 < Nf <250 m = 69 Nf -0.35
(2.39)
(2.39a)
(2.39b)
(2.39c)
(2.39d)
m — 1 n
When the slug motion is inertia dominant, the constant k reduces to
0.345 and if pg « p i, Eq. (2.39) becomes:
Usiug = 0.345 Vg Dp. (2.40)
The range for inertia dominant slug motion is Nf > 300 and Neo > 100. For
viscosity dominant flow, Nf < 2 and Neo > 100. For air-water and mud-gas
systems in kick operations, generally the slug flow is inertia dominant.
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Literature Review 28
The determination of gas fraction a is done by:
a _ Usg _ UsgUg K Um + Ki U0 (2-41)
where
Ki = coefficient that accounts for the change in relative velocity due to the
approaching velocity profile produced by the wake of a preceding bubble.
In slug flow, U0 = U„ since the exponent n = 0. In circular pipes with fully
developed turbulent flow (Nr9 > 8000), K = 1.2 and Ki = 1. In annular sections,
K varies from 1.2 to about 1.125 as a function of the diameter ratio if no
recirculation occurs.
2.2.2.1. LIMITS FOR SLUG FLOW REGIME
According to Taitel et al. (1980) the boundaries for slug flow are bubble
flow, dispersed bubble flow, and annular flow. They consider churn fiow as an
entry region phenomenon that eventually, turns to slug flow if enough pipe
length is available.
The criterion to determine the entry region length of churn flow is based
on observations that a stable slug has a length 8 to 16 times larger than the
pipe diameter, independent of fluid properties. Calculations for the necessary
length to form these stable slugs lead to:
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Literature Review 29
—- m + 8.93 V 9 Dp (2.42)
where Ie = entry length for stable slug formation.
On the other hand, McQuillan and Whalley (1985) claim that churn flow
occurs because of the increase of gas flow rate in the slugs until the flooding of
the falling liquid film around the slug. Their criterion for this transition is more
complex than the previous one since it involves the calculation of slug length
and film thickness.
The transition to annular flow, according to Taitel et al. (1980) occurs
when the gas has enough velocity to suspend the entrained liquid droplets. The
final equation for this transition is
Notice that this transition does not depend on US|.
Again, McQuillan and Whalley (1985) suggest a different criterion for this
transition:
(2.43)
(2.44)
Both equations fit experimental data equally well.
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Literature Review
2.3. TWO-PHASE INCLINED UPWARD FLOW
30
There is not much previous work related to two-phase flow through
inclined pipes. Brill and Beggs (1984) present a summary of correlations for
prediction of liquid holdup and pressure gradient in two-phase inclined flow.
Most of the available work is for near horizontal flow, uphill or downhill. Just two
correlations cover all the inclinations between 0° and 90°: Beggs and Brill
(1973,1984) and Griffith et al. (in Beggs and Brill, 1984).
In inclined flows, it is interesting to know how the gas fraction varies as
the pipe inclination increases. Equations 2.7 and 2.8 predict that when gas
fraction increases, the gas velocity decreases and vice-versa. Beggs and Brill
(1973) show that, for the same air and water flow rates, the air fraction
decreases as the pipe inclination increases from 0° to about 40° from vertical.
After this point, air fraction starts to increase until the pipe reaches the horizontal
position. Bonnecaze et al. (1971) also noticed that the velocity of slugs
increases as the pipe inclination changes from 0° to about 20° from vertical.
From 20° to 80° the velocity decreases until it reaches the same value for the
vertical position. To explain this fact, they proposed an equation based on the
calculated velocity distribution around the bubble. However, comparison of the
data with the proposed equation was not satisfactory.
Wallis (1969) presents other data for slugs confirming the results from the
previous investigators. He performs a more detailed study on this velocity
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Literature Review 31
behavior and concludes that Nf in Eq. (2.39b), Neo in Eq. (2.39c), and 0
(inclination from vertical) are the factors defining Ue / Uo». In general, Wallis'
curves show increasing slug velocity to 40°- 50°, almost stable values around
40°- 60°, and decreasing values from 50°- 70° to horizontal position.
The flow pattern determination is another important point in this study.
The flow patterns observed by Spedding and Nguyen (1980) for inclined and
vertical upwards flow in 45.5 mm pipes, always commenced as bubble or slug
flow regardless of the magnitude of the liquid flow rate for the range of liquid
flow rates studied. According to their results, the transition between bubble and
slug flow practically does not change when the inclination varies from 0° to
almost 90° from vertical.
According to Taitel and Dukler (1976), Barnea et al. (1980), and
Weisman and Kang (1981), stratified flow occurs just in horizontal pipelines.
Stratified flow is defined by both liquid and gas phases being continuous and
separated from each other. When the pipe is slightly inclined upwards,
intermittent flow happens over a wide range of superficial liquid and gas
velocities. Intermittent flow includes elongated bubble and slug flow.
Weisman and Kang (1981) present some data for air-water flow in 25
mm and 45 mm pipes. They showed the occurrence of bubble and churn flows
in vertical pipes. However, inclining the pipe, they noticed that the slug flow
replaced the churn flow. Besides that, the bubble flow region was extended to
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Literature Review 32
somewhat lower gas flow rates as inclination angle was increased. Based on
this result, they modified Taitel and Dukler's (1976) correlation for the transition
bubble-intermittent flow:
where 0 = inclination angle from vertical. Another conclusion from their tests
was that the variation in inclination did not affect substantially the transitions to
annular and dispersed flows. However, we do not expect to have these flow
regimes in normal well control operations.
2.4. TWO-PHASE FLOW THROUGH ANNULAR SECTIONS
Caetano (1986) studied two-phase flow of air-water and air-kerosene
mixtures in an annulus 16 m (52.5 ft) long with 76.2 mm (3 in.) OD x 42.2 mm
(i .66 in.) ID. In his experiments, he defined flow pattern maps for concentric and
fully eccentric geometries. These maps show that the pipe eccentricity causes a
small shift in the transition line between bubble and slug flow towards lower Usg.
Further, the eccentricity affects sensibly the friction factor, which is important for
prediction of pressure gradients. However, in our particular case, friction factor
is not so important since the values of expected flow rates are small. Yet, from
these tests, Caetano noticed that the transition bubble-slug occurred around a =
0.20 in concentric annulus and a = 0.15 in fully eccentric annulus. Also,
Caetano proposed models for calculating liquid holdup and pressure gradient
for each flow pattern. These models are based on Taitel's (1980) equations for
Vg Dp [ Vg DpUsg ^ J Usg + U s l j78 (1 - 0.65 sin 0)
(2.45)
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Literature Review
two-phase flows in vertical pipes.
33
Sadatomi and Sato (1982) also performed some experiments in non
circular channels, including an annular configuration. They defined this relation
for the velocity of slugs:
Usiug = .345 Vg Dep (2.46)
where
Dep = equi-periphery diameter = D0 + Dj,
D0 = external diameter of the annulus,
Dj = internal diameter of the annulus.
This equation fitted well Caetano's (1986) data. Sadatomi and Sato (1982) also
calculated the coefficient K that accounts for the velocity profile and concluded
that this coefficient is approximately equal to the ratio between the maximum
velocity by the mean velocity in the profile. Furthermore, they constructed a flow
pattern map for the tested channels and noticed that the channel geometry does
not have too much influence on the transition from bubble to slug flow and from
slug to annular flow. Another important point was the experimental definition of
the cross-sectional distribution of void fraction. In bubble flow, the bubbles
tended to concentrate in the corners or walls of the cross-sections while in slug
flow, the slugs followed a path through the centra! core.
Rader et al. (1975) did experimental work to determine the essential
factors affecting the rise velocity of large bubbles in annular sections. They
noticed major effects of the annulus geometry (diameters) and of the inclination
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Literature Review 34
angle. The velocity increased according to the same Dep used by Sadatomi
(1982). It also increased with inclination angles to 45° and then, decreased.
This is similar to the results reported by Wallis (1969). But, eccentricity of the
pipe, bubble length, and surface tension did not affect significantly the velocity
of large bubbles.
Later, Casariego (1987) conducted some experiments in an annular
section 161.9 mm (6.375 in.) x 60.3 mm (2.375 in.). He measured bubble rise
velocities for a wide range of gas bubble sizes under different flow conditions
and liquid viscosities. Also, he observed the flow patterns that could occur
during the migration of the gas influx. From these experiments and theoretical
study some important relations were obtained:
1. Drag coefficient Co with Nr6;
2. Nr8 with Karman number N« that relates the bouyancy and viscosity
effects and is defined as
|ie = effective liquid viscosity:
3. Stable diameter for bubbles defined through a relation between Nr0
and N,x (or Morton number, Eq. 2.15);
4. Effect of bubble concentration on the velocity of a swarm of bubbles.
These results were necessary for the evaluation of the size, concentration, and
velocity of the gas zone during well control operations.
NK= 1.155 Vgp|D§ (PI - pg)
Pe (2.47)
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3. EXPERIMENTAL PROGRAM
The experimental program consisted of:
1. Construction of the experimental system:
2. Preliminary tests for adjustment of the apparatus:
3. Definition of the procedure for data collection;
4. Tests with gas-water mixtures at different inclinations:
5. Tests with gas-mud mixtures at different inclinations.
3.1. EXPERIMENTAL APPARATUS
The experimental facility simulates two-phase flow through an annular
section and it is composed of a flow loop and a fluid handling system.
The flow loop is about 14.6 m (48 ft) long (Fig. 3.1) and can be inclined at
any desired angle between vertical and horizontal. During the tests, liquid
injection occurs through the internal pipe of the annulus while gas injection
comes from the bottom of the loop. The two-phase flow develops at the bottom
and returns upward through the annulus. For gas fraction measurements, the
pressure actuated valves (V1, V2, V3, and V4 in Fig. 3.1) act simultaneously.
Valve V1 is a three way valve that diverts the liquid flow directly to the output
when the other valves are closed.
35
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Experimental Program 36
SYPHONBREAK
LINE
- ©
Jd kT
JdpT
FLOW LOOP
GASINPUT
AIRFOR
VALVEACTUATORS
MUDINPUT
TWO-PHASE FLOW OUTPUT
Figure 3.1. Flow Loop.
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Experimental Program 37
The annulus, on the right leg of the loop, is 14m (45.8 ft) long (Fig. 3.2)
and its diameters are 154.0 mm (6.065 in.) by 60.3 mm (2.375 in.). The inner
pipe was locked in the fully eccentric position since this was felt to better
represent the well control conditions in inclined wells. This part of the loop has
sensors to measure the following parameters: loop pressure (P), loop
temperature (T), and differential pressures at three locations (DP sensors).
ANNULAR SECTION
13.97 m
4.84 m
2.06 m
GAS + MUDMUD
14.36 m
External Pipe: 168.3 mm (6 5/8 in.) x 154.0 mm (6.065 in.) Internal Pipe: 60.3 mm (2 3/8 in.) x 52.5 mm (2.067 in.)
Figure 3.2. Detail of the annulus in horizontal position (0 = 90° from vertical).
The fluid handling system (Fig. 3.3) is an integration of three subsystems.
The first deals with the liquid injection into the loop. It is composed of a 47.7 m3
(300 bbl) water/mud tanks, a centrifugal pump, and a triplex pump. The second
subsystem controls the gas flow and includes the air compressor, gas pipeline,
and gas flow meter unit. Lastly, the third subsystem collects the output flow,
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Experimental Program 38
SCHEME OF THE EXPERIMENTAL
FACILITIES CENTRIFUGALPUMP WATER/MUD
TANKS
TRIPLEXPUMP
GASFROM
PIPELINE
AIRCOMPRESSOR DEGASSER
GAS FLOWMETER STATION
FLOW LOOP
-oCDi
PRESSURETANK
a .
Figure 3.3. Experimental apparatus.
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Experimental Program 39
controls the pressure in the loop, and separates the gas and liquid phases. It
has a 47.7 m3 (300 bbl) pressure vessel rated at 5516 KPa (800 psi) working
pressure, the choke valve and a degasser.
The monitored variables in the fluid handling system include: pump
pressure, liquid flow rate, gas pressure and temperature, gas flow rate, storage
tank pressure, liquid level in storage tank, and back pressure in the choke.
3.2. EXPERIMENTAL PROCEDURE
In the first part of this study the design of the experiments were
performed. The goal was to choose the fluid properties and the proper range of
liquid and gas flow rates that could reproduce the conditions in field operations.
The second step involved the development of a test procedure. Here, the
emphasis was on the optimization of the procedure to accomplish accurate and
reliable results in the minimum run time. The last step was the processing of the
data collected during the tests for posterior analysis.
3.2.1. DESIGN OF EXPERIMENTS
The liquid phase adopted in the tests were water and water-base muds.
Two different muds consisting of water-bentonite mixtures were used. The first
mud had a density of 1068 kg/m3 (8.9 Ibm/gal), a plastic viscosity of 0.01
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Experimental Program 40
kg/(m.s) (10 cp) and a yield point of 4.79 Pa (10 lbf/100ft2), and the second one
had 1080 kg/m3 (9.0 Ibm/gal), 0.019 kg/(m.s) (19 cp) and 9.10 Pa (29 lbf/100ft2).
For the tests the gas phase consisted of natural gas with specific gravity 0.6i.
The chosen parameters for the test matrix were the superficial liquid
velocity (Usi) and the superficial gas velocity (Usg). For the delineation of the
range of velocities, it was necessary to consider the normal conditions found
during well control operations. Also, the results from previous works helped in
choosing a proper interval covering bubble and slug flows. The chosen Usi
range was from 0.21 m/s (0.7 ft/s) to 0.52 m/s (1.7 ft/s) while the Usg range was
from 0.03 m/s (0.1 ft/s) to 0.58 m/s (1.9 ft/s). For the range of flow rates selected,
the points of the test matrix were predicted to be in the bubble and slug flow
regions according to two-phase flow maps developed by Caetano (1986) and
by Sadatomi et al. (1982) for vertical flow through an annular section.
3.2.2. TEST PROCEDURE
An experimental run begins with the loop locked in position at the
desired inclination and all the valves in the loop in the open position. The liquid
and gas flow rates are adjusted to the desired values and when the two-phase
flow reaches steady conditions, the computerized data acquisition system
begins recording the desired parameters. About one minute later, the valves are
closed but the data collection continues until complete stabilization of the
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Experimental Program 41
system, which happens when the gas is completely separated from the liquid
phase. These steps are repeated for each point of the test matrix.
During the first tests, it was very difficult to maintain constant liquid and
gas flow rates because of a tendency of pressure variations in the system to
occur. Later, the problem was solved using a process control computer to
control the flow rates automatically. Another difficulty in obtaining the desired
gas flow rate was the difficulty in keeping constant pressure in the system for
each test. During each test, an iterative procedure was performed to keep the
test points as close as possible to those chosen in the test matrix.
During the tests, data was collected at a rate of 3 records/second.
Simultaneously, some variables were displayed on a strip chart, supplying a
real time visual plot of the parameters. Table 3.1 shows all the collected
variables and their limits. After the tests, the collected data were prepared for
analysis and the gas fraction, gas velocity, and related variables were
calculated. Table 3.1 also shows the range of values covered during the tests.
The loop pressure was kept between 1724 kPa (250 psi) and 2896 kPa (420
psi) to avoid the rapid expansion of gas bubbles that would occur between the
bottom and top of the annular test section at low pressures.
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Experimental Program
Table 3.1. Variables collected during the experiments.
42
CALIBRATION TEST RANGE
VARIABLE UNIT HIGH LOW LOW HIGH
Liquid Flow Rate m3/min 0.74 -0 . 0 2 0 . 2 0 0.49
gpm 195.00 -5.00 53.00 130.00Loop Pressure kPa 6894.65 -68.95 1723.66 2896.00
psi 1 0 0 0 . 0 0 -1 0 . 0 0 250.00 420.00
Loop Temperature °C 75.00 -50.00 23.89 35.00
°F 167.00 -58.00 75.00 95.00
Loop DP! kPa 137.89 -41.37
psig 2 0 . 0 0 -6 . 0 0
Loop DP2 kPa 2 0 . 6 8 -6.89
psig 3.00 -1 .0 0
Loop DP3 kPa 48.26 -13.79
psig 7.00 -2 . 0 0
Tank Pressure kPa 6894.65 -34.47 1723.66 3102.59psig 1 0 0 0 . 0 0 -5.00 250.00 450.00
Tank DP cm H20 254.00 -12.70 43.18 177.80
in H20 1 0 0 . 0 0 -5.00 17.00 70.00
Gas Flow Rate m3/min 14.16 0 . 0 0 1.32 11.80
scf/h 30000.00 0 . 0 0 2800.00 25000.00Gas Pressure kPa 6894.65 -34.47 4481.52 4964.15
psig 1 0 0 0 . 0 0 -5.00 650.00 720.00
Gas Temperature °C 93.33 0 . 0 0 2 1 . 1 1 32.22
°F 2 0 0 . 0 0 0 . 0 0 70.00 90.00
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4. EXPERIMENTAL RESULTS
A total of 362 data points were collected, including 139 tests with gas-
water mixtures, 115 with gas-mud1, and 108 with gas-mud2. Mud 1 was an
1068 kg/m3 (8.9 Ibm/gal) water-bentonite mix with plastic viscosity and yield
point equal to 0.01 kg/(m.s) (10 cp) and 4.79 Pa (10 lbf/100ft2). Mud 2 was an
1080 kg/m3 (9.0 Ibm/gal), 0.019 kg/(m.s) (19 cp) and 9.10 Pa (29 lbf/100ft2) mix.
The tests were run at O' (vertical), 10°, 20°, 40°, 60° and 80° from the vertical
position. For a summary of the results, see Appendix A.
The tests were to measure the gas fraction for each combination of liquid
and gas fiowrates. Four ways were used to calculate the gas fraction from each
test. The first three derived values were obtained with the differential pressure
readings during the circulation period. The last one was obtained with the static
differential pressure along the loop after closing the valves and waiting for the
stabilization of the variables. Among these four values, the ones calculated with
DP2 and DP3 were, as expected, the best ones since these sensors were
positioned around the middle section of the flow loop. The results from DP1
sensor were similar to the other two sensors but probably not as good due to
larger entrance effects. Lastly, the values calculated during the static period
were less accurate because of leak problems with the valves. The basic
43
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Experimental Results 44
equations used for analysis of the data are presented in Appendix C.
4.1. RESULTS IN VERTICAL POSITION
Figure 4.1 shows the relation between the gas velocity Ug and the
mixture velocity Um. Figure 4.1.a. suggests that the superficial liquid velocity Usi
does not affect the parameters U0 and K (Eq. 2.7), that is, a single line can
describe the relation in the range of interest. The bubble model from Bourgoyne
and Casariego (1988) predicts K values between 1.13 and 1.15 which is very
close to the regression value of 1 .1 0 shown for the slope in the figure. For U0,
the model predictions fall in the range between 2.44 x 10'2 m/s (0.08 ft/s) and
1.95 x 10' 1 m/s (0.64 ft/s), while the tests show an average value of 1.89 x 10' 1
m/s (0.62 ft/s) for the intercept. Actually, the parameters U0 and K are dependeni
on Usi since for larger Usi, smaller K and larger U0 are obtained. However, in the
turbulent region of the tests, K is almost constant and the contribution of U0 to
the total velocity Ug is small compared to the effect of the [K Um] term.
The situation changes for muds as seen in Figures 1.4.b. and 1.4.c. Now,
the gas velocity is more sensitive to variations of Usi- For larger Usi, the swarm
velocity U0 decreases. The cause for smaller U0 might be the break-up of the
bubbles into smaller sizes due to larger turbulence caused by larger Usi- Also,
the K factor, given by the slope of the lines, is larger than for gas-water mixtures.
The reason for larger K factors is the higher viscosity of the mud compared to
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results 45
the water. Higher viscosities imply smaller Reynolds numbers and larger K
values (Fig. 2.4).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results
1.8 1.6
«T 1.4 1.2
t 1
•§ 0 . 8
m 0.6 O 0.4
0.2 0
1.8 1.6
1.4- 1 . 1.2-
1o 1■i 0 .8 ' « 0.6 O 0.4
0.2 0
1.8 1.6
1* 1.4 .§ .1.2
1o 1 -§ 0 . 8
> 0 . 6
0 0.4.0 . 2
0
* vvJ. 1■
_nPj 1
r
i " 1&lu(JC —1 .1U
0 0 .“T 1 1 2 0 .4 0.
Um i6 0 .(m/s)
8 i.;
J
i
• slope = 1,40^ ------
I 1 10
1 1 "1.2 0
1 i '1.4 0
Um
— i.6 0 .(m/s)
1 "I1" '11 1.8 1 .
Fig. 4.1 .a. Vertical
Gas - Water
□ Usi = 0.24 m/s
O Usi = 0.37 m/s
A Usi = 0.52 m/s
Fig. 4.1 .b. Vertical
Gas - Mud1
□
O
Usi = 0.21 m/s
Usi = 0.52 m/s
3JFig. 4.1 .c. Vertical
Gas - Mud2X?p
w
□
O
Usi = 0.21 m/s |
Usi = 0.52 m/sj
slope_= 1,40
21 1 1
0 01" i" i
2 0 4 0."V"!' 1 6 0 8 1 .
Um (m/s)
Figure 4.1. Gas Velocity x Mixture Velocity in Vertical position.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results 47
For tests in vertical position, figures 4.2 and 4.3 shows how well the
previous bubble model (Bourgoyne and Casariego, 1988) predicts the gas
fraction and gas velocity for gas-water mixtures. In fig. 4.2.a., for gas-water, we
can see that the model works better at the small gas fraction region below 0.3.
The calculated stable bubble size results in good prediction of a in this region,
but the bubble size or the K factor should be larger at the higher gas
concentration region. Assuming larger bubble size or K factor, the gas velocity
would increase and the gas concentration would decrease.
In figure 4.4 it is easier to notice that the model tends to over-predict gas
concentrations in the region of large gas fractions. However, the error is larger
for gas-mud mixtures than for gas-water mixtures.
The last figure for the tests in vertical position gives us an idea of relative
(slip) velocity between the gas and liquid phases (Fig. 4.5). The larger the slip
velocity, the larger is the difference between the experimental gas fraction and
the input gas fraction (non-slip gas fraction). As expected, the slip velocity
decreases for larger superficial liquid velocity and increases for larger
superficial gas velocities or non-slip gas fractions. For non-slip gas fractions
closer to one, the slip velocity tendency should revert, that is, start to decrease
because of flow pattern change. However, this range was not covered in these
tests.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gas
Frac
tion
from
Bubb
le M
odel
Ga
s Fr
actio
n fro
m Bu
bble
Mod
el
Experimental Results
0.7-
48
0 .6 .
0.5
0.4
0.3
0.2
0.1
/•
r '
iY£ &*
4wA*
Fig. 4.2.a. Vertical
Gas - Water
□ Usi = 0.24 m/s
O Usi = 0.37 m/s
A Usi = 0.52 m/s
j T - C ' J c o ' i t i o c D r ^O o o O o O o
Gas Fraction from Experiments-® 0.7
Fig. 4.2.b.Vertifial
Gas - Mud1
□ Usi = 0 .2 1 m/s
O Usi = 0.52 m/s
0.6
£ 0.53“ 0.4'
- 0.3 c o
•■S 0 .2 .ca
£ o.iV)toO 0
0.7
0.6
0.5
0.4
0.3
0.1
0c \ i w ^ I f i
ci O o OCD f"; o o
o
r h
-------- 70
/
ts
0
£ ✓#
/
J1 L 1 ^
H 1 1 1 1
&1 1 1 1
rI T I 1 1 1 1 1 1 1 1 1 T T T T T T T T
O t- O J C O ' ^ I - I D C D N . d o d o d o o
Gas Fraction from Experiments
Fig. 4.2.C.Vertical
Gas - Mud2
□
O
Usi = 0.21 m/s
Usi = 0.52 m/s
Gas Fraction from Experiments
Figure 4.2. Gas Fraction from Experiments and from Bubble Model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results 49
CO
E,"®■oo
©JO-O3mEo
oo©>(0©
0
1.8
1.61.41.2
1
0.80.60.40.2
0
/0 C S m A O 0*✓
01 1 ^ .
Vertical1
'AGss - Water
J r
Aw
1□
O
A
Usi = 0.24 m/s
Usi = 0.37 m/s
Usi = 0.52 m/s-Hr
✓0
T T T T T T ■ i ( T T T T T T T T T T T T T T T
O O O O T“" T” T*“ -
Gas Velocity from Experiments (m/s) ch -8
Fig. 4.3.b. Vertical
Gas - Mud1
Usi = 0.21 m/s
Usi = 0.52 m/s
o 1 -6 o 1 .4® 1-2.Q -O
“ 0.81 0.6
1
©E.1 .8 .
.6
.4
.2
1
8
6
4 2
■§1 1 1
- 1 -O 1-Q3COE 0 .
0 .>,•B o ._o © 0 .CO ©0
/0s0*
r✓ J[p C
0s a h
/* r r
0s B-n
S OH
/0
' - C r -
^0 .4o 0 . 2 ©>co ©0
0
S✓' i i r
Ar l
\
r f rn □
✓0 H
/ - f n J
✓ £P✓
0™ " 7
0* *1 1 1 T T T T T T T T T T T T T T T T T T T T T T T T
CM CO COO CM •M- CO 00O O O O t - t- t- t-
Gas Velocity from Experiments (m/s)
Fig. 4.3.C. Vertical
Gas - Mud2
O O O O -r- -T- -r-' T-
Gas Velocity from Experiments (m/s)
Usi = 0.21 m/s
Usi = 0.52 m/s
Figure 4.3. Gas Velocity from Experiments and from Bubble Model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results 50
0.6
0.5
• | 0.4' 2 0.3'
co 0 .2 .0.1
0
Usi = o.i14 IT/sso
JAJ
<£f
0 . 6 '
0.5'Fig. 4.4.a., b. o « *
Vertical Gas - Water £ 0.3
re 0.2-0
0.10
= 0- >2 IT /$
)
\ Ar
.
O ^ W (0 ^ W (Oo O O O o o
Superficial Gas Velocity (m/s)
o i- cvj eoci o o
in co o o o
Superficial Gas Velocity (m/s)Experiments Model
O
0 . 6 '
■2 0.4 o£ 0.3.
co 0 .2 O
i Usi = 0 -21 n i/5
G 0 ^ o
A U
: 4M— T T T T
p r
T T T T T i l l T T T T T T T T in i
0.6
0.5Fig. 4.4.C., d. o 0 a
Yertteal «Gas - Mud1 £ 0.3
co 0 .2 .
o i- c\| co -<3- m co o o o o o o
Superficial Gas Velocity (m/s)
00.1
U5l = 0 -?2 n i/S
)©0 ^
GjGT
O 1- CM CO 'S- in CD O O CD O ci O
Superficial Gas Velocity (m/s)
o2u_CACOo
iUsI = 0-:!1 rr/s
0 (>o
/ cf3U L
= ous"T11 1 1 nil 'in i
Fig. 4.4.e., f.Vertical
Gas - Mud2
o t- cm n in coo o o o o o
Superficial Gas Velocity (m/s)
0.5-
• I 0.4-
I 0.3-
co 0 .2 - 0
0 . 1-
0 -
If)DTT
TT = 0 - >2 nri/S
•
r©< 6
0 sr —
y Q
O i- CM CO in coo o o o o o
Superficial Gas Velocity (m/s)
Figure 4.4. Gas Fraction x Usg from Experiments and from Bubble Model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results 51
0.7
0.6
c 0.5
§ 0 .4
« 0.3
0.2
0.1
to M;C\J r^o
o CO
Fig. 4.5.a. Vertical
Gas - Water
□ Usi = 0.24 m/s
O Usi = 0.37 m/s
A Usi = 0.52 m/s
Non-slip Gas Fraction
Fig. 4.5.b. Vertical
Gas - Mud1
Usi = 0 .2 1 m/s
Usi = 0.52 m/s
0.7
0.6
c 0.5
§ 0.4
»0 .3cfl O 0.2
0.1
0
*/•
/S
✓/
✓• 1 J$s* f 1i 3 r
" •✓ o cu□ u
✓ o
u. /
0.6
0.5
nO0.2
0.1
0CMo
COd d
LOd
od
Non-slip Gas Fraction Fig. 4.5.C.Vertical
Gas - Mud2
Usi = 0 .2 1 m/s
Usi = 0.52 m/s
t- cm to rt in cqd d d d d d d
Non-slip Gas Fraction
Figure 4.5. Comparison between Slip and Non-slip Gas Fractions.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results
4.2. RESULTS IN INCLINED POSITIONS
52
Starting from vertical, as the inclination increases the gas fraction
decreases until a point where the tendency is reversed and the gas fraction
begins to increase again (Fig. 4.6). This reversal point depends on the fluid type
being used and on the flow conditions. For water, the minimum gas fractions
were obtained around 60’ from vertical while for mud 1 and mud 2 the reversal
point ocurred between 40° and 50°. For the mud mixtures at lower liquid flow
rates (Figs. 5.6.c and 5.6.e), a fast increase in gas fraction occurs after the
reversal point and the gas concentration returns to the same level for the
vertical position. This behavior was not observed in the tests with water.
For the same gas and liquid flow rates, the variations of gas fraction at
different inclinations are resulting from distinct mechanisms governing the gas
velocity in each position. The separation between the phases and the
recirculation of liquid increases with the inclination, affecting the flow pattern
and the flow profile, which control the gas velocity and concentration.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gas
Frac
tion
Gas
Frac
tion
Gas
Frac
tion
Experimental Results 53
0.5'
0.4'
0.3.
0 .2 '
0.1
0
<A
a""~
p,5B
A □
P °
u<A n
f *
* 5 *i
I ’
U5 l = ( .24 n/S
0.4-
c 0.3 oTS ® 0.2u.it)Cfl
^ 0.1
Fig. 4.6 a., b. Gas - Water 0
, *A ^ ' a
o(
A
“ C> 1
J
% *
**
IA £
QI >
§ u?i = ( CMin
w cq n ©o o o oUsg (m/s)
0.5-
0.4-
0.3-
0 .2 -
0 .1 -
Q.
0.5'
0.4-
0.3-
0 .2 -
0 .1 .
0 .
❖ £ I f f
I ] t
Us = c .21 In/s|i k k I
O Tc
1 1 1 1 j 1 V >1 | 1 1 ■ 1- CM CO T5 d o c
Usg (m/s
WVqWlVt in cc 6 o o)
V\ r
T7
, VA ¥ \X r?0T|r e
i ■ >V
dbl = Q.21 /?
&A o0“ 40“
□ *10°
ooto
o V20°
0oCO
o I— CM co ^ Ui COO o O o o O
Usg (m/s)
0.4'
0.3'
0 .2 '
0.1'
Fig. 4.6.C., d. g
i A
AA
6 0□ c ! v ^
r
r*
■l
r v
U s = C 52
O t- C\l CO W ioo ci o o o o
Usg (m/s)
0.4-
c 0.3- oO $
0 .2 -wCu
O v- CM CO -M" LO COd d d d d d
Usg (m/s)
O 0.1 -
Fig. 4.6.e., f. Gas - Mud2
A
5i
u? i=C 52 n/§
o cm co in coo d d o d d
Usg (m/s)
Figure 4.6. Variation of Gas Fraction at Different Inclinations.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results
4.3. RESULTS OF THE PLASTIC PIPE TESTS
54
Qualitative tests were run in a 1 2 . 2 m (40 ft) long transparent section
having a 161.9 mm (6.375 in.) by 60.3 mm (2.375 in.) annulus. The flow rates
were such that the superficial liquid velocity was between 0.21 m/s (0.7 ft/s) and
0.52 m/s (1.7 ft/s) and the superficial gas velocity was between 0.20 m/s (0.65
ft/s) and 0.52 m/s (1.7 ft/s). Therefore, the flow conditions and the geometry were
very similar to the tests in the steel pipe. Also, the adopted inclinations were 0°
(vertical), 10°, 20°, 70° and 8 8 °.
In vertical position, bubble flow occurred in ali the flow rate combinations.
For smaller Usg neither recircuiaiion nor preferential path for the bubbles were
observed. In the range of large Usg some larger bubbles with high speed were
formed, causing some recirculation. However, the flow pattern should still be
considered as bubble flow.
At 10° and 20° the flow behavior was very similar to each other. Again,
bubble flow was present in all the cases. However, at points of larger gas
fraction, some larger bubbles with high speed were produced and apparently,
caused more recirculation than in the vertical tests. It was noticed that the
recirculation induces the bubbles to have variable speed along the annulus,
that is, bubbles are accelerated and decelerated each time they gain and loose
some energy from the surrounding flow.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Experimental Results 55
As the inclination increased, the flow pattern gradually changed since
larger bubbles were formed. At 70° the elongated bubble flow pattern was
predominant. However, the long bubbles became unstable when the liquid flow
rate increased and tended to break-up into smaller bubbles. But, for larger gas
flow rates, very fast elongated bubbles appeared because of the recirculation.
Lastly, at 8 8 ° the elongated bubble flow was more stable than at 70° and
within the flow rate ranges used in the tests, the flow pattern did not change.
The only effect of the input flow rates was on the size of the elongated bubbles.
The larger the gas fraction, the larger were the bubbles.
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5. MODELING OF THE EXPERIMENTAL RESULTS
The new model calculates the gas fraction and the gas velocity in
two-phase flow through a fully eccentric annular section in vertical and slanted
holes. It is expected to work with gas-water and gas-mud mixtures in the range
of flow parameters and fluid properties adopted in the experiments.
5.1. DESCRIPTION OF THE MODEL FOR VERTICAL FLOW
Combining equations (2.7) through (2 .1 2 ), we obtain:
^ £ = 1 1 .(1 -o )n + KUm (5 .1 )
With this equation, it is possible to calculate the gas fraction a if the flow
parameters Usg and Um are known. However, it is also necessary to know the
relations for the single bubble velocity Uco, the swarm effect coefficient n, and the
velocity profile coefficient K (functions of the flow parameters and fluid
properties). These are the relations the model has to provide for the estimation
of gas fraction and gas velocity.
For the determination of the three relations, three sets of experiments
with different equipments would be necessary. The first type of tests would be
for the determination of single bubble velocity in drilling fluids. There has been
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modeling of the Experimental Results 57
a number of previous studies for Newtonian fluids but none for single bubble
velocity in muds. Next, it would be necessary to determine the swarm effect on
the bubble velocity. This would not be an easy task especially with muds. Third,
another set of tests would provide the data for the determination of the K factor.
Since just the last iype of tests was done, it was necessary to assume Equation
5.1 was of the correct form and then search for the values of K, n, and U „ that
would minimize the deviation of the observed and predicted values of gas
fraction and gas velocity.
Seven sets of data were used in the determination of the parameters of
interest. Each set was defined by the liquid phase (water, mud 1, and mud 2)
and by the superficial liquid velocity (0.21 m/s, 0.37 m/s, and 0.52 m/s). For mud
1 and mud 2 just the smallest and the largest superficial liquid velocities were
tested.
Assuming the K factor behaves like shown in Figure 2.4, the following
expression was adopted in the analysis:
K = a-i + a2
N& (5.2)
where ai, a2, and a3 are constants to be defined for each data set. The single
bubble velocity and the n exponent were the other constants to be determined
for each set of data. With a program to search for the minimum deviation
between the observed and predicted values in Equation 5.1, it was possible to
find combinations of the five constants that best fitted each set of experimental
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modeling of the Experimental Results 58
results. The coefficients a1 , a2 and a3 were found to define very similar curves
for the coefficient K in the three cases of gas-water mixtures. Thus, just one
combination of coefficients was chosen to fit the gas-water data. The same
method was adopted for the gas-mud mixtures since those coefficients were
again very similar to each other in the gas-mud cases.
After the definition of the constants a1, a2 and a3 for gas-water mixtures
and gas-mud mixtures, the single bubble velocity and the n exponent were re
defined in a way to minimize the deviation from the experimental data. The
results are summarized in Table 5.1.
Table 5.1. Best fit coefficients for the gas velocity equation.
Usi U oo n a1 a2 a3(m/s) (m/s)
Gas 0.24 0.24 3.60 1.28 0 . 2 0 0.40
Water 0.37 0.24 4.28 1.28 0 . 2 0 0.40
0.52 0.24 5.36 1.28 0 . 2 0 0.40Gas 0.24 0.28 0.71 1.30 2 . 0 0 0.25
Mud 1 0.52 0.16 6.80 1.30 2 . 0 0 0.25Gas 0.24 0.32 1 .0 2 1.30 2 . 0 0 0.25
Mud 2 0.52 0.27 5.40 1.30 2 . 0 0 0.25
The single bubble velocity for gas-water data is constant over the tested
interval and its value is very close to the one given by Harmathy's (1960)
equation:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modeling of the Experimental Results 59
UooH = 1.53 9 ( Pi - Pg) P
Pi" (5.3)
For gas-mud data, the single bubble velocity depended on the liquid Reynolds
number:
I L = -0.161 (log(NLRe)2+0.624 log(NLRe) - 0.284) (5.4)
where:
Uoo is in m/s,
NLR e = MH-eq / r- r-\
and peq is the equivalent viscosity defined by Equations C12 and C13 in
Appendix C. Equation 5.4 is applicable for Nlrs > 100. For smaller Nlrs, the
limiting value of 0.32 m/s is assumed for Uoo.
Although Uoo is kept constant in the tested interval for gas-water mixtures,
the n exponent increases with the superficial liquid velocity. A possible reason
for this result is that the higher turbulence caused by larger superficial liquid
velocities would cause the bubbles to break-up into smaller sizes which would
decrease the bubble Reynolds number and increase the n exponent (Equations
2.30 and 2.31). This variation of n exponent, the constant single bubble velocity,
and the good estimative given by Harmathy’s equation are indications that the
bubble sizes are in region 4 of Figure 2.2 where the bubble velocities are
independent of bubble size.
For the gas-mud cases, the single bubble velocity decreases and the n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modeling of the Experimental Results 60
exponent increases with increasing superficial liquid velocity. Again, the same
explanation given for gas-water mixtures would apply here. Larger turbulence
would break-up the bubbles into smaller sizes and cause smaller Uoo, smaller
bubble Reynolds numbers, and larger n exponent.
2
Tested range Extrapolation1.9
1.8
1.7jGas-Mud mixtures
.0I
1.5
1.4Gas-Water m ixti res
1.3
1.2
1.1
11 0 0 1000001000 10000
Mixture Reynolds
Figure 5.1. Curves of K factor for gas-water and gas-mud mixtures.
Figure 5.1. shows the curves for the K factor as function of the mixture
Reynolds number. For gas-water mixtures, the K factor is practically constant in
the range of experimental data. At lower mixture Reynolds numbers more data
is needed to confirm the behavior of this curve. For gas-mud mixtures, the K
factor was much larger than for gas-water mixtures and it seemed to follow one
unique trend for both muds used in the tests. It should be interesting to confirm
this behavior with muds having viscosities outside the tested range. Also, tests
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modeling of the Experimental Results 61
at lower mixture Reynolds numbers should be useful to check if the K factor still
continues to increase or if it starts to decrease at a certain point.
With the relations obtained thus far, we can calculate the gas fraction a in
Eq. 5.1 and then, obtain the gas velocity Ug from the left hand side of the same
equation. For any combination of flow parameters and fluid properties different
from the experimental cases, the single gas bubble velocity and the n exponent
are calculated by simple interpolation. However, it is unexpected that
extrapolations could work well especially in those cases where the flow
characteristics and fluid properties are very different from the experimental
conditions. However, the experiments were run in conditions very close to the
normal situations expected to be found in practice
5.2. DESCRIPTION OF THE MODEL FOR INCLINED FLOW
From the experiments, a very close to linear relationship was observed
between the gas and mixture velocities for all inclinations and fluid types (water
and muds). Figure 5.2. shows these results for the gas-water case with
superficial liquid velocity equal to 0 .2 1 m/s. Other cases presented similar
behavior. Therefore, a simple way to estimate the gas velocity at any inclination
should be through a factor applied to the velocity in vertical position. This factor
was obtained by adjusting straight lines through the experimental points and
then calculating the ratio between each of these curves with the vertical case.
The coefficients for the curves are shown in table 5.2.
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Modeling of the Experimental Results 62
After obtaining the gas velocity at any inclination, the gas fraction is
calculated using Eq. 2 .1 2 . All the intermediate cases of inclination and flow
parameters are solved again by interpolation. Notice that the order of
calculation is the inverse for the vertical flow. There, the gas fraction was
calculated first and then, the gas velocity. Here, the gas velocity is obtained
before the gas fraction in an attempt to minimize the error in gas velocity, which
is sensitive to small errors in gas fraction for the range of conditions studied.
= 0 °
= 10 °
= 2 0 “O 0 . 8
0.60.40.2
Ga = 60“USl : 0.21 m/s
= 80“
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Um (m/s)
Figure 5.2. Relation between gas and mixture velocities.
The values of the coefficients a and b in Table 5.2 could be related to K
and U0 in Equation 2.7. However, even for vertical position (I = O'), they do not
represent the actual values of these coefficients since either K or U0 are
variables and not constants as represented in the table.
The coefficients a and b were defined just to correlate gas velocities at
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modeling of the Experimental Results 63
different inclination angles based on experimental results. They do not define
the flow profile, the swarm velocity, or even the flow pattern. For a better
definition of the values and trends for a and d as functions of inclination angle
and superficial liquid velocity, more data points are necessary. The flow pattern
change from bubble to slug flow in the vertical position, or from bubble to
elongated bubble in inclined positions, should cause some variations in a and b
not clearly seen yet with the available data.
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Modeling of the Experimental Results
Table 5.2. Coefficients for the equations of Ug = f (Um).
Gas Usl = 0.21 m/s Usl = 0.52 m/sWater Ug = a Um +b
a b a b1 = 0 ° 1 . 0 2 0 . 2 0 0.90 0.37
1 = 1 0 * 1 .0 1 0.39 0.79 0.64
ii N> O e 1.05 0.42 0.83 0.70
eoll 0.93 0.80 0.83 1 . 1 0
0oCOll 1 . 0 0 1 .01 0 . 8 8 1.30
1 = 80° 1 . 1 0 0 . 8 6 0.90 1 . 1 0
GasMud1
Usl = 0.21 m/s Usl = 0.52 m/sUg = a Um +b
a b a b
l = 0 ° 1.44 0.33 1.30 0.28
l = 1 0 ° 1.52 0.44 1.51 0.29ooCMII 1.60 0.51 1.64 0.29
II o 0 1.60 0.65 1.64 0.52
1 = 60° 1.61 0.63 1.64 0.57
1 = 80° 1.44 0.37 1.76 0.33
Gas Usl = 0 .21 m/s Usl = 0.52 m/sMud2 Ug = a Um +b
a b a bl = 0 ° 1.69 0.30 1.56 0.18
1 = 1 0 ° 1 . 6 8 0.34 1.56 0.30
1 = 2 0 ° 1.72 0.39 1.56 0.34
oo•M"II 1.96 0.50 1.54 0.82
II CT) O 0 1.89 0.34 1.54 0.75
0oGOII 1.40 0.30 1.52 0.69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Modeling of the Experimental Results 65
5.3. COMMENTS ABOUT THE MODEL AND HOW IT COMPARES WITH
THE EXPERIMENTAL DATA
One important aspect of this model is that it does not require the
estimation of bubble size. This feature makes the model much easier to use
without sacrificing any accuracy. A representative bubble size could be
estimated for each set of flow conditions and from that it should be possible to
calculate the single bubble velocity. However, this would imply another step in
the calculation and would not bring higher precision or accuracy. The utilization
of bubble diameter in the model would be profitable if experimental data on
single bubble velocity as function of bubble diameter were available for drilling
fluids.
Another point is the application of Harmathy's equation (Eq. 5.3) for the
single bubble velocity in gas-water mixtures. This correlation comes from a
regression with experimental data obtained by several authors and it is
applicable to region 4 in Fig. 2.2, where the bubble velocity is independent of
the bubble diameter. Besides the convenience of being independent of the
bubble size, this equation leads to results very similar to Casariego's (1987)
model and yet, it gives a good approximation for the single bubble velocity in
water according to the present experiments.
The model results were compared with the experimental data and
presented in Figures 5.3. through 5.8.
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Modeling of the Experimental Results
5% Error Lines
« 1.4E,* 1.2
E 1
^ 0.8
Gas-Water, Usl = 0.24 m/s
Gas-Water, Usl = 0.52 m/s
E Gas-rwud1, Usl = 0.21 m/s
Gas-Mud1, Usl = 0.52 m/s
0.4 Gas-Mud2, Usl = 0.21 m/s
0.2 Gas-Mud2, Usl = 052 m/s
O O J ^ CO o o T- N ' t tfl o C\l
Ug from experiments (m/s)
0.6
5% Error Lines0.5
0.4ES Gas-Water, Usl = 0.24 m/s
0.3co•ccoV -
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.21 m/sll- 0 . 2COcoO
Gas-Mud1, Usl = 0.52 m/s
Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
1.1 0.2 0.3 0.4 0.5 0.6Gas Fraction from Experiments
Figure 5.3. Gas Velocity and Gas Fraction for l= 0° (vertical).
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Modeling of the Experimental Results
5% Error Linesvo
® 1 -2 ■o Gas-Water, Usl = 0.21 m/s
Gas-Water, Usl = 0.52 m/s
o 0.8 Gas-Mud1, Us! = 0.21 m/io>
Gas-Mud1, Usl = 0.52 m/s
0.4 Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.8 1.2 1.1 Ug from experiments (m/s)
0.4
0.5
5% Error Lines_ 0.4 o■Oo
§ 0 .3Gas-Water, Usl = 0.21 m/s
eoCO
Gas-Water, Usl = 0.52 m/s
0.2 Gas-Mud1, Usl = 0.21 m/su.<AcoCD
Gas-Mud1, Usl = 0.52 m/s
0.1 Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.1 0.2 0.3 0.4Gas Fraction from Experiments
0.5
Figure 5.4. Gas Velocity and Gas Fraction for l= 10°.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gas
Frac
tion
from
Mod
el
Ug fro
m m
odel
(m
/s)
Modeling of the Experimental Results
2.5
5% Error Lines
Gas-Water, Usl = 0.21 m/s
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.21 m/s
Gas-Mud1, Usl = 0.52 m/s
0.5 Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.5 2.5Ug from experiments (m/s)
0.5
5% Error Lines0.4
0.3 Gas-Water, Usl = 0.21 m/s
Gas-Water, Usl = 0.52 m/s
0.2 Gas-Mud1, Usl = 0.21 m/s
Gas-Mud1, Usl = 0.52 m/s
Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.1Gas Fraction from Experiments
0.2 0.3 0.4 0.5
Figure 5.5. Gas Velocity and Gas Fraction for l= 20°.
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Modeling of the Experimental Results
2.5
5% Error Lines
Gas-Water, Usl = 0.21 m/s
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.21 m/sO)
Gas-Mud1, Usl = 0.52 m/s
0.5 Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.5 2.5Ug from experiments (m/s)
0.4
5% Error Lines
■§ 0.3 o 2ES Gas-Water, Usl = 0.21 m/s
0.2Co
2
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.21 m/sLL .
cn as CD
Gas-Mud1, Usl = 0.52 m/s
Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.1 0.2 0.3Gas Fraction from Experiments
0.4
Figure 5.6. Gas Velocity and Gas Fraction for l= 40°.
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Modeling of the Experimental Results
2.5
5% Error Lines
Gas-Water, Usl = 0.21 m/s
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.21 m/so>
Gas-Mud1, Usl = 0.52 m/s
0.5 Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.5 2.5Ug from experiments (m/s)
0.4
5% Error Lines
-§0.3o
2Eo Gas-Water, Usl = 0.21 m/s
0.2co
2LL
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.21 m/s
C/itoCD
Gas-Mud1, Usl = 0.52 m/s
Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.1 0.2 Gas Fraction from Experiments
0.3 0.4
Figure 5.7. Gas Velocity and Gas Fraction for l= 60".
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Gas
Frac
tion
from
Mod
el
Ug fro
m m
odel
(m
/s)
Modeling of the Experimental Results
2.5
5% Error Lines
Gas-Water, Usl = 0.21 m/s
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.21 m/s
Gas-Mud1, Usl = 0.52 m/s
0.5 Gas-Mud2, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
0.5 2.5Ug from experiments (m/s)
0.5
5% Error Lines0.4
0.3
0.2
0.1 0.2 0.3 0.4 0.5
Gas-Water, Usl = 0.21 m/s
Gas-Water, Usl = 0.52 m/s
Gas-Mud1, Usl = 0.52 m/s
Gas-Mud2, Usl = 0.21 m/s
Gas-Mud1, Usl = 0.21 m/s
Gas-Mud2, Usl = 052 m/s
Gas Fraction from Experiments
Figure 5.8. Gas Velocity and Gas Fraction for l= 80°.
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6. SUBROUTINES FOR APPLICATION OF THE MODEL
The primary type of application for this bubble velocity model is the
simulation of gas kicks in vertical and slanted wells. The simulation has to
estimate the pressure at different points inside the well, the time for gas arrival
at the surface, and the gas distribution along the well. The two purposes of this
type of simulation are the planning of killing procedures and training of
personnel.
This chapter contains a brief description of the subroutines that could be
used in the implementation of the model in gas kick simulators using water base
muds. The input and output variables and their units are described in the
subroutines presented in Appendix B.
The subroutine that interfaces with the simulator is BUBMOD. Other
subroutines are called directly or indirectly by this. The output from this
subroutine are the gas velocity and gas fraction. However, depending on the
situation as explained ahead, just the velocity is returned and the gas fraction is
considered as an input variable.
After the determination of the type of liquid phase and the swarm effect
exponent, three variables are considered: the mixture velocity, the superficial
72
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Subroutines for Application of the Model 73
liquid velocity and the superficial gas velocity. If the mixture velocity is zero, the
K factor is assumed to be equal to zero. For Um * 0, the K fator is calculated by
subroutine KFAC. In the next step, the superficial liquid velocity is checked for
the calculation of the liquid Reynolds number which is used in the
determination of the single bubble velocity. For gas-water mixtures, the single
bubble velocity is simply the Harmathy velocity (Eq. 5.3). For gas-mud mixtures
the velocity is given by Equation 5.4. The third condition tested is for Usg = 0 in
which case just the gas velocity is calculated. For Usg * 0, both gas velocity and
gas fraction are calculated.
The three conditions checked by the subroutine BUBMOD correspond to
the most common situations happening during the gas kicks. First, while drilling,
both gas and liquid flow into the annulus in the beginning of the kick (Um, US|
and Usg are different from zero). Right before the well is shut in, when the
drilling is stopped, just gas continues to flow into the well (Usi = 0). When the
blowout preventer is closed neither liquid nor gas flow into the annulus
anymore (Um, Usi and Usg are all zero). Lastly, during the transportation period,
just liquid is injected if the correct control of the bottom hole pressure is attained,
avoiding more gas influx (Usg = 0 ).
The velocity calculated so far is for vertical wells. In case of inclined
holes, a correction is made to the gas velocity according to Table 5.2 and then,
the gas fraction is recalculated and both variables are returned from the
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Subroutines for Application of the Model
subroutine.
74
The other subroutine.i shown in Appendix B are self-explanatory. Figure
6 .1 shows the calling structure of these subroutines and the purpose of each
one.
InterfaceSubroutine
Gas Velocity in Inclined Well
Single Bubble Velocity
K FactorFinding
BIlWeHLQJJInlAIEiimReynoldsNumber
HarmathyVelocity
Gas Velocity Function
1
y@[El!IVLiquid
Reynolds
Bubble Swarm Exponent
m m mGas Gas
Density ViscosityEquiv. Viscosity
Correction
ip @ y K n r
z Factor 2-D Polinomial Interpolation
PolinomialInterpolation
Figure 6.1. Structure of the subroutines shown in Appendix B.
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7. CONCLUSIONS
During the execution of this project, an experimental apparatus was built
with the goal of studying the gas flow behavior through drilling fluids in vertical
and inclined annular sections of the well. An experimental procedure was
defined and tests were run covering the range of flow parameters normally
found in field operations. Just the case of fully eccentric annulus was tested
since this is the most probable condition found along the major part of the well.
The following results were obiained from the experiments conducted in
the transparent model with air-water mixtures:
a. bubble flow regime was observed to occur for gas fraction up to 50% in
vertical flow;
b. the flow pattern changed gradually from bubble flow to elongated
bubble with increasing inclination;
c. the size and stability of the elongated bubbles were dependent on the
turbulence level in the two-phase flow.
These were the most important results obtained from experiments
conducted in the higher pressure steel model using gas-water and gas-mud
mixtures:
a. the gas velocity increased when inclination was varied from 0 °
75
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Conclusions 76
(vertical) to between 40' and 60' and then began to decrease;
b. the single bubble velocity in gas-water mixtures was constant in the
range of the tests. For gas-mud mixtures, the single bubble velocity was very
sensitive to variations in the liquid Reynolds number. This is an indication of
formation of large bubbles at low turbulence levels and the break-up of these
bubbles when the turbulence increases;
c. bubble swarm effect (velocity decrease owed to gas concentration)
was observed in the cases studied in the experiments. The swarm effect
exponent increases rapidly with the liquid Reynolds number.
With the experimental results, it was possible to evaluate the previous
model for gas velocity and concentration in vertical wells. This model worked
very well for gas-water mixtures.
A new s im p lif ie d model for gas bubble velocity and concentration was
developed including a correction for non-vertical wells. It is expected that the
model would work well within the range of fluid properties and flow parameters
adopted in the tests.
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8. RECOMMENDATIONS
The recommendations are related to three aspects for the continuation of
this research: the experimental apparatus, different muds and flow conditions
for new tests, and improvement of the model.
In the experimemai apparatus, a modification that would improve the gas
fraction measurements is the placement of just two valves in the straight portion
of the annulus, one at the beginning and other at the end (see Fig. 3.1). Also, if
the mixing of both phases happened before the valve at the bottom, it would not
be necessary to have four pressure actuated valves in the loop, but just those
two mentioned above. The valve at the bottom would be the three way valve (Vi
in Fig. 3.1) and the valve at the top would be valve V4 . The main advantages of
these modifications would be the simplification of the system, faster response
when opening and closing the valves, and better values for gas fraction
measurements.
it would be of interest to extend the tests to the following flow conditions:
a. smaller superficial liquid velocities, to check the effect of this parameter
on the single bubble velocity, on the swarm effect exponent, and on the
behavior of K factor at lower mixture Reynolds number (with smaller liquid
Reynolds numbers, it is expected the formation of larger bubbles that would
77
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Recommendations 78
lead to larger single bubble velocities and smaller swarm effect);
b. larger superficial gas velocities for verification of bubble-slug transition
and slug flow;
c. different mud viscosities for the confirmation of the trends of single
bubble velocity and swarm effect exponent.
The improvement of the model is more dependent on the experimental
determination of bubble size distribution, single bubble velocity, swarm effect
exponent and velocity profile coefficient under different flow conditions and
drilling fluids. Presently, the K factor and the single bubble velocity were defined
in a way just to match the experimental gas velocity. However, they were not
directly measured in the experiments. The equipments necessary for the
experimental determination of these parameters are more complex and their
precision and accuracy should be checked if possible. Also, operational
difficulties could restrict the use of these equipments. For practical purposes, it
still might be better just to extend the range of tests as previously suggested and
then to adjust the correlations.
One important point that was not covered in this study is the effect of the
gel of the mud. This mud property would affect the bubble velocity and
concentration when the mud is not flowing. It might even be an important factor
in the spreading of the gas contaminated region in the annulus because just the
larger bubbles would flow through the mud while the smaller ones would stop
with the mud. Since this property varies along the time, the determination of
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Recommendations 79
single bubble velocity in static muds would be a problem. Another difficult
problem would be the measurement of an effective surface tension between
gas and mud because of the combination of gel forces and surface forces at any
mud-gas interface. Considering these problems, one way to evaluate the
spreading of the bubbles during static periods would be through the
experimental determination of bubble size distribution during the flowing period.
According to the bubble size distribution it would be possible to estimate which
ones would still flow in static conditions and at what point in time, if any, it would
stop.
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NOMENCLATURE
a gas fraction
Wns non-slip gas fraction
Yg gas specific gravity (air = 1 )
e/Dp pipe roughness
0 inclination angle from vertical
Pa air viscosity
fie effective liquid viscosity
Peq equivalent viscosity
gas viscosity
Pi liquid viscosity
Pns non-slip viscosity = peq (1 -ans)+ Pg ans
Pp plastic viscosity
Pw water viscosity
Pg gas density
PI liquid density
Pns non-slip density = pi (1-ans)+ Pg ans
Ps slip density
a surface tension
I Fredrickson and Bird (1958) factor
80
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Nomenclature
xy yield point
A cross sectional area of the medium
Cd drag coefficient
Db bubble diameter
Dec distance between centers
Dep equi-periphery diameter = D0 + Dj
Dj internal diameter of the annulus
D0 external diameter of the annulus
Dp pipe diameter
DP-i total differential pressure in the loop
DP2 differential pressure in the DP sensor 2
DP3 differential pressure in the DP sensor 3
DPf frictional pressure loss
DPnyd hydrostatic differential pressure
DP jot differential pressure read by the sensor
Drt diameter ratio = Dj/D0
Ds particle diameter
e eccentricity = 2 DBc/(D0-Dj)
f Fanning friction factor
F drag force
Fca geometric factor for concentric annulus
Fea geometric factor for eccentric annulus
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Nomenclature
Fg geometric factor for calculation of friction factor
g gravity acceleration
9c conversion factor for unit systems
j dimensionless factor in the ellipsoidal bubble velocity equation
K coefficient for the effect of the velocity profile
k' consistency index
Ki effect of the preceding bubble's wake on the bubble velocity
L length along which DPjot is measured
Ie entry length for slug formation
m function of Nf
n exponent for the velocity of a swarm of bubbles
n' flow behavior index
Nji Viscosity number
NE6 Eotvos number
Nf flow number = g Db8/3 pi2/3 (pi. pg) p'4/3 o' 1/3
Nf inverse viscosity number
Nk Karman number = 1.155 [g pi (prpg) Db3 ]1/2 / j i e
NLa Laplace number
Nm Morton number
Nr9 Two-Phase Reynolds number or Mixture Reynolds number
Nbro Bubble Reynolds number
NLRe Liquid Reynolds number
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Nomenclature
Nmrb Modified Reynolds number
CDa:Q.
Z
Particle Reynolds number
Nstr Strouhal number
Nv viscosity group
Nv Velocity number = U„» (Db pi)273 (pi o ) ‘ 1/3
N w e Weber Number
P pressure
P cr critical pressure
Qair airflow rate
^9 gas flow rate
Qgas gas flow rate
qi liquid flow rate
Qstd standard flow rate
t bubble break-up time
T temperature
TCT critical temperature
Tko temperature ratio = 273.16/T(°K)
U0 bubble velocity at inclination 0
Ug bubble velocity
u. liquid velocity = US| / (1 -a)
u m mixture velocity = US| + Usg
U0 bubble swarm velocity in stagnant liquid
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Nomenclature
U r gas velocity with respect to the liquid around it
Usg superficial gas velocity = qg / A
Usl superficial liquid velocity - qi / A
Usiug slug velocity
u„ single bubble velocity
z super-compressibility factor
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REFERENCES
1. Abou-EI-Hassan, M. E., 1986. Correlations for Bubble Rise in Gas-liquid
Systems. In: Cheremisinoff, N. P. (Editor), Encyclopedia of Fluid Mechanics,
Volume 3, Gas-liquid Flows. Gulf Publishing Company, Houston, pp. 110-120.
2. Acharya, A., Mashelkar, R. A., and Ulbrecht, J., 1977. Mechanics of Bubble
Motion and Deformation in Non-newtonian Media. Chemical Engineering
Science, 32: 863-872.
3. Acharya, A., Mashelkar, R. A. and Ulbrecht, J. J., 1978. Bubble Formation in
Non-newtonian Liquids. Ind. Eng. Chem. Fundam., 17 (3): 230-232.
4. Astarita, G., and Apuzzo, G., 1965. Motion of Gas Bubbles in Non-Newtonian
Liquids. AlChE J., 11(5): 815-820.
5. Beggs, H. D. and Brill, J. P., May, 1973. A Study of Two-phase Flow in
Inclined Pipes. Journal of Petroleum Technology, 607-617.
6 . Bhatia, V. K., 1969. Gas Holdup of a Bubble Swarm in Two Phase Vertical
Flow. AlChE J., 15 (3): 466-469.
85
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References 86
7. Bhavaraju, S. M., Mashelkar, R. A. and Blanch, H. W., 1978. Bubble Motion
and Mass Transfer in Non-newtonian Fluids. AlChE J., 24 (6 ): 1063-1076.
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Tech. Conf. and Exhib. of the Society of Petroleum Engineers, Houston, TX,
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10. Brill, J. P. and Beggs, H. D., 1984. Two-phase Flow in Pipes. 3rd ed., Tulsa
University.
11. Buchholz, H., Buchholz, R., Lucke and Schugerl, K., 1978. Bubble Swarm
Behaviour and Gas Absorption in Non-Newtonian Fluids in Sparged Columns.
Chemical Engineering Science, 33: 1061-1070.
12. Buckmaster, J. D., 1972. Pointed Bubbles in Slow Viscous Flow. J. Fluid
Mech., 55, part 3: 385-400.
13. Caetano Filho, E., 1986. Upward Vertical Two-phase Flow Through an
Annulus. Ph.D. Dissertation, Tulsa University.
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References 87
14. Casariego, V., 1981. Experimental Study of Two-phase Flow Patterns
Occurring in a Well During Pressure Control Operations. Ms. Thesis, Louisiana
State University.
15. Casariego, V., 1987. Generation, Migration, and Transportation of Gas
Contaminated Regions of Drilling Fluid. Ph.D. Dissertation, Louisiana State
University.
16. Cox, R. G., 1969. The Deformation of a Drop in a General Time-dependent
Fluid Flow. J. Fluid Mech., 37, part 3: 601-623.
17. Craft, B. C. and Hawkins M. F., 1959. Applied Petroleum Reservoir
Engineering. Prentice-Hall Inc., Englewood Cliffs, NJ, 437 pp.
18. Davies, R. M. and Taylor, G., 1950. The Mechanics of Large Bubbles Rising
Through Extended Liquids and Through Liquids in Tubes. Proceedings of the
Royal Society, Series A, Mathematical and Physical Science, 200 (1062):
375-390.
19. Fernandes, R. C., Semiat, R. and Dukler, A. E., 1983. Hydrodynamic Model
for Gas-liquid Slug Flow in Vertical Tubes. AlChE J., 29 (6 ): 981-989.
20. Fredrickson, A. G. and Bird, R. B., 1958. Non-newtonian Flow in Annuli.
Industrial and Engineering Chemistry, 50 (3): 347-352.
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References 88
21. Grace, J. R. and Harrison, D., 1967. The Influence of Bubble Shape on the.
Rising Velocities of Large Bubbles. Chemical Engineering Science, 22:
1337-1347.
22. Grace, J. R., 1973. Shapes and Velocities of Bubbles Rising in Infinite
Liquids. Trans. Instn. Chem. Engrs., 51:116-120.
23. Grace, J. R., Wairegi, T. and Nguyen, T. H., 1976. Shapes and Velocities of
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX A: EXPERIMENTAL DATA
Test Usg Usl ug a Test Usg Usl ug aCode m/s m/s m/s Code m/s m/s m/s
GAS-WATER I2g1 10 0.06 0.37 0.76 0.08i2g2 10 0.09 0.37 0.78 0.11
iig i 0.07 0.24 0.53 0.14 I2g3 10 0.14 0.37 0.95 0.151192 0.09 0.25 0.60 0.15 I2g4 10 0.20 0.37 1.09 0.18Hg3 0.14 0.25 0.58 0.25 I2g5 10 0.25 0.37 1.17 0.21Hg4 0.20 0.26 0.68 0.30 I2g6 10 0.31 0.38 1.10 0.28Iig5 0.25 0.25 0.68 0.37 I2g7 10 0.36 0.37 1.12 0.32ng6 0.31 0.25 0.74 0.42 I2g8 10 0.42 0.36 1.15 0.36ng7 0.36 0.24 0.80 0.45 I2g9 10 0.47 0.36 1.21 0.3911 g8 0.42 0.25 0.84 0.50 I2g1 Oil 0 0.55 0.37 1.35 0.4111 g9 0.47 0.25 0.92 0.51 I3gl 10 0.06 0.52 1.11 0.06H gio 0.54 0.25 1.08 0.51 I3g2 10 0.09 0.52 1.05 0.09I2gi 0.07 0.37 0.76 0.10 I3g3 10 0.14 0.52 1.16 0.12I2g2 0.10 0.38 0.70 0.14 I3g4 10 0.20 0.53 1.25 0.16I2g3 0.15 0.37 0.76 0.19 I3g5 10 0.25 0.52 1.36 0.18I2g4 0.33 0.37 0.97 0.34 I3g6 10 0.30 0.52 1.27 0.24I2g5 0.26 0.37 0.86 0.30 I3g7 10 0.36 0.52 1.34 0.27I2g6 0.30 0.37 0.92 0.33 I3g8 10 0.41 0.52 1.37 0.30I2g7 0.37 0.37 0.96 0.39 I3g9 10 0.47 0.52 1.46 0.32I2g8 0.42 0.36 1.01 0.42 I3g1 Oil 0 0.53 0.53 1.44 0.37I2g9 0.47 0.37 1.12 0.42 I1g1i20 0.06 0.22 0.72 0.08I2g10 0.53 0.37 1.18 0.45 I1g2i20 0.09 0.22 0.71 0.12I3g1 0.06 0.52 0.94 0.07 Hg3i20 0.15 0.22 0.85 0.17I3g2 0.09 0.53 0.91 0.10 I1g4i20 0.20 0.22 0.86 0.23I3g3 0.14 0.52 0.99 0.15 I1g5i20 0.26 0.21 0.91 0.29I3g4 0.20 0.52 1.03 0.1S I1g6i20 0.31 0.22 0.96 0.33I3g5 0.26 0.52 1.05 0.25 I1g7i20 0.37 0.22 1.05 0.36I3g6 0.31 0.52 1.08 0.29 I1g8i20 0.43 0.22 1.10 0.39I3g7 0.37 0.52 1.15 0.32 I1g9i20 0.48 0.22 1.15 0.42I3g8 0.43 0.52 1.22 0.35 I1g10i20 0.55 0.21 1.23 0.44I3g9 0.46 0.52 1.28 0.36 I3g1i20 0.06 0.53 1.25 0.05I3g10 0.50 0.52 1.33 0.38 I3g2i20 0.09 0.53 1.16 0.08n g i 10 0.07 0.21 0.66 0.10 I3g3i20 0.14 0.52 1.27 0.1111 g2 10 0.10 0.21 0.66 0.15 I3g4i20 0.20 0.53 1.30 0.1511 g3 10 0.15 0.22 0.78 0.19 I3g5i20 0.26 0.53 1.33 0.2011 g4 10 0.20 0.22 0.89 0.23 I3g6i20 0.31 0.52 1.41 0.22n g5 10 0.26 0.22 0.93 0.28 I3g7i20 0.37 0.53 1.44 0.26Hg6 10 0.31 0.22 0.89 0.35 I3g8i20 0.42 0.52 1.48 0.2911 g7 10 0.37 0.21 0.94 0.39 I3g9i20 0.48 0.52 1.53 0.32n g8 10 0.43 0.22 1.04 0.41 I3g10i20 0.54 0.53 1.62 0.33n g9 10 0.48 0.22 1.09 0.44 11 g1 i40 0.06 0.22 1.05 0.06ngiOMO 0.54 0.23 1.18 0.46 I1g2i40 0.09 0.22 1.03 0.09
99
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix A: Experimental Data
TestCode
Usgm/s
Uslm/s
ugm/s
a TestCode
Usgm/s
Uslm/s
Ugm/s
a
I1g3i40 0.15 0.22 1.19 0.12 I3g6i80 0.32 0.53 1.82 0.18I1g4i40 0.20 0.21 1.29 0.16 I3g7i80 0.38 0.53 1.92 0.20I1g5i40 0.26 0.21 1.24 0.21 I3g8i80 0.42 0.52 1.92 0.22I1g6i40 0.32 0.22 1.32 0.24 I3g9i80 0.48 0.52 1.98 0.24I1g7i40 0.37 0.22 1.26 0.29 I3g10i80 0.53 0.52 2.06 0.26I1g8i40 0.42 0.21 1.35 0.31I1g9i40 0.49 0.22 1.44 0.34 GAS-MUD 1I1g10i40 0.55 0.21 1.57 0.35I3g1i40 0.06 0.52 1.67 0.04 m1g 1 0.08 0.21 0.78 0.11I3g2i40 0.09 0.53 1.56 0.06 m1g2 0.11 0.21 0.79 0.14I3g3i40 0.15 0.53 1.77 0.08 m1g3 0.16 0.21 0.84 0.20I3g4i40 0.20 0.53 1.71 0.12 m1g4 0.20 0.21 0.92 0.22I3g5i40 0.26 0.52 1.74 0.15 m1g5 0.26 0.21 1.01 0.26I3g6i40 0.33 0.52 1.80 0.18 m1g6 0.32 0.21 1.15 0.28I3g7i40 0.37 0.53 1.72 0.21 m1g7 0.38 0.21 1.16 0.33I3g8i40 0.43 0.53 1.76 0.24 m1g8 0.44 0.21 1.23 0.36I3g9i40 0.49 0.52 1.90 0.26 m1g9 0.49 0.21 1.31 0.38I3g10i40 0.54 0.53 1.95 0.28 m1g10 0.54 0.21 1.47 0.3711 gl i60 0.06 0.22 1.27 0.05 m3gl 0.08 0.52 1.10 0.08Hg2i60 0.09 0.22 1.22 0.07 m3g2 0.11 0.52 1.07 0.11I1g3i60 0.15 0.21 1.42 0.10 m3g3 0.17 0.52 1.16 0.15I1g4i60 0.20 0.22 1.51 0.14 m3g4 0.20 0.51 1.19 C.17I1g5i60 0.27 0.22 1.55 0.17 m3g5 0.27 0.51 1.26 0.21I1g6i60 0.31 0.22 1.54 0.20 m3g6 0.32 0.52 1.48 0.22I1g7i60 0.37 0.22 1.60 0.23 m3g7 0.38 0.51 1.43 0.27I1g8i60 0.43 0.22 1.63 0.26 m3g8 0.44 0.52 1.51 0.29!1g9i6Q 0.49 0.22 1.71 0.29 m3g9 0.50 0.52 1.59 0.32I1g10i60 0.54 0.21 1.74 0.31 m3g10 0.55 0.52 1.69 0.33I3g1 i60 0.06 0.52 1.80 0.03 m1g1i10 0.14 0.21 0.96 0.14I3g2i60 0.09 0.52 1.77 0.05 m lg lilO b 0.08 0.21 0.92 0.09I3g3i60 0.15 0.53 1.98 0.07 m1g2i10 0.14 0.21 0.95I3g4i60 0.20 0.53 2.01 0.10 m1g2i10b 0.11 0.21 0.92 O.'-LI3g6i60 0.31 0.52 2.03 0.15 m1g3i10 0.17 0.21 1.00 0.17I3g7i60 0.37 0.52 2.05 0.18 m1g3i10b 0.16 0.21 0.86 0.17I3g8i60 0.43 0.52 2.11 0.20 m1g4i10 0.21 0.21 1.12 0.19I3g9i60 0.49 0.53 2.19 0.22 m1g5i10 0.27 0.20 1.19 0.22I3g10i60 0.53 0.53 2.26 0.24 m1g6i10 0.33 0.22 1.28 0.25I1g1i80 0.06 0.21 1.10 0.06 m1g7i10 0.38 0.20 1.34 0.28I1g2i80 0.09 0.21 1.15 0.08 m1g8i10 0.44 0.21 1.42 0.31I1g3i80 0.15 0.21 1.38 0.11 m1g9i10 0.50 0.21 1.52 0.33I1g4i80 0.20 0.21 1.30 0.16 m1g10i10 0.55 0.21 1.61 0.35I1g5i80 0.26 0.21 1.44 0.18 m3g1i10 0.07 0.52 1.20 0.06I1g6i80 0.32 0.21 1.42 0.22 m3g2i10 0.14 0.52 1.25 0.12I1g7i80 0.39 0.22 1.53 0.26 m3g3i10 0.17 0.52 ' 1.27 0.13I1g8i80 0.44 0.22 1.61 0.27 m3g4i10 0.21 0.52 1.43 0.14I1g9i80 0.48 0.22 1.61 0.30 m3g5i10 0.26 0.52 1.51 0.17I1g10i80 0.55 0.21 1.67 0.33 m3g6i10 0.32 0.52 1.58 0.21I3g1 i80 0.06 0.52 1.47 0.04 m3g7i10 0.38 0.52 1.65 0.23I3g2i80 0.09 0.52 1.55 0.06 m3g8i10 0.44 0.52 1.75 0.26I3g3i80 0.15 0.52 1.65 0.09 m3g9i10 0.49 0.52 1.80 0.27I3g4i80 0.20 0.52 1.65 0.12 m3g10i10 0.55 0.52 1.90 0.29I3g5i80 0.26 0.53 1.82 0.14 m1g1i20 0.09 0.21 0.97 0.10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix A: Experimental Data
TestCode
Usgm/s
Uslm/s
ugm/s
a TestCode
Usgm/s
Uslm/s
Ugm/s
a
m1g2i20 0.11 0.20 0.98 0.11 m3g10i60 0.56 0.52 2.32 0.24m1g3i20 0.17 0.20 1.09 0.16 m1g2i80 0.11 0.20 0.76 0.14m1g4i20 0.21 0.20 1.19 0.17 m1g3i80 0.15 0.20 0.77 0.20m1g5i20 0.27 0.20 1.29 0.21 m1g4i80 0.21 0.21 0.98 0.21m1g6i20 0.32 0.20 1.37 0.24 m1g5i80 0.27 0.20 1.01 0.27m1g7i20 0.38 0.20 1.46 0.26 m1g6i80 0.32 0.21 1.12 0.29m1g8 i20 0.44 0.20 1.53 0.29 m1g7i80 0.38 0.21 1.20 0.32m1g9i20 0.50 0.21 1.61 0.31 m1g8i80 0.44 0.20 1.28 0.35m1g10i20 0.55 0.20 1.71 0.32 m1g9i80 0.50 0.20 1.40 0.36m3g1i20 0.09 0.52 1.26 0.07 m1g10i80 0.56 0.20 1.53 0.37m3g2i20 0.11 0.52 1.29 0.08 m3g2i80 0.10 0.52 1.38 0.08m3g3i20 0.17 0.52 1.37 0.12 m3g3i80 0.15 0.52 1.46 0.10m3g4i20 0.21 0.52 1.50 0.14 m3g4i80 0.20 0.52 1.67 0.12m3g5i20 0.27 0.52 1.60 0.17 m3g5i80 0.27 0.52 1.68 0.16m3g6i20 0.32 0.52 1.70 0.19 m3g6i80 0.33 0.52 1.82 0.18m3g7i20 0.38 0.52 1.80 0.21 m3g7i80 0.39 0.52 1.99 0.19m3g8i20 0.44 0.52 1.88 0.23 m3g8i80 0.44 0.52 2.06 0.22m3g9i20 0.50 0.52 1.95 0.26 m3g9i80 0.50 0.52 2.09 0.24m3g10i20 0.55 0.52 2.03 0.27 m3g10i80 0.56 0.52 2.18 0.26m1g2i40 0.11 0.20 1.10 0.10m1g3i40 0.17 0.20 1.20 0.14 GAS-MUD 2m1g4i40 0.21 0.20 1.31 0.16m1g5i40 0.27 0.20 1.44 0.18 m1g2 0.12 0.22 0.89 0.14m1g6i40 0.32 0.20 1.53 0.21 m1g3 0.17 0.22 0.95 0.18m1g7i40 0.38 0.20 1.61 0.24 m1g4 0.23 0.21 1.05 0.22m1g8i40 0.44 0.20 1.66 0.27 m1g5 0.27 0.21 1.12 0.24m1g9i40 0.50 0.20 1.75 0.28 m1g6 0.32 0.21 1.17 0.28m1g10i40 0.56 0.20 1.84 0.31 m1g7 0.38 0.21 1.27 0.30m3g2i40 0.11 0.52 1.51 0.07 m1g8 0.44 0.21 1.34 0.33m3g3i40 0.16 0.52 1.59 0.10 m1g9 0.50 0.21 1.40 0.36m3g4i40 0.21 0.52 1.73 0.12 m1g10 0.56 0.21 1.51 0.37m3g5i40 0.27 0.52 1.87 0.14 m3g2 0.12 0.54 1.32 0.09m3g6i40 0.32 0.52 1.92 0.17 m3g3 0.17 0.54 1.31 0.13m3g7i40 0.38 0.52 2.03 0.19 m3g4 0.23 0.54 1.43 0.16m3g8i40 0.44 0.52 2.10 0.21 m3g5 0.27 0.52 1.42 0.19m3g9i40 0.50 0.52 2.16 0.23 m3g6 0.32 0.52 1.48 0.21m3g10i40 0.55 0.52 2.25 0.25 m3g7 0.38 0.52 1.59 0.24m1g3i60 0.17 0.21 1.17 0.14 m3g8 0.44 0.52 1.63 0.27m1g4i60 0.21 0.20 1.34 0.15 m3g9 0.50 0.52 1.76 0.29m1g5i60 0.27 0.20 1.38 0.20 m3g10 0.55 0.49 1.76 0.32m1g6i60 0.32 0.20 1.51 0.21 m1g2i10 0.11 0.21 0.87 0.13m1g7i60 0.39 0.20 1.59 0.24 m1g3i10 0.17 0.23 1.00 0.17m1g8i60 0.44 0.20 1.64 0.27 m1g4i10 0.23 0.21 1.08 0.21m1g9i60 0.50 0.21 1.79 0.28 m1g5i10 0.27 0.20 1.18 0.22m1g10i60 0.55 0.21 1.82 0.30 m1g6i10 0.33 0.21 1.28 0.26rnSgSiSC 0.16 0.52 1.61 0.10 mig7i10 0.38 0.21 1.35 0.28m3g4i60 0.20 0.52 1.81 0.11 m1g8i10 0.44 0.21 1.41 0.31m3g5i60 0.26 0.52 1.83 0.14 m1g9i10 0.51 0.20 1.54 0.33m3g6i60 0.32 0.52 1.97 0.16 m1g10i10 0.55 0.21 1.61 0.34m3g7i60 0.38 0.52 2.08 0.18 m3g2i10 0.11 0.54 1.28 0.09m3g8i60 0.44 0.52 2.15 0.20 m3g3i10 0.17 0.54 1.27 0.14m3g9i60 0.50 0.51 2.18 0.23 m3g4i10 0.23 0.54 1.37 0.17
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Appendix A: Experimental Data
TestCode
Usgm/s
Uslm/s
ugm/s
a TestCode
Usgm/s
Uslm/s
ugm/s
a
m3g5i10 0.27 0.52 1.50 0.18 m3g3i60 0.17 0.51 1.83 0.09m3g6i10 0.32 0.52 1.59 0.20 m3g4i60 0.23 0.51 1.90 0.12m3g7i10 0.38 0.52 1.66 0.23 m3g5i60 0.29 0.51 2.01 0.15m3g8i10 0.44 0.52 1.76 0.25 m3g6i60 0.35 0.51 2.02 0.17m3g9i10 0.49 0.52 1.81 0.27 m3g7i60 0.41 0.50 2.17 0.19m3g10i10 0.56 0.52 1.95 0.29 m3g8i60 0.47 0.50 2.17 0.22m1g2i20 0.11 0.21 0.91 0.12 m3g9!CC A C C
W . k / v 0.51 2.37 0.22m1g3i20 0.17 0.22 1.03 0.17 m3g10i60 0.59 0.50 2.47 0.24m1g4i20 0.23 0.21 1.13 0.20 m1g2i80 0.12 0.20 0.89 0.13m1g5i20 0.27 0.21 1.23 0.22 m1g3i80 0.17 0.20 0.92 0.19m1g6i20 0.33 0.21 1.37 0.24 m1g4i80 0.23 0.20 0.97 0.24m1g7i20 0.40 0.21 1.49 0.27 m1g5i80 0.29 0.20 1.08 0.27m1g8 i20 0.45 0.21 1.56 0.29 m1g6i80 0.36 0.20 1.11 0.32m1g9i20 0.51 0.21 1.61 0.31 m1g7i80 0.41 0.20 1.16 0.35m1g10i20 0.55 0.21 1.61 0.35 m1g8i80 0.47 0.20 1.18 0.40m3g2i20 0.11 0.52 1.35 0.08 m1g9i80 0.54 0.20 1.35 0.40m3g3i20 0.17 0.52 1.37 0.13 m1g10i80 0.60 0.20 1.35 0.44m3g4i20 0.23 0.52 1.45 0.16 m3g2i80 0.11 0.51 1.66 0.07m3g5i20 0.26 0.51 1.55 0.17 m3g3i80 0.17 0.51 1.71 0.10m3g6i20 0.32 0.52 1.67 0.19 m3g4i80 0.23 0.51 1,78 0.13m3g7i20 0.38 0.51 1.80 0.21 m3g5i80 0.29 0.51 1.93 0.15m3g8i20 0.45 0.52 1.85 0.24 m3g6i80 0.35 0.51 1.99 0.18m3g9i20 0.50 0.52 1.94 0.26 m3g7i80 0.41 0.51 2.11 0.19m3g10i20 0.56 0.52 1.97 0.29 m3g8i80 0.46 0.51 2.13 0.22m1g2i40 0.11 0.20 1.13 0.10 m3g9i80 0.53 0.51 2.23 0.24m1g3i40 0.17 0.20 1.22 0.14 m3g10i80 0.59 0.50 2.40 0.25m1g4i40 0.23 0.21 1.36 0.17m1g5i40 0.29 0.21 1.51 0.19m1g6i40 0.35 0.20 1.57 0.22m1g7i40 0.41 0.20 1.73 0.24m1g8i40 0.47 0.20 1.80 0.26m1g9i40 0.53 0.20 1.94 0.27m1g10i40 0.61 0.20 2.10 0.29m3g2i40 0.11 0.51 1.73 0.07m3g3i40 0.17 0.51 1.80 0.10m3g4i40 0.23 0.51 1.90 0.12m3g5i40 0.29 0.51 2.05 0.14m3g6i40 0.35 0.51 2.12 0.17m3g7i40 0.41 0.51 2.29 0.18m3g8i40 0.47 0.50 2.33 0.20m3g9i40 0.54 0.50 2.49 0.22m3g10i40 0.59 0.50 2.57 0.23m1g2i60 0.11 0.20 0.96 0.12m1g3i60 0.17 0.20 1.02 0.17m1g4i60 0.23 0.20 1.16 0.20m1g5i60 0.29 0.20 1.29 0.23m1g6i60 0.35 0.19 1 <IC■ • V k / 0.26m1g7i60 0.42 0.20 1.53 0.27m1g8i60 0.47 0.20 1.59 0.30m1g9i80 0.52 0.20 1.70 0.31m1g10i60 0.60 0.20 1.88 0.32m3g2i60 0.11 0.51 1.69 0.07
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX B: SUBROUTINES FOR IMPLEMENTATION OF THE
BUBBLE MODEL
The main program just need to call the subroutine BUBMOD. All the
other subroutines are connected to this first one.
PROGRAM LISTING
SUBROUTINE BUBMOD(FLUID,RO,VISC,PLN,PLK,SGFG,P,T,Dl,D2, : INC, USL, USG, UM, K, UOO, UGVERT, UG, GF)
cccp
THIS SUBROUTINE CALCULATES GAS VELOCITY (UG) .cp
INPUT:c FLUID = 1 - WATER; 2 - MUDc RO = LIQUID DENSITY (PPG)c vise = LIQUID VISCOSITY (CP)/~1o PLN = n' FROM POWER LAW MODELc PLK = k' FROM POWER LAW MODEL (EQ. CP)c SGFG = GAS SPECIFIC GRAVITYc P — PRESSURE (PSI)c T = TEMPERATURE (deg. F)c Dl = INNER DIAMETER OF THE ANNULUS (IN)c D2 = OUTER DIAMETER OF THE ANNULUS (IN)c INC = WELLBORE INCLINATION (degrees)c USL = SUPERFICIAL LIQUID VELOCITY (FT/S)c USG = SUPERFICIAL GAS VELOCITY (FT/S)cp
UM = MIXTURE VELOCITY (FT/S)cp
OUTPUT:c K = FLOW PROFILE COEFFICIENTc UOO = SINGLE BUBBLE VELOCITY (FT/S)c UGVERT= GAS VELOCITY FOR VERTICAL WELL (FT/S)c UG = GAS VELOCITY AT INCLINATION INCccc
GF GAS FRACTION AT INCLINATION INC
103
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oAppendix B: Subroutines for Implementation of the Bubble Model 104
REAL RO,VISC,PLN,PLK,SGFG,P,T,Dl,D2,INC,USL,USG,UM,K,UOO,UGVERT, : UG,GF,Q,HLNS,WG,GVIS,DENNS,VISNS,VEL,EQVIS,REY,UOOH,NEXP,G,
ST,UGI,URATIO,UH,REYL,RTSAFE,FNEXP,FUOO INTEGER FLUIDNEXP=FNEXP(FLUID,VISC,USL)IF (UM.EQ.O.) THEN
K=0.ELSE
Q=UM*(D2**2-D1**2)*2.448 HLNS=USL/UMCALL GASDEN(P,T,SGFG,WG)CALL GASVIS(T,SGFG,P,GVIS)CALL REYNOL(FLUID,Q,HLNS,RO,WG,VISC,GVIS,Dl,D2,PLN,PLK,
: DENNS,VISNS,VEL,REY,EQVIS)CALL KFAC(FLUID,REY,K)
ENDIFIF (USL.EQ.O.) THEN
CALL GASDEN(P,T,SGFG,WG)UOO=FUOO(FLUID,0.,RO,WG)
ELSECALL LIQREY(RO,USL,Dl,D2,EQVIS,REYL)UOO=FUOO(FLUID,REYL,RO,WG)
ENDIFIF (USG.EQ.O.) THEN
UGVERT=UOO*(1-GF)**NEXP+K*UM ELSE
GF=RTSAFE(.001,1.,.001,UOO,NEXP,USG,USL,K)UGVERT=USG/GF
ENDIFIF (INC.EQ.0.) THEN
UG=UGVERT ELSE
CALL UGINC(UM,INC,USL,VISC,UGVERT,UG)IF (USG.NE.0.) GF=USG/UG
ENDIFRETURNEND
FUNCTION FNEXP(FLUID,VISC,USL)
CALCULATES THE SWARM EFFECT EXPONENT NINPUT: FLUID = 1 - WATER; 2 - MUD
VISC = LIQUID VISCOSITY (CP)USL = SUPERFICIAL LIQUID VELOCITY (FT/S)
OUTPUT: FNEXP
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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ooAppendix B: Subroutines for Implementation of the Bubble Model 105
cREAL USL, VISC,USLTW (3) , USLTM(2), VISCT (2) , NTW (3) ,NTM1 (2) ,NTM2 (2) , : AUX (2),DUINTEGER FLUID DATA USLTW/0.7,1.2, 1.7/DATA USLTM/0.7,1.7/DATA VISCT/10.,19./DATA NTW/3.60,4.28,5.36/DATA NTM1/0.71,6.80/DATA NTM2/1.02,5.40/
CIF (FLUID.EQ.l) THEN
CALL POLINT(USLTW,NTW,3,USL,FNEXP,DU)FNEXP=AMAX1(FNEXP,NTW(1))
ELSECALL POLINT(USLTM,NTM1,2,USL,AUX(1),DU)AUX(1)=AMAX1(AUX(1),NTM1(1) )CALL POLINT(USLTM,NTM2,2,USL,AUX(2),DU)AUX(2)=AMAX1(AUX(2),NTM2(1))CALL POLINT(VISCT,AUX,2,VISC,FNEXP,DU)FNEXP=AMAX1(FNEXP,AUX(1))
ENDIFC
RETURNEND
SUBROUTINE FUNCD(X,F,DF,UOO,NEXP,USG,USL,K)
CALCULATES THE VALUE OF THE FUNCTION F AND ITS DERIVATIVE. THIS FUNCTION RESULTS FROM COMBINATION OF EQUATIONS 2.7 TO 2.12 IN THE DISSERTATIONINPUT:X = GAS FRACTIONUOO = SINGLE GAS BUBBLE VELOCITY (FT/S)NEXP = SWARM EFFECT EXPONENTUSG = SUPERFICIAL GAS VELOCITY (FT/S)USL = SUPERFICIAL LIQUID VELOCITY (FT/S)K = FLOW PROFILE COEFFICIENTOUTPUT: F,DF
REAL X,F,DF,UOO,NEXP,USG,USL,KF=UOO*(1.-X)**NEXP-USG/X+K*(USG+USL) DF=-UOO*NEXP*(1.-X)**(NEXP-1.)+USG/X/XRETURNEND
FUNCTION FUOO(FLUID,REYL,RO,WG)
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Appendix B: Subroutines for Implementation of the Bubble Model 106
SINGLE BUBBLE VELOCITY INPUT:FLUID = 1 - WATER; 2 - MUD REYL = LIQUID REYNOLDS NUMBER RO = LIQUID DENSITY (PPG)WG = GAS DENSITY (PPG)OUTPUT:FUOO = SINGLE BUBBLE VELOCITY (FT/S)
REAL REYL,RO,WG INTEGER FLUIDIF (FLUID.EQ.l) THEN
CALL UHARMA(RO,WG,FUOO)ELSE
IF (REYL.GT.100.) THENFUOO=-0.528*(ALOGIO(REYL))**2+2.047*ALOG10(REYL)-0.932 ELSEFUOO=l.05
ENDIF ENDIFIF (FUOO.LT.O.) FUOO=0.RETURNEND
SUBROUTINE GASDEN(P,T,SGFG,WG)
Gas Density INPUT:P - pressure (psig)T - temperature (deg.F)SGFG - specific gravityOUTPUT:WG - gas density (Ibm/gal)
REAL P,T,SGFG, WG, ZC
CALL ZFACST(T,P,SGFG,Z)WG=.361*(P+14.7)/Z/(T+460.)*SGFG
CRETURN
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noAppendix B: Subroutines for Implementation of the Bubble Model 107
END
SUBROUTINE GASVIS (T,SGFG,P,GVIS)
CALCULATE VISCOSITY OF HYDROCARBON GASES USING THE LEE ET AL CORRELATION (TRANSACTIONS AIME, 1966, PG. 997).INPUT:T = TEMPERATURE (deg. F)P = PRESSURE (PSI)SGFG = GAS SPECIFIC GRAVITYOUTPUT:GVIS = GAS VISCOSITY (CP)
REAL TABS,W,AK,X,Y,Z,RHOGTABS=T+460.W=SGFG*29.AK=(9.4+.02*W)*(TABS**1.5)/ (209.+19.*W+TABS)X=3.5+(986./TABS)+.01*WY=2.4-.2*XCALCULATE GAS DENSITY, GM/CC.CALL ZFACST (T,P,SGFG,Z)RHOG=(P+14.7)*W/(10.72*Z*TABS*62.4)CALCULATE GAS VISCOSITY,CP.GVIS=AK*EXP(X*RHOG**Y)/10000.RETURNEND
SUBROUTINE KFAC(FLUID,REY, K)
CALCULATES THE FLOW PROFILE COEFFICIENT K INPUT:FLUID = 1 - WATER; 2 - MUD REY = REYNOLDS NUMBEROUTPUT:K = FLOW PROFILE COEFFICIENT
REAL REY,K,AW(3) ,AM(3) INTEGER FLUID
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Appendix B: Subroutines for Implementation of the Bubble Model 108
DATA AW/1.28,0.20,0.40/DATA AM/1.30,2.00,0.25/IF (FLUID.EQ.l) THEN
K=AW(1)+AW(2)/(REY**AW(3)) ELSE
K=AM(1)+AM(2) / (REY**AM(3) ) ENDIFRETURNEND
SUBROUTINE LIQREY(RO,USL,D1,D2,EQVIS,REYL)
SUBROUTINE TO CALCULATE LIQUID REYNOLDS NUMBER INPUT:RO = LIQUID DENSITY (PPG)Dl,D2 = ANNULAR SECTION (IN)EQVIS = MIXTURE EQUIVALENT VISCOSITY (CP)OUTPUT:REYL = LIQUID REYNOLDS NUMBER
REAL RO,USL,D1,D2,EQVIS,REYLREYL= 928.*RO*USL*(D2-D1)/EQVISRETURNEND
FUNCTION PLFUNC(PLN,K)
CALCULATE THE CORRECTION FOR POWER LAW FLUIDS INPUT:PLN = n1 FROM POWER LAW MODEL OUTPUT:K = CORRECTION FACTOR
REAL S,PLN,K,TABLE(13,11),STAB(13),KTAB(11),DY,PLFUNCDATA STAB/0.,.25,.5,1.,2.,3.,4.,5.,6.,7.,8.,9.,10./ DATA KTAB/.01,.1,.2,.3,.4,.5,.6,.7,.8,.9,1./DATA TABLE/.5050,.5312,.5566,.6051,.6929,.7468,.7819,
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix B: Subroutines for Implementation of the Bubble Model 109
.8064,.8246 .8388 .8502,.8595,.8673,
.5500,.5606 .5710 .5908,.6270,.6547,.6755,
.6924,.7046 .7150 .7235,.7306,.7367,
.6000,.6062 .6122 .6237,.6445,.6612,.6736,
.6838,.6919 .6987 .7042,.7089,.7130,
.6500,.6539 .6577 .6649, .6781,.6882,.6966,
.7030,.7084 .7128 .7164,.7195,.7222,
.7000,.7024 .7048 .7094,.7179,.7246,.7297,
.7342,.7372 .7401 .7418,-7446,.7462,
.7500,.7516 .7531 .7560,.7611,.7651, .7685,
.7711,.7732 .7751 .7760,.7770,.7778,
.8000,.8009 .8018 .8034,.8064,.8081,.8107,
.8128,.8144 .8160 .8168,.8176,.8184,
.8500,.8504 .8509 .8517,.8533,.8546, .8551,
.8568,.8577 .8585 .8594,.8602,.8611,
.9000,.9002 .9004 .9008,.9015, .9022, .9027,
.9032,.9036 .9041 .9045,.9050, .9054,
.9500,.9500 .9501 .9502,.9504,.9506,.9508,
.9510,.9511 .9513 .9515,.9517,.9519,1., 1., 1., 1. l.,l. l.,l.,l.,l.,l.,l.,l./
cS=1./PLNCALL POLIN2(STAB,KTAB,TABLE,13,11,S,K,PLFUNC,DY)
CRETURNEND
CC ----------------------------------------------------------
SUBROUTINE POLIN2(X1A,X2A,YA,M,N,XI,X2,Y,DY)CC ----------------------------------------------------------------------------------------------------------C INTERPOLATION IN 2 DIMENSIONSC FROM PRESS ET AL. (1988)CC INPUT: X1A,X2A,YA,M,N,XI,X2CC OUTPUT: Y,DYC ----------------------------------------------------------------------------------------------------------C
INTEGER J,K,M,N,NMAX,MMAX PARAMETER (NMAX=11,MMAX=13)REAL X1A(M) ,X2A(N) ,YA(M,N), YNTMP (NMAX) , YMTMP (MMAX) ,X2,DY,X1,YCDO 12 J=1,M DO 11 K=1,NYNTMP(K)=YA(J,K)
11 CONTINUECALL POLINT (X2A, YNTMP,N,X2, YMTMP (J) ,DY)
12 CONTINUECALL POLINT (X1A, YMTMP,M, XI, Y,DY)RETURNEND
CC ----------------------------------------------------------
SUBROUTINE POLINT(XA,YA,N,X,Y,DY)C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix B: Subroutines for Implementation of the Bubble Model 110
c ----------------------------------------------------------------------------------------------------------------C INTERPOLATION IN 1 DIMENSIONC FROM PRESS ET AL. (1988)CC INPUT: XA,YA,N,XCC . OUTPUT: Y,DYC -------------------------------------------------------------C
INTEGER I,N,NS,M,NMAX PARAMETER (NMAX=13)REAL XA (N),YA(N),C (NMAX), D (NMAX),: DIF,X,Y,DY,DIFT,HO,HP,DEN,WNS=1DIF=ABS(X-XA(l))DO 11 1=1,NDIFT=ABS(X-XA(I))IF (DIFT.LT.DIF) THEN NS=IDIF=DIFT
ENDIF C (I) =YA(I)D (I) =YA (I)
11 CONTINUE Y=YA(NS)NS=NS-1DO 13 M=1,N-l DO 12 1=1,N-M
HO=XA(I)-X HP=XA(I+M)-X W=C(I+1)-D(I)DEN=HO-HPIF(DEN.EQ.O.)PAUSE DEN=W/DEN D(I)=HP*DEN C(I)=HO*DEN
12 CONTINUEIF (2 *NS.LT.N-M)THEN DY=C(NS+1)
ELSEDY=D(NS)NS=NS-1
ENDIFY=Y+DY
13 CONTINUE RETURN END
CC --------------------------------------------------------------
SUBROUTINE REYNOL(FLUID,Q, HLNS, RO,ROG,VISC,GVIS,Dl,D2,PLN,PLK,: DENNS,VISNS,VEL,REY,EQVIS)
Cc ----------------------------------------------------------------------------------------------------------------C REYNOLDS NUMBER FOR NEWTONIAN AND POWER LAW FLUIDSC IN PIPES AND ANNULAR.C
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Appendix B: Subroutines for Implementation of the Bubble Model 111
cp Input Variables:c FLUID - 1 = Newtonian Fluidc 2 = Power Law Fluidc Q - flow Rate (gpm)c HLNS - non-slip liquid holdupc RO - density of the fluid (lbm/gal)c ROG - density of the gas (lbm/gal)c Dl - external diameter of the internal pipe (in)c D2 - internal diameter of the external pipe (in)c VISC - newtonian fluid viscosity (cp)c GVIS - gas viscosity (cp)c PLN - flow behavior index for Power Law fluidcc
PLK - consistency index for Power Law fluid (eq.cp)Vs
cp
Output Variables:c DENNS - non-slip mixture density (lbm/gal)c VISNS - non-slip mixture viscosity (cp)c VEL - average flow velocity (ft/s)c REY - Reynolds numbercc
EQVIS - equivalent viscosity for Power Law fluids (eq. <cp Other Variables:Vs
c K - pipe diameter ratioc FACTOR - dimensionless number, function of pipe diameterccc
ratio and PLN, used for Power Law fluids.
REAL Q, HLNS, RO, ROG, Dl, D2, VISC, GVIS, VEL,REY, K, FACTOR, PLN,PLK, EQVIS, : AUX1,AUX2,AUX3,PLFUNC,DENNS,VISNSINTEGER FLUID
CC Pipe Diameter Ratio
K=D1/D2VEL=Q/(D2*D2-Dl*Dl)/2.448 DENNS=RO* HLNS+ROG*(l.-HLNS)IF (FLUID.EQ.l) THEN
C Newtonian Fluid-----EQVIS=VISCVISNS=VISC*HLNS+GVIS*(1.-HLNS)
ELSEIF (FLUID.EQ.2) THEN
C Power Law Fluid-----IF (K.EQ.O.) THEN
C Pipe Flow-----EQVIS= PLK*(D2/VEL)**(1-PLN)/96.*((3+1./PLN)/.0416)**PLN
ELSEC ---- Annular Flow-----
AUX1=1+KAUX2=1-KAUX3=1+K*K-AUX1*AUX2/ALOG(1./K)FACTOR=PLFUNC(PLN,K)EQVIS= (VEL/D2)**(PLN-1)* (AUX1*((1./PLN+2.)/.0416))**PLN*
: PLK*AUX3/96./AUX2**(1+PLN)/FACTOR* *PLN
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onAppendix B: Subroutines for Implementation of the Bubble Model 112
ENDIFVISNS=EQVIS*HLNS+GVIS*(1.-HLNS)
ELSEPRINT 13
13 FORMAT (' ','TYPE OF THE FLUID NOT SPECIFIED')STOP ENDIF
ENDIFC
REY = 928*DENNS*VEL*D2*(1-K) /VISNSC
RETURNEND
FUNCTION RTSAFE(XI,X2,XACC,UOO,NEXP,USG,USL,K)
FUNCTION TO CALCULATE GAS FRACTION (BETWEEN XI AND X2) USING THE RELATION SHOWN IN SUBROUTINE FUNCD.
REAL XI,X2,XACC,UOO,NEXP,USG,USL,K,FL,FH,DF,XL,XH,SWAP, : DXOLD,DX,F,TEMPINTEGER J,MAXIT PARAMETER (MAXIT=100)CALL FUNCD(XI,FL,DF,UOO,NEXP,USG,USL,K)CALL FUNCD(X2,FH,DF,UOO,NEXP,USG,USL,K)IF(FL*FH.GE.0.) PAUSE 'root must be bracketed1 IF(FL.LT.0.)THEN XL=X1 XH=X2
ELSE XH=X1 XL=X2 SWAP=FL FL=FH FH=SWAP
ENDIFRTSAFE=.5*(X1+X2)DXOLD=ABS(X2-X1)DX=DXOLDCALL FUNCD(RTSAFE,F,DF,UOO,NEXP, USG,USL, K)DO 11 J=1,MAXIT
IF(((RTSAFE-XH)*DF-F)*((RTSAFE-XL)*DF-F).GE.0.* .OR. ABS(2.*F).GT.ABS(DXOLD*DF) ) THEN
DXOLD=DX DX=0. 5* (XH-XL)RTSAFE=XL+DX IF(XL.EQ.RTSAFE)RETURN
ELSEDXOLD=DX DX=F/DF TEMP=RTSAFE RTSAFE=RTSAFE-DX
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Appendix B: Subroutines for Implementation of the Bubble Model 113
IF (TEMP.EQ.RTSAFE) RETURN ENDIFIF(ABS(DX).LT.XACC) RETURN CALL FUNCD(RTSAFE,F,DF,UOO,NEXP,USG,USL,K) IF (F.LT.O.) THEN XL=RTSAFE FL=F
ELSEXH=RTSAFE FH=F
ENDIF 11 CONTINUE
PAUSE 'RTSAFE exceeding maximum iterations'C
RETURNEND
SUBROUTINE UGINC(UM,INC,USL,VISC,UGVERT,UGI)
GAS VELOCITY FOR INCLINED WELLBORES INPUT:UM = MIXTURE VELOCITY (FT/S)INC = INCLINATION (degrees)USL = SUPERFICIAL LIQUID VELOCITY (FT/S)VISC = LIQUID VISCOSITY (CP)UGVERT= GAS VELOCITY IN VERTICAL WELL (FT/S)OUTPUT:UGI = GAS VELOCITY IN INCLINED WELL (FT/S)
REAL UM,INC,USL,VISC,UGVERT,UGI,UG(3,2,6) ,AUXI(6),AUX(3,2), : ANGLE(6),USLT(2),VISCT(3),DU, VERTINTEGER I, J, KDATA ANGLE/0.,10.,20.,40.,60., 80./DATA USLT/0.7,1.7/DATA VISCT/1.,10.,19./VERT=1.02l*UM+0.6526 UG (1,1,1) =1.UG(1,1,2)=(1.010*UM+1.289)/VERT UG(1,1,3)=(1.054*UM+1.373)/VERT UG(1,1,4)=(0.9295*UM+2.63)/VERT UG(1,1,5)=(0.9976*UM+3.326)/VERT UG(1,1,6)=(1.104*UM+2.821)/VERTVERT=0.9041*UM+1.213 UG (1, 2,1) =1.UG(1,2,2)=(0.7921*UM+2.108)/VERT UG(1,2,3)=(0.8310*UM+2.309)/VERT UG(1,2,4)=(0.8310*UM+3.600)/VERT
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oAppendix B: Subroutines for Implementation of the Bubble Model 114
UG(1,2, 5) = (0.8770*UM+4.281)/VERT UG(1,2,6)=(0.9040*UM+3.600)/VERT
CVERT=1.438*UM+1.091 UG(2,1,1)=1.UG(2,1,2)=(1.521*UM+1.460)/VERT UG(2,1,3) = (1.600*UM+1,662)/VERT UG(2,1,4)=(1.596*UM+2.121)/VERT UG(2,l,5)=(1.609*UM+2.057)/VERT UG(2,1,6)=(1.438*UM+1.200)/VERT
CVERT=1.304*UM+0.9228 UG (2,2,1) =1.UG(2,2,2)=(1.513*UM+0.9520)/VERT UG(2,2,3)=(1.644*UM+0.9600)/VERT UG(2,2,4)=(1.644*UM+1.696)/VERT UG(2,2,5)=(1.637*UM+1.882)/VERT UG(2,2,6)=(1.756*UM+1.091)/VERT
CVERT=1.685*UM+1. 000 UG(3,1,1)=1.UG(3,1,2)=(1.683*UM+1.126)/VERT UG(3/1,3)=(1.715*UM+1.290)/VERT UG(3,1,4)=(1.957*UM+1.654)/VERT UG(3,1,5)=(1.892*UM+1.121)/VERT UG(3,1,6)=(1.400*UM+1.000)/VERT
CVERT=1.560*UM+0.600 UG(3,2,1)=1.UG(3,2,2)=(1.560*UM+1.000)/VERT UG(3,2, 3) = (1.560*UM+1.Ill)/VERT UG(3,2,4)=(1.540*UM+2.700)/VERT UG(3,2,5)=(1.539*UM+2.474)/VERT UG(3,2,6)=(1.521*UM+2.271)/VERT
CDO 20 1=1,3
DO 20 J=l,2 DO 10 K=l,6
AUXI (K) =UG (I, J, K)10 CONTINUE
CALL POLINT(ANGLE,AUXI,6,INC,AUX(I,J),DU) 20 CONTINUE
CALL POLIN2(VISCT,USLT,AUX,3,2,VISC,USL,UGI,DU) UGI=UGI*UGVERT
CRETURNEND
SUBROUTINE UHARMA(RO,WG,UH)
HARMATHY'S BUBBLE VELOCITY INPUT:
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Appendix B: Subroutines for Implementation of the Bubble Model 115
RO = LIQUID DENSITY (PPG)WG = GAS DENSITY (PPG)UH = HARMATHY'S SINGLE BUBBLE VELOCITY (FT/S)
REAL RO,WG,UHUH=1.53*(981.*(RO-WG)*70./(RO*RO)*8.33)**.25*0.03281RETURNEND
SUBROUTINE ZFACST (T,P,SGFG,Z)
Z-FACTOR INPUT:P = PRESSURE (PSIG)T = TEMPERATURE (deg. F)SGFG = GAS SPECIFIC GRAVITYOUTPUT:Z = SUPERCOMPRESSIBILITY FACTOR
REAL T,P,SGFG,Z,TC,PC,TR,PR,A,B,C,D,E,F,GCALCULATE CRITICAL AND REDUCED TEMPERATURE AND PRESSURE.TC=169.0+314.0*SGFGPC=708.75-57.5*SGFGTR=(T+460.0)/TCPR=(P+14.7)/PCA=1.39*(TR-.92)**.5-.36*TR-.101B=(.62-.23*TR)*PRC=(.066/(TR-.86)-.037)*PR**2D=(.32/(10.**(9.*(TR-1.))))*PR**6E=B+C+DF=(.132-.32*ALOG10(TR))G=10.**(.3106-.49*TR+.1824*TR**2)Z=A+(1.-A)*EXP(-E)+F*PR**GRETURNEND
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APPENDIX C: BASIC EQUATIONS USED IN THE ANALYSIS OF
THE EXPERIMENTAL DATA
The following equations were used to calculate the superficial liquid and
gas velocities, and the gas fraction in the annular section.
a. Gas density (Craft and Hawkins, 1959)
where
z = super-compressibility factor,
yg = gas specific gravity (air = 1 ),
P = pressure (psig),
T = temperature (°F),
pg = gas density (Ibm/gal);
for air, the critical constants for calculation of z factor are Pcr = 547 psia and Tcr =
.361 Yg(P + 14.7) Pg z(T + 460) (C.1)
239 °R.
b. Gas flow rate
Pg (C.2)
Qgas — 3.496 z Qstd (T + 460) (P + 14.7) (C.3)
116
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Appendix C: Basic Equations used in the Analysis of the Experimental Data 117
where
Qair = air flow rate (bbl/min),
Qgas = Qas flow rate (bbl/min),
pg = gas or air density (Ibm/gal),
Qstd = standard flow rate (MMscf/d).
c. Air-water surface tension
a = 75.6531 - 0.14145 T - 0.00026 T2, (C.4)
where
o = air-water surface tension (d/cm),
T = temperature (°C).
d. Air viscosity (White, 1986)
pa = 0.0171 Tko'0-7, (C.5)
where
pa = air viscosity (cp),
Tko = 2.73-16 = temperature ratio
T = °K.
e. Water viscosity (White, 1986)
pw = 1.792 EXP[-1 .94 - 4.8 Tko + 6.74 Tko2] (C.6)
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Appendix C: Basic Equations used in the Analysis of the Experimental Data 118
where
|iw = water viscosity (cp).
f. Conversion from Bingham parameters to Power Law
n' = 3.32 log10-2Mp + ^Pp + ly
1 . 510 [Pp + Xy]511"'
where
lip = plastic viscosity (cp),
xy = yield point (lbf/1 0 0 ft2),
n'= flow behavior index,
k'= consistency index (eq. cp).
g. Two-Phase Reynolds Number
m 928 Pns Um Do (1 ’ Dft)
N r8 = i £ -------------
where
pns = non-slip density (Ibm/gal) = pi (1- ans)+ Pg « n s .
Um = mixture velocity (ft/s),
Dj = annulus inside diameter (in),
D0 = annulus outside diameter (in),
D rt = diameter ratio = Dj/D0,
Pns = non-slip viscosity (cp) = peq (1-ans)+ Pg ans:
(C.7)
(C.8 )
(C.9)
(C.10)
(C.11)
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Appendix C: Basic Equations used in the Analysis of the Experimental Data
for Newtonian fluids, the equivalent viscosity psq is the liquid viscosity m
Power Law fluids, there are two expressions for peq:
g.1. Pipe flow
1,1 nO-nOIX l - r g n + ± \n'
' n'J^ 8q 96 U$‘n ) .0416°'
g.2. Annular flow
H-eq 2 + ^jn*
96 Do(n''1) (1 - Drt) (1 + n’)Ln'
where
Fg = 1 + D i - ( l ^ |In
LDrt.
S = Fredrickson and Bird (1958) factor.
h. Friction Factors
h.1. Pipe flow
h.1.1. Laminar (N r0 < 2100):
h.1 .2 . Turbulent
h.1.2.1. Newtonian fluid (Colebrook's equation)
4= = -4 log10f.269 - f - + J-25SLVf 1 Dp N ReVf.
where e / Dp = pipe roughness;
h. 1.2.2. Power Law fluid (Dodge-Metzner's equation)
119
and for
(C.12)
(C.13)
(0-14)
(0.15)
(C.16)
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Appendix C: Basic Equations used in the Analysis of the Experimental Data
J_ = —4 _ |0 g(NRe f(1*-5n’)) -,325.Yf n- 75 Re } n’ 1-2 (C. 17)
120
h.2. Annular flow
h.2.1. Laminar (Nr0 < 3000)
h.2.1.1. Concentric Annulus
N r 0
where
(C. 18)
Fca =16 (1 - Dft)2 In[ 1
Drt.
(1 + D2) In 1LDrtJ
- (1 - D2)
h.2.1.2. Eccentric Annulus (Snyder and Goldstein, 1965)
Fef = ea
(C. 19)
(C. 20)NRa
where
c 4 (1 - D2) (1 - Dft)2 r ea - ;--------------
<{> sinh4 /□ (C. 2 1 )
0 = f(/o,/i),
/ 0 = f(Drt)e)I
f\ = f(Dn.e),
e = eccentricity = 2 DBc/(D0-Dj), (C.22)
Dec = distance between centers;
h.2 .2 . Turbulent
For Newtonian fluids, we can apply the Colebrook's Eq. (C.16) with the
correction factor developed by Gunn and Darling (1963) for non-circular
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Appendix C: Basic Equations used in the Analysis of the Experimental Data 121
geometries. For Power Law fluids, we apply the Dodge-Metzner Eq. (C.17) with
the same correction factor.
i. Gas Fraction
The gas fraction is not obtained directly during the experiment but
calculated using the collected data. There are two ways to calculate gas
fraction: by the differential pressures in each sensor while circulating, and
through the total differential pressure in the loop after the valves are closed.
The first method while circulating, uses the readings in each of the three
differential pressure sensors. With these values and the flow rates, we can
calculate the friction loss, the mixture density and the gas fraction:
DPjot = DPHyd + DPf,
DPHyd = [0.052 (8.33 - ps) L] COS 0 (0.24)
(C.23)
(C. 25)
then,
Ps =- DPjpt + 0.433 L cos 9
(C.26)
Finally,
(C. 27)
In the above equations,
DPiot = differential pressure read by the sensor (psi),
DPhyd = hydrostatic differential pressure (psi),
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Appendix C: Basic Equations used in the Analysis of the Experimental Data 122
DPf = frictional pressure loss (psi),
L = length along which DPjot is measured (ft),
ps = slip density (Ibm/gal),
f = Fanning friction factor.
During the tests, the friction losses for gas-water mixtures were in the
range of 0.2% to 2% of the total differential pressure. For gas-mud1 mixtures,
the friction increased to 0.7% to 7% of the total differential pressure and for gas-
mud2 the friction losses were between 2 % and 2 0 % of the total differential
pressure.
The second method to calculate a uses the stabilized value of DP-i
(differential pressure in the loop). The main problem with inis method is ine
need to estimate the error due to the presence of the two curved sections below
and above the annular section in the loop. This is the equation, with the
correction already included:
a = .002 (20.77 - .148 0) + .019 [ p| L + „ r „DPl— - ] ,1 Pi .052 p, cos 0 J (C. 28)
where:
0 = angle from vertical position (degrees),
L = distance between points where DP is measured (ft),
DP-i = total differential pressure in the loop (psi),
pi = liquid density (Ibm/gal).
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VITA
Edson Yoshihito Nakagawa, son of Kichisaburo Nakagawa and Celia
Emsko Nakagawa, was born in Araras, Sao Paulo, Brasil, in April 15, 1957.
In 1979, he graduated in Civil Engineering at Escola de Engenharia de
Piracicaba, in Sao Paulo state. Right after that, he joined Petroleo Brasileiro
S.A., PETROBRAS. In 1980 and 1981 he took the Course of Petroleum
Engineering sponsored by PETROBRAS and worked in exploratory drilling in
the Amazon jungle. After that, he worked for one year in offshore plataforms in
Campos Basin, Rio de Janeiro. With two years of experience in the field, he
went to PETROBRAS' Research Center (CENPES) in Rio de Janeiro to work in
projects related to drilling optimization: drilling models, drilling bits, and
cementing operations. In August of 1984, he entered Universidade de Ouro
Preto, where he received his Master's degree in Petroleum Engineering in
March of 1986. Then he returned to CENPES for one year. Finally, he came to
Louisiana State University in August of 1987 to pursue the Ph.D. degree in
Petroleum Engineering.
He is married to Albertina Nakagawa and they have two children: Carlos
Renato (7 years old) and Aline Sayuri (4 years old).
123
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DOCTORAL EXAMINATION AND DISSERTATION REPORT
Candidate: E dson Y o s h ih i to Nakagawa
Major Field: P e tro le u m E n g in e e r in g
Title of Dissertation: Gas K ick B e h a v io r D u rin g W ell C o n tro l O p e ra t io n s i n V e r t i c a l and S la n te d W e lls
Approved:
M a j o r P r o f e s s o r a n a ' C h a i r m a n
D e a n o f t h e G r a d u a t e S c h o o l
EXAMINING COMMITTEE:
Date of Examination: ___________________________________ _
November 8 , 1990
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