gillermo michel. modeling of multiphase flow in wells
DESCRIPTION
Thesis: MODELING OF MULTIPHASE FLOW IN WELLS UNDER NONISOTHERMAL AND NONEQUILIBRIUM CONDITIONSTRANSCRIPT
UNIVERSITY OF OKLAHOMA
GRADUATE COLLEGE
MODELING OF MULTIPHASE FLOW IN WELLS UNDER
NONISOTHERMAL AND NONEQUILIBRIUM CONDITIONS
A THESIS
SUBMITTED TO THE GRADUATE FACULTY
In partial fulfillment of the requirements for the
Degree of
MASTER OF SCIENCE
By
GUILLERMO GERMAN MICHEL VILLAZÓN Norman, Oklahoma
2007
MODELING OF MULTIPHASE FLOW IN WELLS UNDER NONISOTHERMAL AND NONEQUILIBRIUM CONDITIONS
A THESIS APPROVED FOR THE MEWBOURNE SCHOOL OF PETROLEUM AND GEOLOGICAL
ENGINEERING
BY
______________________________
Faruk Civan, Chair
______________________________
Roy Knapp
______________________________
Robert Hubbard
©Copyright GUILLERMO GERMAN MICHEL VILLAZÓN 2007 All Rights Reserved.
“Science without religion is lame, religion without science is blind”
Albert Einstein (1879 - 1955)
"Science, Philosophy and Religion: a Symposium", 1941
To my parents, Ricardo and Stina, for their unconditional and unlimited support
and faith.
To my lovely wife, Alejandra, for her infinite love, kindness and care.
ACKNOWLEDGMENTS
I wish to acknowledge and thank many people for their cooperation during the
course of my studies at the University of Oklahoma.
In particular, I would like to express my most sincere gratitude to Dr. Faruk
Civan, chairman of my committee, for his advice and assistance in completing the
present work, for his patience and guidance, and for the trust he put in my work.
I acknowledge the time and dedication given by the members of my committee
Dr. Roy Knapp and Mr. Robert Hubbard.
I would like to thank to the ConocoPhillips Company for providing a fellowship
during my graduate studies.
I am grateful to our Creator, for all the blessings received in the path that he has
chosen for me.
iv
TABLE OF CONTENTS ACKNOWLEDGMENTS ..................................................................................... iv TABLE OF CONTENTS........................................................................................ v LIST OF FIGURES .............................................................................................. vii LIST OF TABLES................................................................................................ vii ABSTRACT......................................................................................................... viii 1. INTRODUCCION .......................................................................................... 1
1.1. OVERVIEW ................................................................................................ 1 1.2. DESCRIPTION OF THE PROBLEM......................................................... 2 1.3. PRESENT STUDY...................................................................................... 7 1.4. ORGANIZATION OF THE THESIS.......................................................... 9
2. LITERATURE REVIEW ............................................................................. 12 2.1. OVERVIEW .............................................................................................. 12 2.2. THE ANSARI ET AL. APPROACH ....................................................... 13 2.3. THE ASHEIM APPROACH ..................................................................... 14 2.4. THE AYALA AND ADEWUMI APPROACH ........................................ 15 2.5. THE DOWNAR-ZAPOLSKI ET AL. APPROACH................................. 16 2.6. THE BADUR AND BANASZKIEWICZ APPROACH........................... 17 2.7. THE FEBURIE ET AL. APPROACH....................................................... 18 2.8. THE CIVAN APPROACH........................................................................ 19 2.9. SUMMARY............................................................................................... 20
3. DETERMINATION AND CONSTITUTIVE EQUATIONS OF THE PHYSICAL PROPERTIES .................................................................................. 21
3.1. OVERVIEW .............................................................................................. 21 3.2. PHYSICAL PROPERTIES OF A MULTIPHASE FLUID ...................... 22 3.3. STANDARD CONSTITUTIVE EQUATIONS........................................ 26 3.4. PROPOSED MODEL FOR LIQUID HOLDUP ....................................... 28 3.5. RELAXATION TIME FOR PRODUCING WELLS................................ 32
4. DESCRIPTION OF THE MULTIPHASE FLOW OF A RESERVOIR FLUID IN WELLS ............................................................................................... 34
4.1. OVERVIEW .............................................................................................. 34 4.2. MODELING MULTIPHASE FLOW IN WELLS.................................... 35 4.3. STEADY-STATE MODEL UNDER NONISOTHERMAL AND NONEQUILIBRIUM CONDITIONS.............................................................. 42
5. SIMULATION OF THE MULTI-PHASE FLOW OF A RESERVOIR FLUID IN WELLS ............................................................................................... 45
5.1. OVERVIEW .............................................................................................. 45 5.2. SELECTION OF THE NUMERICAL DIFFERENTATION SCHEME.. 46 5.3. SCHEME OF THE NUMERICAL DIFFERENTATION......................... 47 5.4. COMPUTING THE CHANGE OF STATE.............................................. 48 5.5. COMPUTATIONAL PROCEDURE ........................................................ 52
6. VALIDATION AND APPLICATION......................................................... 56 6.1. OVERVIEW .............................................................................................. 56
v
6.2. DATA SELECTION.................................................................................. 57 6.3. SIMULATOR VALIDATION .................................................................. 58 6.4. UPWARD MOTION OF BLACK-OIL IN VERTICAL WELLS ............ 62 6.4.1 CASE 1: TWO-PHASE FLOW OF HEAVY CRUDE OIL ................... 63 6.4.2 CASE 2: TWO-PHASE FLOW OF LIGHT CRUDE OIL ..................... 68 6.4.3 CASE 3: THREE-PHASE FLOW OF A HEAVY CRUDE OIL............ 73
7. DISCUSSION AND CONCLUSIONS ........................................................ 78 7.1. DISCUSSION............................................................................................ 78 7.2. CONCLUSIONS........................................................................................ 82
REFERENCES ..................................................................................................... 83 APPENDIX A: A DISCUSSION OF THE EQUILIBRIUM CONDITION FOR A MULTIPHASE FLUID SYSTEM........................................................................ 86 APPENDIX B: CORRELATIONS AND BASIC RELATIONSHIPS ................ 90 APPENDIX C: NOMENCLATURE.................................................................... 98
vi
LIST OF FIGURES
Figure 1-1 : Schematic of the motion in wells........................................................ 3 Figure 1-2 : Phases distribution for a cross-sectional area ..................................... 4 Figure 4-1 : Schematic for local properties in a conduit....................................... 35 Figure 4-2 : Example of local velocity, pressure and temperature distributions .. 36 Figure 5-1 : Segmentation of the production pipe in a well ................................. 48 Figure 5-2 : Flowchart for simulating multiphase flow in wells .......................... 54 Figure 6-1: Correlation for Case 1 ........................................................................ 60 Figure 6-2 : Correlation for Case 2 ....................................................................... 61 Figure 6-3 : Correlation for Case 3 ....................................................................... 61 Figure 6-4 : Pressure drop for Case 1 ................................................................... 64 Figure 6-5 : Void fraction for Case 1.................................................................... 65 Figure 6-6 : Temperature drop for Case 1 ............................................................ 66 Figure 6-7 : Temperature difference for Case 1.................................................... 66 Figure 6-8 : Dryness gradient for Case 1 .............................................................. 67 Figure 6-9 : Relaxation time for Case 1................................................................ 68 Figure 6-10 : Pressure drop for Case 2 ................................................................. 69 Figure 6-11 : Void fraction for Case 2.................................................................. 69 Figure 6-12 : Temperature drop for Case 2 .......................................................... 70 Figure 6-13 : Temperature difference for Case 2.................................................. 71 Figure 6-14 : Dryness gradient for Case 2 ............................................................ 71 Figure 6-15 : Relaxation time for Case 2.............................................................. 72 Figure 6-16 : Pressure drop for Case 3 ................................................................. 73 Figure 6-17 : Void fraction for Case 3.................................................................. 74 Figure 6-18 : Temperature drop for Case 3 .......................................................... 75 Figure 6-19 : Temperature difference for Case 3.................................................. 75 Figure 6-20 : Dryness gradient for Case 3 ............................................................ 76 Figure 6-21 : Relaxation time for Case 3.............................................................. 77
LIST OF TABLES Table 2-1 : Literature Review............................................................................... 20 Table 6-1 : Data considered for application.......................................................... 57 Table 6-2 : Adjustable parameters and Coefficient of Determination.................. 62
vii
ABSTRACT
The multiphase flow of reservoir fluids in producing wells has been a subject of
investigation in various previous studies. In general, the motion of reservoir fluids
undergoing a gas separation along the well has been modeled by using empirical
correlations. Recently, however, the emphasis has shifted to theoretical modeling.
The present study provides a rigorous theoretical approach for modeling of the
upward motion of reservoir fluids considering the gas separation phenomenon in
the production wells.
The reservoir fluid is represented as a mixture of three phases, consisting of the
gas, oil, and water phases. A homogenous fluid model is formulated for general
purposes for describing the upward motion of a multiphase fluid system in pipes.
But, its application is demonstrated for well operations under the steady-state
conditions. The upward motion is considered under the non-isothermal and non-
equilibrium conditions by taking into account the irreversible loss in energy. The
loss in energy is mainly due to the interaction of the system with the
surroundings. The homogeneous model is simplified for the steady-state motion
in pipes having constant and circular cross-sectional areas.
The separation of the gas phase is considered to cause a non-equilibrium effect in
the upward motion. The non-equilibrium effect occurs when the phase velocities
are not equal. Two approaches are presented for describing the non-equilibrium
viii
effect on the bases of the prediction of the liquid holdup and the estimation of the
relaxation time occurring in the gas phase separation.
A new improved model for prediction of the liquid holdup is formulated. The
liquid holdup is predicted by the means of a constitutive equation. The
constitutive equation is based on the mixture density and the slip ratio. The
proposed holdup model provides a closure for the developed homogenous model
and it is employed for the application in the present study.
A practical means for solving the resulting differential equations is developed. A
series of simulated case studies are performed using the selected data. The data
was acquired from producing vertical wells and published in a previous study.
After validating the output data of the simulations, the motion of the studied cases
is described and characterized. The characterization includes the behavior of the
relaxation time occurring in the gas phase separation. The model developed here
provides important improvements over the existing models, which do not take
into account accurately the effects of the relaxation phenomenon and the liquid
holdup.
ix
1. INTRODUCCION
CHAPTER 1
INTRODUCTION
1.1. OVERVIEW
The particular phenomenon of concern of this thesis is the upward motion of
reservoir fluids in producing wells. In this chapter, the motivation and the scope
of the present study are established. A description of the fluid flow in petroleum
wells in terms of the governing physical phenomena is addressed. Then, the
specific objectives of the present study are defined. The specific objectives are
considered to accomplish the solution of the main problem. At the end, the
organization of the study towards the fulfillment of objectives is presented.
1
1.2. DESCRIPTION OF THE PROBLEM
In general, hydrocarbon fluids present in reservoirs contain a large number of
various substances. Each of these substances has different physical properties and
behavior affecting in specific ways the properties of the fluid phases. Moreover,
the interfaces or surface borders between the fluid phases have physical properties
and behavior on their own. Consequently, large amounts of measurements have to
be done in order to determine the required properties by means of a detailed
model. For that reason, theoretical models of fluid dynamics for reservoir fluids in
producing wells have been proposed in various types and successes.
Typically, the reservoir fluid consists of three distinct phases1,22. These are the
gas, oil, and water phases. Thus, the flow of the reservoir fluid in wells can be
modeled as the flow of a multiphase-fluid system of several phases.
For a producing well, the motion of the reservoir fluid is depicted in figure 1-1.
By considering the well fluid as a single multiphase-fluid system containing gas,
oil, and water phases, the flow in the production pipe can be described by the
fundamental equations governing the flow of fluids in conduits.
2
Oil WaterGas
Figure 1-1 : Schematic of the motion in wells
In the present modeling approach, it is assumed that the three fluid phases (gas,
oil, and water) are homogeneous and uniformly distributed over a cross-sectional
area (figure 1-2a). As the multi-phase fluid flows upward along the pipe from the
well-bore to the wellhead, an interface mass transfer is considered to occur across
the gas and liquid (oil and water) interphases14. The mass transfer may be
bidirectional. However, only the separation of the gaseous phase (gas) from the
liquid phases (oil and water) is considered in this study. Because the pressure
continuously decreases in the upward motion of the fluid, there is no dissolution
of the gas phase into the liquid phases occurring during flow. Within a particular
cross-sectional area, the multiphase fluid has a distribution of the mass fraction
3
for the various phases set by the local state of properties. While moving upward,
the multiphase fluid of various phases undergo a change in mass fraction
distribution along the well (figure 1-2b and figure 1-2c).
Multiphase Water Oil Gas
(a) (b) (c)
Figure 1-2 : Phases distribution for a cross-sectional area
Usually, depending on the prevailing conditions in a pipe, the interface mass
transfer between the liquid and gaseous phases occurs without reaching an
equilibrium state when the flow is sufficiently fast. Hence, it is reasonable to
consider that the mass transfer between the various phases occurs at a non-
equilibrium state17 (flashing) process. This means that the mass transfer occurs
dynamically backward and forward between the various phases. Unfortunately,
there is no well-proven and satisfactory model available for such cases involving
the flashing hydrocarbons.
A generalized model for flashing fluids has been developed in a limited number
of previous studies6,17. This flashing model considers a relaxation in time for gas
4
separation from the liquid phases due to the slow mass transfer between the gas
and liquid phases. Consequently, a unidirectional and cumulative mass transfer
from the liquid phases to the gaseous phase is assumed for the present study.
The mass transfer from the liquid phases to the gaseous phase begins when the
multiphase-fluid system pressure drops to below the bubble-point pressure. As the
multiphase-fluid flows along the pipe length, the pressure and temperature of the
fluid system decrease. The motion of the multiphase-fluid causes a pressure drop.
Simultaneously, the heat transfer by conduction and convection, the effect of the
fluid expansion and the effect of friction cause a temperature change. The
temperature change by expansion is referred to as the Joule-Thompson effect. The
heat transfer can be computed knowing the temperature of the surroundings. The
surrounding temperature is set mainly by the insulation technique of the conduit
and the geothermal gradient of the surrounding rock formation.
Another approach to modeling the motion of the multiphase fluid system is to
estimate the volumetric fraction of the liquid phases, referred to as liquid holdup.
Several studies have been performed for predicting the liquid holdup in wells.
These studies model the deviation from equilibrium in terms of a slippage
occurring between the gas phase and the liquid phases (oil and water) rather than
as a flashing process.
5
However, the slippage phenomenon have been proven to be complex enough to
be modeled by a single correlation for the liquid holdup9,18. All developed models
use a set of these correlations for predicting the liquid holdup. Usually, different
correlations are employed depending on the local conditions along the producing
pipe.
In the field facilities, the hydrocarbon fluid can be separated into three
components1,22 (gas, oil, and water). They are called the pseudo-components.
These components are at atmospheric conditions and behave differently than the
phases flowing through the conduit. Therefore, the gas, oil, and water pseudo-
components are different substances than the gas, oil, and water phases.
Because both the pressure and temperature are changing along the pipe, it is
impractical to measure directly all the physical properties of the reservoir fluid
phases during flow. Hence, several correlations have been developed for
estimating the properties of these phases. In general, the properties of the pseudo-
components and the conditions of the local state are required for these
correlations. Thus, by knowing the afore-mentioned properties, the physical
properties of the various phases can be estimated.
6
1.3. PRESENT STUDY
The scope of the present study is to develop an improved model for the flow of a
reservoir fluid as a multiphase fluid system in wells producing under steady-state
conditions. The flow is assumed to be non-adiabatic and non-isothermal
considering the convective and conductive heat transfer as the energy losses and
the effect of the friction. The reservoir fluid is flowing along the production pipe
with a non-equilibrium mass transfer across the interface between the liquid
phases and the gas phase. The study cases are considered based on the published
data for producing vertical wells.
Then, the fundamental laws of mass, momentum and energy conservation are
applied to describe the change in velocity, pressure, and temperature of reservoir
fluids flowing through the wells. However, the change in density of the fluid
cannot be obtained by predicting the previously mentioned changes alone because
the gas mass transfer from the liquid phases to the gas phase is not at equilibrium
during flow. Therefore, the flashing process occurring inside the production pipe
has to be modeled by other means.
The velocities of the phases are equal when the system has reached an
equilibrium9,18 as shown in Appendix A. For this reason, the deviation from
equilibrium is predicted by estimating the phase velocities. In this study, a new
7
equation for obtaining the ratio of the gas phase velocity to the liquid phase
velocity is presented. With this velocity ratio, the liquid holdup can be obtained
accurately as well as the flowing density.
By using the fundamental laws of conservation and the proposed method for
liquid holdup prediction, a series of simulations are then performed to accurately
predict the drop in pressure and temperature along the wells of each study case.
Both the relaxation time and the liquid holdup models describe the same
phenomenon satisfactorily which is the deviation from the equilibrium. Thus, the
behavior of the relaxation along the pipe is estimated with the data yielded by the
simulations.
The fundamental laws of conservation are formulated in their differential forms.
Thus, all the properties of the multiphase fluid system are either spatially
averaged in nature or homogenous. A numerical method is developed for solving
the differential equations given by the conservation laws. This numerical method
is extensively described.
The main objective of the present study is to model and characterize the flow of a
reservoir fluid in producing wells. The main objective is accomplished by the
following specific objectives:
8
• Develop a technique for estimating the properties for a multiphase fluid
system.
• Introduce a new improved model for estimating the liquid holdup as a
deviation from equilibrium.
• Develop a homogenous model applicable to the flow of reservoir fluids in
wells under non-isothermal and non-adiabatic conditions.
• Prove the relaxation time as a property that characterizes the deviation
from equilibrium for flowing reservoir fluids.
• Solve the developed homogenous model for simulating the flow with a
numerical scheme.
• Validate the results of the simulations by using a correlation developed for
experimental measurement of the void fraction.
1.4. ORGANIZATION OF THE THESIS
The contents of this thesis are organized and reported in seven chapters and two
appendixes as described in the following.
The current chapter, Chapter One, provides an overview of the problem of interest
and presents the scope of the present study. Chapter Two presents a
comprehensive review of the relevant literature.
9
Chapter Three provides a technique to approximate the properties of a multiphase
fluid system under non-equilibrium conditions. The reservoir fluid is represented
as a multiphase fluid system.
Chapter Four describes the modeling of a reservoir fluid in motion in wells. The
laws of mass, momentum, and energy conservation are expressed in differential
forms. A homogenous model for pipes with circular cross-sectional area is
developed. The cross-sectional area can be either constant or variable.
Chapter Five presents the numerical method developed in order to perform the
simulations using the technique specified in Chapter Three and the homogenous
model developed in Chapter Four.
Chapter Six shows the relevant results obtained by the simulations. The results are
validated with a model developed in the literature for correlating experimental
data. The application is illustrated by means of three study cases for the upward
motion of the gas/oil/water mixtures in wells.
Chapter Seven contains the discussion and conclusions after analyzing the results
obtained for the application.
10
Appendix A shows that a multiphase fluid system is at equilibrium condition if
the phase velocities are equal.
Appendix B presents a collection of correlations required for estimating the
properties of the gas, oil, and water phases as well as the wall surface properties
of the pipe.
Appendix C illustrates the adopted nomenclature for the various properties,
parameters and variables employed by the formulations in the present study.
11
2. LITERATURE REVIEW
CHAPTER 2
LITERATURE REVIEW
2.1. OVERVIEW
In this chapter, a review of the relevant studies about the flow and behavior of the
flashing fluids at steady-state is presented. These studies describe the flashing
phenomenon with different approaches. The description of each approach is
properly addressed towards detailing the features of interest for the present study.
Usually, some simplifications were made in order to enable the measurement of
the pertinent properties. At the end, a table summarizes the key features covered
by the current and the reviewed studies involved in modeling the flashing fluids.
12
2.2. THE ANSARI ET AL. APPROACH
The mechanistic approach proposed by Ansari et al. 2 modeled the upward flow of
reservoir fluids in pipes. The model compiled and systematized the use of several
correlations for predicting the liquid holdup and flow pattern distribution along
the well. Separate models and correlations were proposed for each flow pattern.
Consequently, the estimation of the flowing density is not continuous when a
change in flow pattern is predicted.
The chosen correlations were selected to minimize the error as it was
demonstrated in the error analysis section of the previous studies. The validation
of the model was executed with data measured in producing vertical wells
although the formulations can be applied for all angles of inclination.
In this study, the multiphase fluid system is defined as a mixture of phases
flowing within a pipe having a constant and circular cross-sectional area.
However, this approach is not a homogeneous model because the velocity of the
mixture is set equal to the volumetric flux even though the system is not at
equilibrium.
The pressure drop is estimated mainly by the prediction of the liquid holdup.
However, there is no specification on how to incorporate the effect of a
13
simultaneous drop in pressure and temperature. Thus, it is implied that the system
is isothermal having taken the average between the inlet and outlet temperatures.
2.3. THE ASHEIM APPROACH
The mathematical approach proposed by Asheim3 modeled the slippage occurring
in an upward motion of reservoir fluids in producing wells with constant diameter
and variable inclination. The deviation from the equilibrium is modeled by the
liquid holdup prediction.
The prediction of the liquid hold up is achieved by assuming a linear relationship
between the velocity of the gas phase and the velocity of the liquid phase. The
linear parameters have to be assumed a priori for the phase velocity relationship.
This assumption resulted in a quadratic relationship between the velocity of the
various phases and the liquid holdup. Therefore, there is no assurance for a
continuous estimation of the flowing density when the mixture is undergoing a
change from the saturated to the unsaturated fluid conditions. In the error
analysis, it was proven that this approach minimizes the error in history matching.
Although the multiphase fluid system is defined as a mixture of phases, this
approach is not a homogeneous model because the velocity of the mixture is set
equal to the volumetric flux.
14
In this study, the liquid holdup prediction is mainly set by the pressure drop. The
flow is assumed to be isothermal having taken the average between the inlet and
outlet temperatures.
2.4. THE AYALA AND ADEWUMI APPROACH
The multi-fluid approach proposed by Ayala and Adewumi4 modeled the flow of
gas and condensates along a transmission pipeline with constant diameter. The
multiphase fluid system is defined as a collection of two completely separated
phases undergoing mass transfer across the interface.
The pressure drop is mainly set by the liquid hold up and the mechanical loss of
momentum occurring at the interface. The mechanical loss of momentum is the
free term in the modeling that provides closure in the pressure, temperature, gas
velocity, and liquid velocity formulations. The gas density and liquid density are
obtained by the equation of state for each phase. Furthermore, the mass transfer is
estimated by a numerical scheme based on the gas density equation of state.
The mechanical loss of momentum is estimated by the correlations describing
several flow pattern distributions. For that reason, there is no continuous
15
estimation of the liquid phase velocity when the mixture is undergoing a change
from a saturated to an unsaturated fluid.
2.5. THE DOWNAR-ZAPOLSKI ET AL. APPROACH
The homogenous approach proposed by Downar-Zapolski et al.17 modeled the
flow of water and steam along a horizontal conduit with variable cross-sectional
area. However, the effect caused by the change in the cross-sectional area was
only considered in the velocity formulation. The cross-sectional area effect was
omitted in the pressure and temperature formulations.
The multiphase fluid system is defined as a mixture of water and steam phases
flowing at the non-equilibrium and adiabatic conditions. The critical flow of the
mixture in pipes with small diameter is the key testing condition.
The deviation from the equilibrium is described by the means of a relaxation time
occurring in the steam separation. The pressure drop and the void fraction are
known a priori by experimental measurement. Then, a correlation is developed to
estimate the relaxation time by using the experimental data at various flow rates.
16
2.6. THE BADUR AND BANASZKIEWICZ APPROACH
The homogenous approach proposed by Badur and Banaszkiewicz5 modeled the
flow of water and steam along a conduit with small cross-sectional area. The main
feature is testing and describing the mass transfer of the gas phase by the means
of a constitutive equation. The multiphase fluid system is defined as a mixture of
the water and steam phases where the flowing conditions cause a deviation from
the equilibrium.
The pipe is horizontal with a variable cross-sectional area. The effect of a variable
area is omitted in the pressure and temperature formulations but it is considered in
the velocity formulation.
The homogenous model is closed by a constitutive equation for the flowing fluid
quality. This constitutive equation includes the relaxation time as a coefficient by
assuming that the flow is adiabatic. The remaining constant parameters of this
equation can be correlated by analyzing the experimental data.
An adequate correlation was developed for two different flow rates. The predicted
and experimental pressure drops are compared by the means of a plot as well as
the predicted and experimental void fractions.
17
2.7. THE FEBURIE ET AL. APPROACH
The homogenous approach proposed by Feburie et al.19 modeled the flow of
steam and water derived from a multi-fluid model. The model was applied to the
flow of steam/water mixtures along horizontal conduits with variable and small
cross-sectional area.
The multiphase fluid system is defined as a mixture of superheated water,
saturated water and saturated steam. Although the phases are assumed to flow at
equilibrium conditions, the deviation from the equilibrium was addressed by
partitioning the water phase into the superheated water phase and the saturated
water phase.
The homogenous model is closed by a constitutive equation for the relaxation in
the mass transfer occurring at the interface between the superheated water and the
saturated steam/water mixture. The temperature change is formulated by the
change in entropy considering irreversible heat transfer towards the surroundings.
However, the effect caused by the variable cross-sectional area is omitted in the
pressure and entropy formulations.
The validity of the constitutive equation was tested by comparing the predicted
pressure drop with the experimental pressure drop at various flowing conditions.
18
2.8. THE CIVAN APPROACH
The mechanistic approach proposed by Civan14 modeled the upward flow of
reservoir fluids in wells at non-equilibrium conditions. The flow is assumed to be
isothermal. It was implied that the constant temperature considered in this model
is the average between the inlet and outlet temperatures.
The key feature of the study is to demonstrate that the deviation from the
equilibrium in producing wells can be modeled by means of the relaxation time
concept even though this property was originally developed for tubes with small
diameter. It was shown that the law of conservation for the gas phase can be used
to give closure in a homogenous model for producing wells. Nonetheless, this
approach is not a homogeneous model because the velocity of the mixture is set
equal to volumetric flux.
The multiphase fluid system is defined as a mixture of gas, oil and water flowing
within a pipe having a constant and circular cross-sectional area. Although the
model was formulated for all angles of inclination, the application only
considered a vertical well.
The relaxation time is estimated by a correlation developed for the steam/water
mixture flowing in small tubes. The mass transfer of the gas phase and the density
19
of the mixture are set mainly by the relaxation time. Consequently, the pressure
drop and the quality gradient are set by this property as well.
2.9. SUMMARY
The table 2-1 summarizes the main attributes of the present study and all the
mentioned approaches.
Table 2-1 : Literature Review
Modeling Attributes
Ansari et al. (1994)
Asheim (1986)
Ayala and Adewumi.
(2003)
Downar-Zapolsky
et al. (1996)
Badur and Banaszkiewicz
(1998)
Febuire et al.
(1993) Civan (2006)
Present Study
Multiphase effect
consideration Mixture Mixture Multi-fluid Mixture Mixture
Mixture and
Multi-fluid Mixture Mixture
Homogeneous mass flux along well
No No No Yes Yes Yes No Yes
Nonequilibrium model
consideration Holdup Holdup Holdup Relaxation Relaxation Relaxation Relaxation
Holdup and
Relaxation
Orientation angle
Near Vertical
Near Vertical Topography Horizontal Horizontal Horizontal Vertical Vertical
Fluid type Reservoir Reservoir Reservoir Water Water Water Reservoir Reservoir
Slip ratio consideration No No No No No No Yes Yes
Thermal effect consideration --- Isothermal Adiabatic Adiabatic Adiabatic Non-
adiabatic Isothermal Non-adiabatic
20
3. DETERMINATION AND CONSTITUTIVE EQUATIONS OF THE
PHYSICAL PROPERTIES
CHAPTER 3
DETERMINATION AND CONSTITUTIVE EQUATIONS OF THE
PHYSICAL PROPERTIES
3.1. OVERVIEW
In this chapter, a procedure for estimating the properties of a flowing reservoir
fluid is presented. The reservoir fluid is modeled as a multiphase fluid system.
Several properties are defined when the flow is under non-equilibrium conditions.
The equilibrium condition is defined as the ideal state where all the phases of a
multiphase fluid system flow at the same velocity9,18 as shown in Appendix A.
For instance, a multiphase fluid is considered at equilibrium when it has been
static for an adequate lapse of time. Two approaches are introduced for describing
the non-equilibrium effect on the bases of the liquid holdup prediction and the
21
relaxation time elapsed before equilibrium is attained. A new improved
relationship for predicting the liquid holdup is proposed.
3.2. PHYSICAL PROPERTIES OF A MULTIPHASE FLUID
This section reviews a technique required to determine the properties of the
multiphase fluid system considered for modeling of the present phenomenon. The
objective is to estimate the properties from the mean cross-sectional pressure (P),
the mean cross-sectional temperature (T) and the constant properties of the
pseudo-components.
The mass flow rate is the main property for the mass balance equation. The
multiphase fluid density is set by this property. Knowing the pseudo-component
specific gravities of the gas and oil phases (γG and γO) and volumetric flow rates
of the gas, oil, and water phases at standard conditions ( , and ), the
overall mass flow rate
sGV& s
OV& sWV&
1 ( ) for the system can be determined by: m&
sW
sW
sO
sWO
sG
saG VVVm &&&& ρργργ ++= …………………...….……….………(3-1)
The water and air density at standard conditions (ρas and ρw
s) are constants. It is
assumed that there is no loss in mass during flow in the system. Thus, the mass
flow rate is constant at any point in the system.
22
It is necessary to define the volumetric flow rate of each phase1 ( , and )
by using the volumetric flow rates of the pseudo-components.
gV& oV& wV&
gs
Wwgs
Oogs
Gg BVRVRVV )( //&&&& −−= ……...………………………..............(3-2)
os
Oo BVV && = …………………………………………………….………...(3-3)
ws
Ww BVV && = ………………………………………………….…………..(3-4)
The formation volume factors for each phase, i.e. gas, oil, and water, (BBg, BoB and
BBw) can be estimated by the correlations given in Appendix B. The equations ,
and are consistent with the black-oil model for reservoir modeling. With
the volumetric flow rates of the gas, oil, and water phases, the multiphase-fluid
volumetric flow rate (V ) and the gas, oil, and water phase fractional flows (S
3-2
3-3 3-4
1 & 1g,
So and Sw) can be calculated as the following:
wog VVVV &&&& ++= ……………….……………………...………………..(3-5)
VV
S gg &
&= ……………………………….………………….……...…….(3-6)
VVS o
o &
&= …………………………………...………….………………..(3-7)
VVS w
w &
&= ………………………………………………………..………(3-8)
The sum of the oil fractional flow and the water fractional flow is denoted as the
liquid fractional flow (SL). By combining the previous equations, it is observed
that:
23
1=++ wog SSS ……………………………………………………….(3-9)
gL SS −= 1 …………………………………………………………...(3-10)
Note that while the mass flow rate is constant, the volumetric flow rate changes as
the pressure and the temperature change. The fractional flow of the various phases
can be used as the weighting factors in weighted averages for other properties.
A mass fraction of a phase is the mass of that phase per unit mass of the mixture
where both masses are flowing across the local cross-sectional area. The quality
or dryness of a multiphase fluid system is defined as the mass fraction for the gas
phase. The determination of the actual quality of the multi-phase fluid (x) is
discussed later on in section 3.4. The quality of the multiphase-fluid system in
equilibrium state14 (xst) between the liquid phases and the gaseous phase is given
by equation 3-11. This equilibrium quality is defined as the theoretical quality that
the flowing fluid would have if it was static; i.e. not flowing:
mVRVRV
xsaG
sWwg
sOog
sG
st &
&&& ργ)( // −−= ……….…………………..……..(3-11)
The gas-in-solution ratios (Rg/o and Rg/w) can be estimated with the correlations
presented in Appendix B.
24
The equilibrium density16 (ρst) of the multiphase-fluid system is calculated by
combining equations 3-1 and 3-5. It is the theoretical density that the flowing
fluid would have if it was static.
Vm
st &&
=ρ ……………………...……………………………………....(3-12)
Knowing the volumetric flow rate and the internal pipe diameter (D), the
volumetric flux 16 (u) in a circular pipe can be obtained as:
2
4DVu
π
&= …………………………...…………………………….…...(3-13)
For a homogeneous fluid, the viscosity14 (μ) can be estimated by weighting the
phase viscosities by their own fractional flow.
wwoogg SSS μμμμ ++= ………………………………………………(3-14)
The viscosities for each phase, i.e. gas, oil, and water phases, (μg, μo and μw) can
be estimated by the correlations given in Appendix B.
The Joule-Thompson coefficient (η) can be obtained by the fundamentals of
thermodynamics11.
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛∂∂
−−=Pp T
Tc
υυη 1 …………..………..……….………………..(3-15)
25
The specific volume (υ) is known as the reciprocal of density. The specific heat
(cp) for reservoir fluids can be estimated by a correlation given in Appendix B.
The thermal effect on the compressibility is neglected. Therefore, the isothermal
compressibility (c) is adopted and it is defined in the following:
TP
c ⎟⎠⎞
⎜⎝⎛∂∂
=ρ
ρ1 …………………………………………………………(3-16)
3.3. STANDARD CONSTITUTIVE EQUATIONS
Having defined the main properties needed to characterize a fluid flowing in a
conduit, the Reynolds number16 (Re) can be calculated.
μρ vD
=Re ………………………………………………………..….(3-17)
In the previous equation, the property v stands for the actual or flowing density of
the multiphase fluid system. The determination of this property is discussed later
in section 3.4.
The wall shear stress along the perimeter of a circular, cross-sectional area9 (τw) is
given by:
26
2
81 vf Mw ρτ = …………..……………………..…...………………...(3-18)
The Moody wall friction factor (ƒM) can be estimated using the correlation given
in Appendix B. The wall shear stress defines the effect of the friction in both the
momentum and energy balance equations.
The heat flux for a cross-sectional area in a circular pipe is stated by equation
3-19.
)(4sf TT
DUQ −= ……………..……..……………..………………..(3-19)
The overall heat transfer coefficient (U) can be estimated by a correlation given in
Appendix B. The heat flux accounts for the conduction and convection heat
transfer occurring between the pipe and its surroundings. The external
temperature (Ts) is considered to be an apparent temperature accountable for the
surroundings. It is assumed to change with a constant slope (αs) along the pipe9.
This slope is usually referred to as the geothermal or thermal gradient.
ϕα sin0 lTT sss += ………………...………..…………....…………...(3-20)
The initial external temperature (Ts0) is the external temperature at the surface.
The external temperature is also set by the position in the pipe (l) and the local
inclination (ϕ). Note that the heat exchange can be a gain or loss depending on the
27
sign of the difference between the fluid temperature and the surroundings
temperature.
3.4. PROPOSED MODEL FOR LIQUID HOLDUP
Usually, the flow in producing pipes is not under equilibrium conditions. A
volumetric fraction of a phase is the volume of that phase per unit volume of the
mixture where both volumes are flowing across the local cross-sectional area. It
was observed experimentally9,18 that the volumetric fraction of the liquid phases
or liquid holdup (HL) is greater than the summation of their fractional flows. The
behavior of this deviation was extensively studied9,18. It was determined that the
phase distribution varies in nature according the local conditions.
Several pattern distributions may take place inside the multiphase fluid system9,18
while flowing from saturated liquid to saturated gas. In general, the behavior of
the deviation from equilibrium, usually referred to as the prediction of the liquid
holdup, was investigated for each flow pattern separately.
Several correlations were developed in order to determine the flow pattern. The
motion of a gas/oil/water mixture in wells might involve with more than one flow
pattern. Thus, this motion can be modeled more precisely with different
techniques. However, the change in modeling the motion from one flow pattern to
another yield a discontinuity in predicting the multiphase fluid properties. This
28
discontinuity might lead to substantial errors in the prediction of the fluid
behavior and numerical instability during numerical solution of the relevant
equations.
All liquid holdup models9,18 propose the following constitutive equations for the
actual or flowing fluid density:
LLgg HH ρρρ += ………………………………………...……….(3-21)
1=+ Lg HH ………………..………………………………………(3-22)
There, the property Hg stands for the volumetric fraction for the gas phase or void
fraction which is similar to the liquid holdup definition.
It is assumed that the liquid phases are flowing at the same velocity. Then, the
subsequent equations apply for estimating the gas (ρg) and liquid density (ρL).
g
ststg S
xρρ = ………………………………………………………..(3-23)
L
ststL S
x )1( −= ρρ …………………………………………………...(3-24)
The densities of the various phases largely differ in oil and gas wells. Thus, the
phases flow at different velocites9 because these phases coexist inside a closed
environment, such as a pipe or conduit. However, the difference in velocity is
negligible at some specific conditions. This difference induces a slippage of the
29
gas phase past the liquid phases. The actual velocities9,18 of the gas (vg) and liquid
phases (vL) can be obtained with the following.
g
gg H
uSv = …………………………………………………….………(3-25)
L
LL H
uSv = ………………………………………………….…………(3-26)
Having determined the velocities of the phases, the actual or flowing velocity18 for
the multiphase fluid system can now be stated by the next equation.
LLgg
LLLggg
HHvHvH
vρρρρ
+
+= ……………………….…………………….(3-27)
Because the slippage occurring in the flow is accountable for the deviation from
equilibrium, the slip ratio is introduced for the measurement of the slippage. The
slip ratio is defined as the ratio of the gas phase velocity to liquid phase velocity.
L
g
vv
=λ ………………………………………………………….……(3-28)
By combining equations 3-25, 3-26 and 3-28, the void fraction and the liquid
holdup can be determined if the value for the local slip ratio is known.
gL
LL SS
SH+
=λλ ………………………………………………..…….(3-29)
gL
gg SS
SH
+=λ
…………………………………………….……….(3-30)
30
In Appendix A, it is shown that a multiphase fluid system in equilibrium yields18:
• A liquid holdup equal to the liquid fractional flow.
• A void fraction equal to the gaseous fractional flow.
• A flowing density equal to the equilibrium density.
• A flowing velocity equal to the volumetric flux.
Consequently, a deviation from equilibrium occurs when the value of the slip
ratio is different than the unity. Thus, the equilibrium is represented when the
value of the slip ratio is equal to the unity in the present modeling.
The actual or flowing dryness of the multiphase fluid is calculated by:
ρρ g
gHx = ………………………………………………..………….(3-31)
In this study, a new constitutive equation is proposed for modeling of the slip
ratio and the slippage as the follows:
( )( )( )( )
( )( )( )( )
( )( )( )( )gLstL
gststst
Lstgst
Lstgst
Lgstg
Lststst
ρρρρρρρρ
λρρρρρρρρ
ρρρρρρρρ
λ
−−
−−+
−−
−−+
−−−−
=
0
0
0000
0
……………..…(3-32)
Hence, the slippage can be modeled along the pipe by using an apparent slip ratio
at the surface (λ0) and the value of the equilibrium density at the surface (ρst0).
31
The slip ratio is defined to be dependant only upon the equilibrium density. In
essence, the procedure is a Lagrange interpolation12 based on three points or states
of physical properties. The first state is the saturated gas set to be represented by
ρst = ρg and λ = 1. The second point or state is represented by the actual state at
the surface represented by ρst = ρst0 and λ = λ0. The third point or state is the
saturated liquid set to be represented by ρst = ρL and λ = 1.
By assuming equilibrium in the transition from saturated fluid to under-saturated
fluid, the model predicts continuous trends in all properties. Furthermore, the
model predicts a continuous liquid holdup while changing the type of flow pattern
because the slip ratio is set to be a continuous function and independent of flow
patterns. Hence, the deficiencies of the previous models have been alleviated.
3.5. RELAXATION TIME FOR PRODUCING WELLS
Another approach for modeling flow under non-equilibrium conditions is to
consider the flowing fluid as a flashing fluid17. This means that the separation of
the gas phase does not occur instantaneously. The theoretical time required for a
complete separation to take place is called the relaxation time of separation.
32
Bilicki and Kestin6 approximated the relaxation time for the cumulative
separation of the gas phase by using the first two terms of the Taylor series
expansion over the substantial time derivative. Downar-Zapolski et al17 applied
this principle for flashing fluids to express deviation from the equilibrium.
xDtDxxst θ+= ………………………………………………………(3-33)
Note that equilibrium or static quality is taken as the upper limit for a complete
gas phase separation. After expanding the substantial time derivative and
rearranging equation 3-33, the next expression is obtained for the determination of
the relaxation time.
( )1−
⎟⎠⎞
⎜⎝⎛
∂∂
+∂∂
−=lxv
txxxstθ ……………………………..……………..(3-34)
For a steady-state flow regime, the relaxation time is approximated by
disregarding the change of quality in time.
dldxv
xxst −=θ …………………………………………………….……..(3-35)
33
4. DESCRIPTION OF THE MULTIPHASE FLOW OF A RESERVOIR
FLUID IN WELLS
CHAPTER 4
DESCRIPTION OF THE MULTIPHASE FLOW OF A RESERVOIR
FLUID IN WELLS
4.1. OVERVIEW
In this chapter, the motion of a multiphase fluid system with a constant and
circular cross-sectional area is modeled. The laws of mass, momentum, and
energy conservation are employed in a homogenous model. In transport
phenomena, a homogenous model is derived by a spatial averaging performed for
all phases within the control volume of a multiphase fluid system18. Therefore, the
laws of conservation are expressed in differential forms. The spatial averaging can
be implemented in volume, area, and thickness. The adopted homogenous model
is an Eulerian area-averaged model over the cross-sectional area. At the end, the
developed homogenous model is simplified to its steady-state form.
34
4.2. MODELING MULTIPHASE FLOW IN WELLS
The laws of conservation for mass, momentum, and energy are applied to describe
the flow in conduits for a multiphase-fluid system. The mathematical formulation
of these three fundamentals laws is described by the transport phenomena
models7. According to the chosen assumptions, a model can be classified as
microscopic, multiple gradient, maximum gradient, or macroscopic18,21.
min
Pin
mout
Pout
τw
Qf
ϕ
Δl
(a) (b)
A
Figure 4-1 : Schematic for local properties in a conduit
The infinitesimal element for a conduit is its local cross-sectional area (A) as
depicted in figure 4-1a. A schematic of the fundamental elements16 to be
considered for any cross-sectional area along the pipe are shown in figure 4-1b.
35
These are mass coming in, the mass going out, the pressure at the inlet, the
pressure at the outlet, the wall shear stress, the heat transfer with the surroundings
and the local inclination effect. The flow to be modeled is considered to be
upward. This means that this movement has to over come the gravitational force
for all acute and obtuse angles represented by ϕ.
The wall shear stress causes friction between the fluid and the pipe. Hence, the
velocity and pressure changes over the cross-sectional area16. Because of the
thermal conduction and convection, the temperature changes over the cross-
sectional area11 also. For simplicity, a uniform distribution for pressure, an
average velocity, and an average temperature over the elemental cross-sectional
are considered in the following formulations (figure 4-2).
v P T
(a) (b) (c) Figure 4-2 : Example of local velocity, pressure and temperature
distributions
It is assumed that the pipe is stationary and has a constant and circular cross-
sectional area. Consequently, there are neither depositions nor deformations in the
36
pipe. The thermal radiation along the pipe length is omitted. Thus, the present
model can be classified as a macroscopic model21.
Considering that there is no mass accumulation inside the conduit, the equation
for the mass balance is expressed as18:
( ) ( ) 0=∂∂
+∂∂ ρρ vA
lA
t…………………………………………..…(4-1)
After expanding the derivatives and rearranging the terms, the next expression is
obtained.
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=∂∂
+∂∂
+∂∂
lAv
tA
Alv
lv
tρρρρ …………….………………….(4-2)
Assuming a pipe with a constant and circular cross-sectional area, the final form
for the law of mass balance is shown below:
0=∂∂
+∂∂
+∂∂
lv
lv
tρρρ ……….....…………………..………………….(4-3)
The loss of momentum by the fluid motion is compounded by the wall shear
stress, gravitational force and the drop of pressure as depicted in figure 4-1b
where CA stands for perimeter of the cross-sectional area. Thus, the momentum
balance equation takes the following expression18.
37
( ) ( ) (APl
gACuvvAl
vAt wA ∂
)∂−−−=
∂∂
+∂∂ ϕρτρρ sin ...….……......(4-4)
After expanding the derivatives and rearranging the terms, the next expression is
obtained.
( ) ( )
( )APl
gAC
vAl
At
vlvvA
tvA
wA ∂∂
−−−=
⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+∂∂
+∂∂
ϕρτ
ρρρρ
sin……...……………………...(4-5)
By replacing the mass balance equation (equation 4-1) and expanding the
derivatives, equation 4-5 reduces to:
lA
AP
lPg
AC
lvv
tv A
∂∂
−∂∂
−−−=∂∂
+∂∂ ϕρτρρ sin ………………………(4-6)
For a pipe with constant and circular cross-sectional area, the momentum balance
equation takes the final form:
ϕρτ
ρρ sin4
gDl
Plvv
tv w −−=
∂∂
+∂∂
+∂∂ ….……...…...………………..(4-7)
For the present model, the loss or gain of energy is due to the motion of the fluid
and the heat transfer by conduction and convection. The frictional effect due to
the wall shear stress causes a change in temperature but it does not induce a direct
38
loss or gain of energy. Therefore, the frictional term is not considered in the
energy balance equation initially18.
( ) ( ) ( ) fTT AQvAPl
evAl
eAt
−∂∂
−=∂∂
+∂∂ ρρ ……..………...………..(4-8)
By expressing the total energy in terms of enthalpy, kinetic energy, and potential
energy, the equation 4-8 takes the form:
( ) fAQuAPl
gzvPhvAl
gzvPhAt
−∂∂
−=⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ ++−
∂∂
+
⎥⎦⎤
⎢⎣⎡ ⎟
⎠⎞⎜
⎝⎛ ++−
∂∂
221
221
ρρ
ρρ…...……..(4-9)
After expanding the derivatives and rearranging the terms, the energy balance is
set in the following convenient expression.
( ) ( )
( ) ( ) ( ) ( ) fAQAPt
vAl
At
gzvh
gzvhl
vAgzvht
A
−∂∂
=⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+++
++∂∂
+++∂∂
ρρ
ρρ
221
2212
21
……....……..(4-10)
By replacing the mass balance equation (equation 4-1) and expanding the
derivatives, the equation 4-10 is conveniently expressed as:
( ) fAQgvAAPt
lvvA
tvAv
lhvA
thA
−−∂∂
=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂
ϕρ
ρρρρ
sin
…………………..………….(4-11)
39
Note that it is assumed that the conduit is stationary so that there is no change of
potential energy over time. The change of potential energy over the length is set
by the local inclination angle (ϕ). By replacing the momentum balance equation
(equation 4-7) and rearranging the terms, the energy balance equation is expressed
in terms of enthalpy.
fw Q
lA
APv
tA
AP
Dv
lPv
tP
lhv
th
−∂∂
+∂∂
+=∂∂
−∂∂
−∂∂
+∂∂ τ
ρρ4
…...……...(4-12)
By expressing the enthalpy in terms of temperature as recommended by Brill and
Mukherjee9, the energy balance equation takes the form:
( ) ( )
fw
pppp
QlA
APv
tA
AP
Dv
lPcv
tPc
lTcv
tTc
−∂∂
+∂∂
+=
∂∂
+−∂∂
++∂∂
+∂∂
τ
ηρηρρρ
4
11……….…….(4-13)
Because it is assumed that the pipe has constant and circular cross-sectional area,
the energy balance takes the final form:
( ) ( )
fw
pppp
QD
v
lPcv
tPc
lTcv
tTc
−=
∂∂
+−∂∂
+−∂∂
+∂∂
τ
ηρηρρρ
4
11…………….…(4-14)
Generally; the homogenous model consisted of the mass balance in equation 4-3,
momentum balance in equation 4-7 and energy balance in equation 4-14 are
employed to model the flow in conduits. However, if the system is not at
equilibrium then this homogeneous model is not closed.
40
In reality, once the actual pressure in the system falls below the bubble-point
pressure, the multiphase-fluid can be considered as a flashing liquid. Thus, this
fluid system generates gas with a local mass flux (Γ ) under non-equilibrium state
conditions. Therefore, it is necessary to define the mass balance equation for the
gas phase to give closure to the homogenous model18.
( ) ( ) Γ=∂∂
+∂∂ AxvA
lxA
tρρ …………………………………………(4-15)
After expanding the derivatives and rearranging the terms, the mass balance for
the gas phase is set conveniently as:
( ) ( ) Γ=⎥⎦
⎤⎢⎣
⎡∂∂
+∂∂
+∂∂
+∂∂ AvA
lA
tx
lxvA
txA ρρρρ ……….……………(4-16)
By replacing the mass balance equation (equation 4-1) and expanding the
derivatives, the equation 4-16 is takes the form:
ρΓ
=∂∂
+∂∂
lxv
tx ………..........………………………………………...(4-17)
Bilicki and Kestin6 replaced the interface transfer term Γ /ρ by using the
definition of the relaxation time stated in equation 3-33.
θstxx
lxv
tx −
−=∂∂
+∂∂ ………………..…...……………………….......(4-18)
41
The set compounded by the equations 4-3, 4-7, 4-14 and 4-18 is called as the
Homogeneous Relaxation Model (HRM) by Downar-Zapolski et al17.
4.3. STEADY-STATE MODEL UNDER NONISOTHERMAL AND
NONEQUILIBRIUM CONDITIONS
Having defined a technique to estimate the multiphase-fluid properties and the
governing equations for its flow along a conduit in the preceding section, the
developed homogenous model can be simplified to the present model.
Considering a steady-state flowing regime, the conservation laws for constant
diameter pipes can be expressed by the next set of differential equations17.
0=+dldv
dldv ρρ ………………………………………………...…....(4-19)
ϕρτ
ρ sin4
−−=+Ddl
dPdldvv w ………………...………….…………..(4-20)
( ) fw
pp QD
vdldPcv
dldTcv −=+−
τηρρ
41 …….……………………….(4-21)
θstxx
dldxv
−−= ………..……………...……….…………..………...(4-22)
The equation 4-19 implies that the term ρv is equal to a constant value. This
constant can be obtained from the basics of fluid mechanics. Thus, the mass
balance equation is reduced to a non-differential equation.
42
2
4Dm
Amv
πρ
&&== ……...……..…………………...……..……………...(4-23)
It is shown in Appendix A that the previously defined flowing density and the
flowing velocity (equations 3-21 and 3-27 respectively) satisfy the relationship
stated by equation 4-23.
By substituting the wall shear stress defined in equation 3-18 and introducing the
equation 4-19, the momentum conservation equation is expressed as:
ϕρρρ sin21 2
2 gDvf
dldv
dldP
M −−=− ……………………………….(4-24)
The momentum equation takes its final form by rearranging the kinetic
momentum term and replacing the isothermal compressibility defined in equation
3-16.
2
2
1
sin21
vP
gDvf
dldP
T
M
⎟⎠⎞
⎜⎝⎛∂∂
−
−−=
ρ
ϕρρ………..…..…………….……..……...(4-25)
For simplicity, the partial derivative of the density with respect to the pressure is
computed using the equilibrium density. This partial derivative is approximated
by taking a numerical derivative of the equilibrium density over small change in
pressure.
43
The energy equation is rearranged as the following:
vQ
DdldP
dldPc
dldTc fw
Pp ρρτ
υη −++=4 ………………...…………….…(4-26)
The energy equation takes its final form by introducing the wall shear stress
defined in equation 3-18, the Joule-Thompson coefficient defined in equation
3-15 and the heat flux defined in equation 3-19.
)(21 2
spP
MPP
TTmcDU
Dcvf
dldP
TcT
dldT
−−+⎟⎠⎞
⎜⎝⎛∂∂
=&
πυ …...………..…...…..(4-27)
For simplicity, the partial derivative of the specific volume over the temperature
is computed using the equilibrium density. This partial derivative is approximated
by taking a numerical derivative of the reciprocal of the equilibrium density over
small change in temperature.
By solving the gas phase balance equation for the change of quality in length, it
can be rearranged as follows:
θvxx
dldx st−
−= ………...…………………………………………….(4-28)
44
5. SIMULATION OF THE MULTI-PHASE FLOW OF A RESERVOIR
FLUID IN WELLS
CHAPTER 5
SIMULATION OF THE MULTI-PHASE FLOW OF A RESERVOIR
FLUID IN WELLS
5.1. OVERVIEW
In this chapter, a numerical scheme is developed in order to solve the set of
differential equations stated by the present homogenous model at steady-state.
There is no known analytical procedure for solving these differential equations
simultaneously. Therefore, the pipe is segmented into numerous partitions in this
scheme in order to approximate a solution having the wellhead and the well-bore
as the integration limits. A succession of calculations is performed for solving
each individual partition towards computing the global solution. Then, the
selected numerical method is implemented into an algorithm in order to perform
simulations of the present phenomenon.
45
5.2. SELECTION OF THE NUMERICAL DIFFERENTATION SCHEME
In the present study, the equations for the steady-state model are a set of
differential equations to be solved by numerical differentiation. These differential
equations account for the changes in pressure and temperature over the pipe
length and they are expressed in equations 4-25 and 4-27, respectively.
The analytic expression is known for each of these differential equations. It is
assumed that these equations are continuous in all their higher order derivatives.
Thus, both equations can be numerically solved by using a Taylor series
approach.
The Runge-Kutta numerical differentiation achieves a higher accuracy of a Taylor
series approach without requiring the calculation of higher order derivatives12.
For this reason, the Runge-Kutta numerical differentiation (RK) is chosen for
solving the actual system of ordinary differential equations (ODE). Explanation of
its principles and its various forms is beyond the scope of the present work.
In general, the gain in accuracy is offset by the computational effort beyond the
fourth-order RK methods12. Thus, the fourth-order RK methods are the most
efficient. Among all forms of these methods, the classical fourth-order of Runge-
Kutta method (RK4) is selected.
46
5.3. SCHEME OF THE NUMERICAL DIFFERENTATION
In the selected scheme, the pipe is divided into small portions. All of these
segments have a length of Δl and a characteristic angle of inclination. This angle
of inclination is given by the positioning of the pipe. The flow of the multiphase-
fluid along the pipe occurring under this scheme is depicted in figure 5-1.
Consequently, The rate of change in pressure (il
PΔΔ ), temperature (
ilTΔΔ ), and
quality (il
xΔΔ ) over the pipe length are calculated by using the data from the
previous segment. Then, the local pressure, temperature, and quality are computed
for each segment as described in equations 5-1 and 5-2.
llPPP
iii Δ
ΔΔ
+=+1 ………………………………………………....….(5-1)
llTTT
iii Δ
ΔΔ
+=+1 ………………………………………………….....(5-2)
Hence, the knowledge of the inlet values for pressure (P0) and temperature (T0)
allows generating a set of values for each segment from the first segment to the
last one. Moreover, it allows the current scheme of numerical differentiation to
describe the behavior of the pressure, temperature, and remaining properties along
the pipe. The accuracy of the numerical differentiation increases by increasing the
number of segments.
47
OUTLET
INLET
ϕ1
ϕ2
ϕ3
i = 1i = 2i = 3i = 4
i = 0
i = N i = N-1
Figure 5-1 : Segmentation of the production pipe in a well
5.4. COMPUTING THE CHANGE OF STATE
Because equations 4-25 and 4-27 were formulated for an elemental area, both
equations are differential equations used for computing the change of state along
48
the pipe length. In addition, they are dependent on each other. Hence, they are
suitable for solution by the simultaneous scheme of the RK4.
The rate of change for pressure and temperature over the pipe length is
determined by taking a weighted average of four intermediate rates of change as
detailed in equations 5-3 and 5-4.
⎟⎟⎠
⎞⎜⎜⎝
⎛
ΔΔ
+ΔΔ
+ΔΔ
+ΔΔ
=ΔΔ
4,3,2,1,
2261
iiiii lP
lP
lP
lP
lP …………………..….(5-3)
⎟⎟⎠
⎞⎜⎜⎝
⎛
ΔΔ
+ΔΔ
+ΔΔ
+ΔΔ
=ΔΔ
4,3,2,1,
2261
iiiii lT
lT
lT
lT
lT ……………………..(5-4)
The first set of intermediate rates of change is obtained with the equations 4-25
and 4-27 describing pressure and temperature, respectively. Consequently, all the
necessary multiphase-fluid properties have to be computed for the previous
segment (i). These properties are computed by using the values of the pressure,
temperature, quality, external temperature, and inclination at this segment and the
system’s constants with the approach extensively described in Chapter 2. The
current form of this first set of rates of change is formulated in the equations 5-5,
and 5-6.
2
2
1, )(1
sin)(
21
ii
iii
iiM
i vP
gD
vf
lP
∂∂
−
−−=
ΔΔ
ρ
ϕρρ…………...……...……….(5-5)
49
)()(
21 2
1,is
ipip
iiM
iiip
i
i
TTmcDU
Dc
vf
lP
Tc
TlT
−−+ΔΔ
∂∂
=ΔΔ
&
πυ …………...(5-6)
For the calculation of the second set of rates, an intermediate value for both,
pressure, and temperature, have to be computed. These intermediate values are
approximated by taking a half increment of the length Δl and assuming the first
set of estimations as the actual change over the length.
llPPP
iii Δ
ΔΔ
+=1,
1, 21 ………………………………………...………(5-7)
llTTT
iii Δ
ΔΔ
+=1,
1, 21 ………………………………………….……..(5-8)
The second set of rates is also obtained by employing the equations 4-25 and
4-27. However, the intermediate pressure Pi,1 and temperature Ti,1 are used for the
approximation of the required properties along with all the system’s constants.
This second set is formulated in the equations 5-9 and 5-10.
21,
1,
1,
21,
1,1,
2, )(1
sin)(
21
ii
iii
iiM
i vP
gD
vf
lP
∂∂
−
−−=
ΔΔ
ρ
ϕρρ…………….…..……(5-9)
)()(
21
1,1,
21,
1,1,1,1,
1,
2,is
ipip
iiM
iiip
i
i
TTmcDU
Dc
vf
lP
Tc
T
lT
−−+ΔΔ
∂∂
=ΔΔ
&
πυ .…..(5-10)
50
Similarly to the procedure of calculating the second set, intermediate values for
pressure and temperature have to be computed in order to determine the third set
of rates.
llPPP
iii Δ
ΔΔ
+=2,
2, 21 …………………………………………….….(5-11)
llTTT
iii Δ
ΔΔ
+=2,
2, 21 …………………………………………..……(5-12)
In the same manner for estimating the previous sets, the third set of rates is
formulated by the equations 5-13 and 5-14.
22,
2,
2,
22,
2,2,
3, )(1
sin)(
21
ii
iii
iiM
i vP
gD
vf
lP
∂∂
−
−−=
ΔΔ
ρ
ϕρρ…......……………...(5-13)
)()(
21
2,2,
22,
2,2,2,2,
2,
3,is
ipip
iiM
iiip
i
i
TTmcDU
Dc
vf
lP
Tc
T
lT
−−+ΔΔ
∂∂
=ΔΔ
&
πυ .....(5-14)
For the fourth set of rates, the intermediate pressure and temperature are obtained
by taking a full increment of the length Δl.
llPPP
iii Δ
ΔΔ
+=3,
3, ………………………………….……………….(5-15)
llTTT
iii Δ
ΔΔ
+=3,
3, ………………….……………………………….(5-16)
51
Then, the required properties are estimated by using the pressure Pi,3 and the
temperature Ti,3 in order to obtain the forth set of rates. This forth set is
formulated by the equations 5-17 and 5-18.
23,
3,
3,
23,
3,3,
4, )(1
sin)(
21
ii
iii
iiM
i vP
gD
vf
lP
∂∂
−
−−=
ΔΔ
ρ
ϕρρ……..…..………….(5-17)
)()(
21
3,3,
23,
3,3,3,3,
3,
4,is
ipip
iiM
iiip
i
i
TTmcDU
Dc
vf
lP
Tc
T
lT
−−+ΔΔ
∂∂
=ΔΔ
&
πυ ..…(5-18)
Note that the external temperature and the angle of inclination are not functions of
any multiphase-fluid property. These parameters are only dependant of the pipe
length. Thus, they are considered as constant for each segment i.
5.5. COMPUTATIONAL PROCEDURE
A collection of known properties is required in order to execute the procedure
described in the previous section,. These known properties or inputs are the
starting values for executing the selected numerical solution scheme. They are
necessary to estimate the properties employed by equations 4-25 and 4-27 for
computing the rate of change of state. The inputs of the model are listed as
follows:
52
• Volumetric rate of production for the gas, oil and water pseudo-
components.
• Specific gravity for the gas and oil pseudo-components.
• Salinity for the water pseudo-component.
• Pipe shape, length, diameter and roughness.
• Wellhead pressure and temperature for the multi-phase fluid.
• External temperature at the wellhead.
• Well-bore pressure and temperature for the multi-phase fluid.
Because the geothermal gradient is assumed to be constant, this property can be
determined at the surface as stated in the next equation.
LTTN
s0−
=α …………………………………………..…………….(5-19)
For a well, the starting values or inputs are known at the outlet, which is the
wellhead. Hence, the model has to be adjusted for computing in counter-flow.
The scheme is successfully adjusted by considering a negative increment in the
elevation.
In the figure 5-2, the proper pressure and temperature drops are achieved by
performing the shooting method12. The values of the initial slip ratio and the
53
initial external temperature are changed simultaneously until the desired pressure
and temperature drop are obtained.
INPUT System’s Constants s
GV& , sOV& , s
WV& , γG, γO, ξ
INPUT Surface Data P0, T0
CALCULATE Properties below bubble-point pressure
SET l = 0
INPUT System’s Constants L, D, ε, ϕ, Δl, TN, PN
SET l = l -Δ l
l ≥ -L
GUESS λ0, Ts0
CALCULATE αs
0≥sgV&
SOLVE the change in state with liquid hold up
SOLVE the change in state
CALCULATE Properties below bubble-
point pressure
CALCULATE Properties above
bubble-point pressure
NN TTandPP ==
END
BEGIN
Yes
No
Yes
No
No Yes
Figure 5-2 : Flowchart for simulating multiphase flow in wells
54
A good initial guess is based on assuming an equilibrium for the initial slip ratio
(λ0=1) and a perfect insulation for the initial external temperature (Ts0=T0).
The presence of the gas phase ( ) indicates that the multiphase fluid system
might not be at equilibrium. Therefore, the change of state is computed by using
the developed homogenous model in conjunction with the proposed liquid holdup
model. Otherwise ( ), the fluid is not flashing and it is considered to be at
equilibrium.
0≥sgV&
0≤sgV&
55
6. VALIDATION AND APPLICATION
CHAPTER 6
VALIDATION AND APPLICATION
6.1. OVERVIEW
In this chapter, the data selected for performing the simulations is presented. This
data is suitable for implementation into the designed algorithm presented in the
previous chapter. Then, the simulations are performed and validated with a
general model for correlating the void fraction. The data is organized in three
study cases for gas/oil/water mixtures flowing upwardly in vertical wells. The
relevant results are presented in the form of a series of plots. These plots illustrate
the pressure drop, the deviation from equilibrium, the temperature drop, the
external temperature effect, the gas phase generation, and the relaxation time
behavior.
56
6.2. DATA SELECTION
A suitable set of data is required for the simulator in order to achieve the
objectives of the present study. Moreover, this set of data has to be taken from the
producing wells with a broad range in production rates, types of gas/oil/water
mixtures and pressure/temperature drop. Following the previous criteria, the data
published by Chierici13 et al. is selected for the analysis. From this published
data, ten samples, presented in table 6-1, are considered for application. The value
of the pipe roughness is assumed to be ε = 0.00015 ft. for all samples9.
Table 6-1 : Data considered for application
Nº γo P0 D T0 TN γg ξ L PN
[Mscf/d] [stb/d] [stb/d] [psia] [in.] [ºF] [ºF] [wt%] [ft.] [psia]7 0.9826 160.2 0.761 761.1 569.4 5.000 104.0 222.8 1.268 3.0 10171 4514.28 0.9826 231.8 1.101 1100.7 589.3 5.000 107.6 226.4 1.268 3.0 10171 4546.8
10 0.9516 2104.5 0.000 4128.0 769.6 5.000 124.9 188.6 0.708 0.0 7648 3322.811 0.9390 1954.7 0.000 3834.3 813.7 2.875 118.9 186.8 0.708 0.0 7579 3439.212 0.9390 2154.8 0.000 4226.8 661.7 2.875 132.8 187.9 0.708 0.0 7579 3288.713 0.9390 3501.6 0.000 6868.5 428.8 2.875 138.9 187.9 0.708 0.0 7579 3244.722 0.8236 180.7 0.000 191.8 1370.3 2.875 92.1 167.0 0.750 0.0 8038 3568.523 0.8236 311.7 0.000 330.8 1341.9 2.875 89.6 167.0 0.750 0.0 8038 3503.124 0.8236 452.7 0.000 480.5 1306.4 2.875 89.4 167.0 0.750 0.0 8038 3430.725 0.8236 1060.7 0.000 1125.9 1171.5 2.875 98.6 167.0 0.750 0.0 8038 3222.0
sGV& s
OV&sWV&
The samples are grouped in three study cases:
• Case 1 includes samples 10, 11, 12 and 13 for representing two-phase flow
of heavy crude oil.
57
• Case 2 includes samples 22, 23, 24 and 25 for representing two-phase flow
of light crude oil.
• Case 3 includes samples 6 and 7 for representing three-phase flow of a
heavy crude oil.
After running a simulation for each sample, several property and coefficient
values are collected as they vary along the length of the producing pipe.
6.3. SIMULATOR VALIDATION
A simulator was built using the Visual Basic for Applications (VBA) environment
and programmed to perform the numerical differentiation as described by the
flowchart given in figure 5-2.
The properties that mainly describe the pressure drop are the liquid holdup and
the void fraction. Thus, the prediction of these properties is paramount for
modeling the multi-phase flow in wells. Because both properties are related as
shown in equation 3-22, the simulator output is validated by testing the void
fraction prediction alone.
Butterworth8 proposed a model for describing the void fraction after comparing
several correlations obtained with experimental data. In that study, it was
58
observed that the void fraction of several fluids can be correlated successfully by
the model expressed in equation 6-1. Then, Butterworth8 suggested that the void
fractions can be expressed in this form for all fluids and flowing conditions.
s
g
L
r
L
gq
st
st
g
xx
C
H
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ −+
=
μμ
ρρ1
1
1 …………..….…………………..(6-1)
The equation 6-1 is rearranged in order to perform a power law correlation as
indicated in the following.
s
g
L
r
L
gq
st
st
g xx
CH ⎟
⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ −=−
μμ
ρρ1
11 ……………………….……….….(6-2)
Then, the void fraction values obtained by the simulator for each sample are
correlated with the Butterworth’s model assuming s =0. The subsequent plots
illustrate how well all three cases correlate. A perfect correlation is graphically
represented by a straight line and mathematically represented by a coefficient of
determination equal to one. Figures 6-1, 6-2 and 6-3 represent the result of cases
1, 2 and 3, respectively.
59
0
9
18
27
36
45
54
0 4 8 12 16 20 2(1/xst-1)q(ρg/ρL)r
1/H
g-1
4
Sample 10Sample 11Sample 12Sample 13
Figure 6-1: Correlation for Case 1
These plots show a direct relationship between the expression 11 −gH and the
expression ( ) ( )rqx L
g
st ρρ11 − which means the void fraction prediction is in good
agreement with the Butterworth’s model. The values of the adjustable parameters
q and r are obtained by the least-squares errors method. These values are shown
in table 6-2. The values of the parameters λ and Ts0 that yield the proper pressure
and temperature drop are also shown in this table.
60
0
10
20
30
40
50
60
0 10 20 30 40 50 6(1/xst-1)q(ρg/ρL)r
1/H
g-1
0
Sample 22Sample 23Sample 24Sample 25
Figure 6-2 : Correlation for Case 2
0
10
20
30
40
50
60
0 30 60 90 120 150 180(1/xst-1)q(ρg/ρL)r
1/H
g-1
Sample 8
Sample 7
Figure 6-3 : Correlation for Case 3
61
Table 6-2 : Adjustable parameters and Coefficient of Determination
Nº λ 0 T s0 [ºF] C q r R2
7 1.800 77.74 0.4915 0.9823 0.4528 0.999548 3.800 76.01 0.2882 0.9317 -0.1661 0.99937
10 2.945 109.84 12.7908 0.4939 1.0161 0.9997111 1.817 98.44 2.1830 0.8182 0.8734 0.9984812 2.247 117.79 9.9014 0.5787 1.0897 0.9990713 1.024 117.27 1.1154 0.9804 1.0085 0.9979522 1.672 81.61 0.8377 0.9391 0.5482 0.9981223 1.345 73.10 0.8656 0.9685 0.7247 0.9977224 1.175 67.36 0.9269 0.9828 0.8542 0.9975425 1.152 65.22 0.9877 0.9818 0.9031 0.99641
The value of the coefficient of regression is calculated by using equation 6-3. This
coefficient indicates how well the Butterworth’s model correlates the predicted
void fraction compared to estimating the void fraction with the fractional flow of
the gas phase. Thus, this fractional flow is employed as the reference base in
evaluating a each sample correlation.
( ) ( )[ ]( )∑
∑
=
=
−−
−⎭⎬⎫
⎩⎨⎧ +−
−= N
i gg
N
i
rqx
xg
SH
cHR
Lg
stst
02
0
211
21
1ρ
ρ
………….…………….(6-3)
6.4. UPWARD MOTION OF BLACK-OIL IN VERTICAL WELLS
In all cases, the pressure drop behaves almost linear. However, the pressure drop
is greater than the predicted pressure drop by the models assuming equilibrium
62
conditions. This is because the liquid holdup phenomenon is present for all cases.
Furthermore, the deviation from the equilibrium becomes more evident with a
series of plots matching the void fraction against the gaseous fractional flow. In
these plots, the equilibrium is represented by a straight line coming from the
origin with a slope of one.
It is shown that the difference between the fluid temperature and the external
temperature is considerably high. This difference makes the temperature drop to
be nonlinear. However, all cases present the same trend in the temperature
gradient. This proved not to be true for the dryness gradient. This apparent lack of
trend is accountable mainly for the deviation from the equilibrium. It is shown
that the relaxation time is a good measure of how much the local conditions
deviate from the equilibrium. A plot describing the nature of the change in
relaxation time against the gaseous fractional flow is presented for all samples.
This is because all flow regimes deviate from the equilibrium.
6.4.1 CASE 1: TWO-PHASE FLOW OF HEAVY CRUDE OIL
For this study case, the samples 10, 11, 12 and 13 of the Chierici13 et al. data were
selected. There is a two-phase flow along the pipe length as shown by the dryness
gradient plot.
63
The pressure drop in this case is almost linear. However, there is a slight increase
in the pressure gradient when the fluid is approaching the surface as seen in figure
6-4.
400
710
1020
1330
1640
1950
2260
2570
2880
3190
3500
0 770 1540 2310 3080 3850 4620 5390 6160 6930 7700Length [feet]
Pres
sure
[psi
a]
Sample 10Sample 11Sample 12Sample 13
Figure 6-4 : Pressure drop for Case 1
Because the liquid phase is nearly incompressible, the pressure drop of the oil/gas
mixture rich in liquid phase is expected to be linear. Under these conditions, the
void fraction is less than the gaseous fractional flow. Furthermore, a near constant
pressure gradient implies a substantial deviation from the equilibrium as depicted
in figure 6-5. Because the sample 13 very slightly deviates from the equilibrium,
it is the only sample that has a nonlinear pressure drop.
64
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Gaseous Fractional Flow [fraction]
Void
Fra
ctio
n [fr
actio
n]
Sample 10Sample 11Sample 12Sample 13
Figure 6-5 : Void fraction for Case 1
The temperature drop in this case is clearly nonlinear. All samples have the same
trend for the temperature gradient as described in figure 6-6. The temperature
change is mainly accountable by the heat transfer magnitude. The heat transfer
seems to increase when the fluid is leaving the bottom-hole. Then, it tends to
increase with a constant rate. This becomes evident when observing the difference
between the fluid temperature and the surroundings temperature as shown in
figure 6-7. The heat transfer is directly related to this difference.
65
119
126
133
140
147
154
161
168
175
182
189
0 770 1540 2310 3080 3850 4620 5390 6160 6930 7700Length [feet]
Tem
pera
ture
[ºF]
Sample 10Sample 11Sample 12Sample 13
Figure 6-6 : Temperature drop for Case 1
0
2.2
4.4
6.6
8.8
11
13.2
15.4
17.6
19.8
22
0 770 1540 2310 3080 3850 4620 5390 6160 6930 7700Length [feet]
Tem
pera
ture
Diff
eren
ce [
ºF]
Sample 10Sample 11Sample 12Sample 13
Figure 6-7 : Temperature difference for Case 1
66
Under equilibrium, the dryness gradient is expected to be close to a constant. In
this case, the increase in the dryness is nonlinear except for sample 13 as
presented in figure 6-8.
0.0%
0.7%
1.4%
2.1%
2.8%
3.5%
4.2%
4.9%
5.6%
6.3%
7.0%
0 770 1540 2310 3080 3850 4620 5390 6160 6930 7700Length [feet]
Dry
ness
[fra
ctio
n]
Sample 10Sample 11Sample 12Sample 13
Figure 6-8 : Dryness gradient for Case 1
The deviation of the dryness gradient from equilibrium is measured by the
relaxation time. This coefficient increases as the mixture departs from saturated
oil and decreases as the mixture approaches the saturated gas, thus it attains a
maximum value. The relaxation time reaches a maximum even for sample 13
which is practically under equilibrium as shown in figure 6-9. The magnitude of
the maximum relaxation time is higher for the samples that deviate further from
equilibrium.
67
0.1
1
10
100
1000
10000
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Gaseous Fractional Flow [fraction]
Rel
axat
ion
Tim
e [s
]
Sample 10Sample 11Sample 12Sample 13
Figure 6-9 : Relaxation time for Case 1
6.4.2 CASE 2: TWO-PHASE FLOW OF LIGHT CRUDE OIL
For this study case, the samples 22, 23, 24 and 25 of the Chierici13 et al. data were
selected. In the motion of these samples, there is a single phase region and a two-
phase region. These regions can be clearly distinguished by observing the dryness
gradient plot sown in figure 6-14.
In this case, the pressure drop is practically linear in the single phase region and it
is close to linear in the two-phase region. The pressure gradient slightly increases
when the fluid is approaching the surface as seen in figure 6-10.
68
1100
1350
1600
1850
2100
2350
2600
2850
3100
3350
3600
0 810 1620 2430 3240 4050 4860 5670 6480 7290 8100Length [feet]
Pres
sure
[psi
a]
Sample 22Sample 23Sample 24Sample 25
Figure 6-10 : Pressure drop for Case 2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Gaseous Fractional Flow [fraction]
Void
Fra
ctio
n [fr
actio
n]
Sample 22Sample 23Sample 24Sample 25
Figure 6-11 : Void fraction for Case 2
69
As stated before, a near constant pressure gradient implies a substantial deviation
from the equilibrium. The deviation for each sample is depicted in figure 6-11.
All samples have a nonlinear temperature drop as described in figure 6-12. The
temperature gradient is mainly set by the heat transfer. As the fluid departs from
the bottom-hole, the heat transfer effect seems to increase asymptotically.
88
96
104
112
120
128
136
144
152
160
168
0 810 1620 2430 3240 4050 4860 5670 6480 7290 8100Length [feet]
Tem
pera
ture
[ºF]
Sample 22Sample 23Sample 24Sample 25
Figure 6-12 : Temperature drop for Case 2
However, this asymptotical trend is not observed in the difference between the
fluid temperature and the surroundings temperature as shown in figure 6-13.
Conversely, the heat transfer is increasing non-monotonically.
70
0
3.6
7.2
10.8
14.4
18
21.6
25.2
28.8
32.4
36
0 810 1620 2430 3240 4050 4860 5670 6480 7290 8100Length [feet]
Tem
pera
ture
Diff
eren
ce [
ºF]
Sample 22Sample 23Sample 24Sample 25
Figure 6-13 : Temperature difference for Case 2
0.0%
0.9%
1.8%
2.7%
3.6%
4.5%
5.4%
6.3%
7.2%
8.1%
9.0%
0 810 1620 2430 3240 4050 4860 5670 6480 7290 8100Length [feet]
Dry
ness
[fr
actio
n]
Sample 22Sample 23Sample 24Sample 25
Figure 6-14 : Dryness gradient for Case 2
71
The dryness increases monotonically for all samples as presented in figure 6-14.
Moreover, all samples have a maximum value for the dryness gradient at
saturated oil conditions.
The relaxation time increases as the mixture departs from saturated oil as shown
in figure 6-15. The relaxation time seems to be reaching a maximum and starting
to decrease as the mixture is closer to be a saturated gas condition. The values of
the relaxation time are higher for the samples that deviate further from the
equilibrium.
0.1
1
10
100
1000
10000
0 0.1 0.2 0.3 0.4 0.5 Gaseous Fractional Flow [fraction]
Rel
axat
ion
Tim
e [s
]
Sample 22Sample 23Sample 24Sample 25
Figure 6-15 : Relaxation time for Case 2
72
6.4.3 CASE 3: THREE-PHASE FLOW OF A HEAVY CRUDE OIL
For this study case, the samples 7 and 8 of the Chierici13 et al. data were selected.
The motion of these samples involves a two-phase region and a three-phase
region. These regions can be clearly distinguished by observing the dryness
gradient plot shown in figure 6-20.
As seen in figure 6-16, the pressure drop in this case is clearly linear. However,
there is a very slight decline in the pressure gradient in the three-phase region
when the fluid is approaching the surface.
500
910
1320
1730
2140
2550
2960
3370
3780
4190
4600
0 1020 2040 3060 4080 5100 6120 7140 8160 9180 10200Length [feet]
Pres
sure
[psi
a]
Sample 8
Sample 7
Figure 6-16 : Pressure drop for Case 3
73
Because a near constant pressure gradient implies a substantial deviation from the
equilibrium, this deviation is expected to be higher than the previous cases as
depicted in figure 6-17.
0
0.03
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.27
0.3
0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3 Gaseous Fractional Flow [fraction]
Void
Fra
ctio
n [fr
actio
n]
Sample 8
Sample 7
Figure 6-17 : Void fraction for Case 3
The temperature drop in this case is clearly nonlinear for both samples. The
temperature change is mainly accountable to the heat transfer magnitude. As the
fluid departs from the bottom-hole, the heat transfer seems to increase
asymptotically as described in figure 6-18. However, this asymptotical trend is
not observed in the difference between the fluid temperature and the surroundings
temperature as shown in figure 6-19. This break in the tendency is due to the
change from two-phase to three-phase flow.
74
100
113
126
139
152
165
178
191
204
217
230
0 1020 2040 3060 4080 5100 6120 7140 8160 9180 10200Length [feet]
Tem
pera
ture
[ºF]
Sample 8
Sample 7
Figure 6-18 : Temperature drop for Case 3
0
3.3
6.6
9.9
13.2
16.5
19.8
23.1
26.4
29.7
33
0 1020 2040 3060 4080 5100 6120 7140 8160 9180 10200Length [feet]
Tem
pera
ture
Diff
eren
ce [
ºF]
Sample 8
Sample 7
Figure 6-19 : Temperature difference for Case 3
75
The dryness increases monotonically for both samples as presented in figure 6-20.
Moreover, both samples have a maximum value for the dryness gradient at
saturated oil conditions.
The relaxation time increases as the mixture departs from saturated oil as shown
in figure 6-21. The relaxation time seems to be reaching a maximum and then
starting to decrease as the mixture is closer to a saturated gas condition. The
values of the relaxation time are higher for the sample that deviates further from
the equilibrium.
0.0%
0.2%
0.4%
0.6%
0.8%
1.0%
1.2%
1.4%
1.6%
1.8%
2.0%
0 1020 2040 3060 4080 5100 6120 7140 8160 9180 10200Length [feet]
Dry
ness
[fra
ctio
n]
Sample 8
Sample 7
Figure 6-20 : Dryness gradient for Case 3
76
0.1
1
10
100
1000
10000
100000
0 0.05 0.1 0.15 0.2 0.25 0.3 Gaseous Fractional Flow [fraction]
Rel
axat
ion
Tim
e [s
]
Sample 8
Sample 7
Figure 6-21 : Relaxation time for Case 3
77
7. DISCUSSION AND CONCLUSIONS
CHAPTER 7
DISCUSSION AND CONCLUSIONS
7.1. DISCUSSION
The motion of a reservoir fluid along the production pipe from the well-bore to
the wellhead was modeled by means of a homogenous model with liquid holdup
and applied to three study cases. The study cases considered the flow of gas/oil
and gas/water/oil mixtures in vertical wells. Consequently, the reservoir fluid was
considered as a multiphase fluid system. For this reason, all the properties of this
fluid were predicted by estimating the properties of its phases.
78
A homogenous model applicable for reservoir fluids flowing along a pipe with
constant cross-sectional area was developed. This model was simplified for
flowing fluids across a circular pipe at steady state. The differential conservation
laws of mass, momentum, and energy were adopted.
By estimating the multiphase fluid properties, the differential laws of
conservation were solved to compute the change in pressure, the change in
temperature and the flowing velocity. The change in pressure was computed
considering the gravitational force, the friction loss and the fluid compressibility.
The change in temperature was computed considering the change in pressure, the
friction effect, and the energy dissipation towards the surroundings. The velocity
is computed by knowing the fluid density.
Because the flow might not be at equilibrium, the density of the multiphase fluid
system was not obtained from the conservation laws. Two approaches were
presented to determine the density. One is the differential conservation law of
gaseous mass that can be solved by estimating the relaxation time, and the other is
predicting the liquid holdup by estimating the slip ratio. The later was used for the
application.
A new approach for predicting the liquid hold up was introduced. The liquid
holdup was associated to the deviation from equilibrium. It was related to the slip
79
ratio which is the ratio of the phase velocities. The slip ratio was computed by
interpolation. The interpolation is based on a Lagrange’s polynomial of the
second order relating the slip ratio with the equilibrium density. The slip ratio for
saturated oil and saturated gas is set to be the unit which represents the
equilibrium. An apparent slip ratio is assumed at the surface. The equilibrium
density is computed by knowing the system pressure and temperature.
The simulation of the reservoir fluid motion was executed by applying both the
homogenous model and liquid holdup model in conjunction. The homogenous
model was solved by using the classical fourth-order of Runge-Kutta method.
Because the slip ratio and the external temperature are not known at the wellhead,
the pressure and the temperature of the multiphase fluid at the well-bore were
used to give a closure to the system. The shooting method was applied for this
purpose. Thus, the slip ratio and the external temperature at the wellhead were
guessed until the pressure and temperature of the multiphase fluid at the well-bore
are predicted by the simulation. The simulation results were validated by a
generalized model for void fraction prediction.
Having validated the simulation results, the relaxation time was computed. The
relaxation time is the main property that delineates the conservation law for the
gas phase. This law completes the homogenous model in order to compute the
flowing density.
80
The assumption made of a constant geothermal gradient and a quadratic
relationship between the slip ratio and the equilibrium density did not introduce a
significant error for modeling the present phenomenon. The deviation of the
results from the Butterworth’s model8 was negligible for all samples.
The pressure drop tended to be linear even for the samples with considerable
deviation from equilibrium.
The temperature drop tended to be non-linear for all samples. The motion of the
fluid proved to be quick enough to delay the heat dissipation towards the
surroundings. This became evident when showing the difference between the
fluid temperature and the apparent external temperature.
It was proven that the relaxation time characterize the deviation from the
equilibrium for flowing fluids in wells. It was suggested by Downar-Zapolski et
al.17 that the relaxation time is a fluid property. In their study, they developed a
single correlation for the relaxation time of flashing water. However, the
relaxation time presented a unique curve for each sample of the study cases. Thus,
the present formulation of this property stills depends on the prevailing conditions
of the flowing reservoir fluid. Nevertheless, the relaxation time curves proved to
be a family of curves. This suggests that the present formulation of the relaxation
time can be adjusted towards becoming a fluid property for the reservoir fluids.
81
7.2. CONCLUSIONS
Having analyzed the results presented in the applications, the following
conclusions concerned with the present study cases have been reached:
1. Assuming a constant geothermal gradient does not introduce a significant
error.
2. The proposed approach predicts a continuously varying liquid-holdup by
means of interpolating the slip ratio.
3. The heat dissipation to the surroundings and the fluid expansion and the
friction effect, cause a non-linear temperature drop.
4. The upward motion of reservoir fluids in producing wells can be
successfully modeled by the developed homogenous model in conjunction
with the proposed model for liquid holdup prediction.
5. The relaxation time of gas separation proved to be an adequate property
for characterizing the deviation from the equilibrium for reservoir fluids.
6. The conservation law for the gas phase and the relaxation time of gas
separation from the liquid phases can be applied in order to achieve a
closure in the area-averaged homogenous model.
82
REFERENCES
[1] Amyx, J., Bass, D., Whiting, R., “Petroleum Reservoir Engineering: physical properties”, McGraw-Hill, USA, pp. 211-472, 1960
[2] Ansari, A.M., Sylvester, N.D., Sarica, C., Shoham, O., Brill, J.P., “A Comprehensive Mechanistic Model for Upward Two-Phase Flow in Wellbores”, SPE Production & Facilities, pp. 143-152, May, 1994
[3] Asheim, H., “MONA, An Accurate Two-Phase Well Flow Model Based on Phase Slippage”, SPE Production Engineering, pp 221-230, May 1986
[4] Ayala, L. F., Adewumi, M. A., “Low-Liquid Loading Multiphase Flow in Natural Gas Pipelines”, J. of the Energy Resources Technology, Vol. 125, pp. 284-293, 2003.
[5] Badur, J., Banaszkiewicz, M., “A Model of two-phase flow with relaxation-gradient microstructure”, Third International Conference on Multiphase Flow, held in Lyon, France, June 8-12, 1998.
[6] Bilicki, Z., Kestin, J., “Physical Aspects of the Relaxation Model in Two-Phase Flow”, Proceedings of the Royal Society of London, Series A, Mathematical and Physical Sciences, Vol.428, No. 1875, pp 379-397, Apr. 9, 1990.
[7] Bird, R., Stewart, W., Lightfoot, E., “Transport phenomena”, Wiley, USA, pp. 71-110, pp. 310-342, 1960
[8] Butterworth, D., “A Comparison of Some Void-Fraction Relationships for Cocurrent Gas-Liquid Flow”, International Journal of Multiphase Flow, Vol. 1, pp 845-850, 1975
[9] Brill, J. P., Mukherjee, H., “Multiphase Flow in Wells”, SPE, Richardson, p.16, pp. 102-122, 1999.
83
[10] Cazaraez-Candia, O., Vásquez-Cruz, M., “Prediction of Pressure, Temperature and Velocity Distribution of Two-Phase Flow in Oil Wells”, Journal of Petroleum Science and Engineering, Vol. 46, pp. 195-208, 2005.
[11] Cengel, Y., Boles, M., “Thermodynamics: an engineering approach”, McGraw-Hill, USA, pp. 150-155, pp. 603-626, 2002.
[12] Chapra, S., Canale, R., “Numerical Methods for Engineers”, McGrw-Hill, USA, pp. 675-718, 1998
[13] Chierici, G.L., Ciucci, G.M., Sclocchi, G., “Two-Phase Vertical Flow in Oil Wells – Prediction of Pressure Drop”, SPE Journal of Petroleum Technology, pp. 927-938, August 1974.
[14] Civan, F., “Including Non-equilibrium Effects in Models for Rapid Multiphase Flow in Wells”, SPE Paper 90583, the 2004 SPE Annual Technical Conference and Exhibition, held in Houston, Texas, 26-29 September 2004.
[15] Civan, F., “Including Non-equilibrium Relaxation in Models for Rapid Multiphase Flow in Wells”, SPE Production&Operations Journal, pp. 98-106, February 2006.
[16] Crowe, C., Elger, D., Roberson, J., “Engineering fluid mechanics”, Wiley, USA, pp. 368-434,2005
[17] Downar-Zapolski, P., Bilicki, Z., Bolle, L. and Franco, J., “The Non-equilibrium Relaxation Model for One-Dimensional Flashings Liquid Flow”, International J. Multiphase Flow, Vol. 22, No. 3, pp. 473-483, 1996.
[18] Faghri, A., Zhang, Y., “Transport phenomena in multiphase systems”, Elsevier Academic Presss, pp. 238-320, pp. 853-945, 2006
[19] Feburie, V., Goit, M., Granger, S., Seyhaeve, J. M., “A Model for Chocked Flow through Cracks with Inlet Subcooling”, International J. Multiphase Flow, Vol. 19, No. 4, pp 541-562, 1993
84
[20] Hagoort, J., “Prediction of wellbore temperatures in gas production wells”, J. of Petroleum Science and Engineering, Vol. 49, pp. 22-36, 2005.
[21] Himmelblau, D., Bischoff, K., “Process analysis and simulation: deterministic systems”, USA, Wiley, 1967, pp 9-37
[22] Lee, J. and Wattenbarger, R. A., “Gas Reservoir Engineering”, SPE, Richardson, TX, pp. 1-28, 2004.
[23] Pattillo, P.D., Bellarby, J.E., Ross, G.R., Gosch, S.W., McLaren, G.D., “Thermal and Mechanical Considerations for Design of Insulated Tubing”, paper SPE 79870 presented at IADC/SPE Drilling Conference, Amsterdam, 19-21 February 2003.
[24] Ros, N. C. J., “Simultaneous Flow of Gas and Liquid as Encountered in Well Tubing”, J. of Petroleum Technology, pp. 1037-1049, October 1961.
[25] Yoshioka, K., Zhu, D., Hill, A.D, Dawkrajai, P., Lake, L.W., “A Comprehensive Model of Temperature Behavior in a Horizontal Well”, paper SPE 95656 presented at the 2005 SPE Annual Technical Conference and Exhibition, Dallas, 9-12 October 2005.
85
APPENDIX A: A DISCUSSION OF THE EQUILIBRIUM CONDITION FOR A MULTIPHASE FLUID SYSTEM
This section shows that a multiphase fluid system is at equilibrium condition if
the velocities of the phases are equal. A multiphase fluid system is defined to be
at equilibrium conditions when18:
• A liquid holdup equal to the liquid fractional flow.
• A void fraction equal to the gaseous fractional flow.
• A flowing density equal to the equilibrium density.
• A flowing velocity equal to the volumetric flux.
Recall the definitions for the mixture density, velocity, volumetric flux and
quality:
LLgg HH ρρρ += …………………………………………….……..(A-1)
LLgg
LLLggg
HHvHvH
vρρρρ
+
+= ………………………………………………(A-2)
AVu&
= ………………………………………………………………....(A-3)
gg Hxρρ
= ……………………………………………………………(A-4)
The mass flow rate can be written as:
LLLggg vAHvAHm ρρ +=& …………………………………………...(A-5)
86
By rearranging the equation A-5 and combining the equations A-1 and A-2, the
next relationships are obtained:
LLLggg vHvHv ρρρ += ………….……………………..………..…..(A-6)
Amv&
=ρ …………………………………………………………...…..(A-7)
The volumetric flow rate of the multiphase fluid system, the gas phase and the
liquid phases can be defined as:
…………………………………………………………...(A-8) Lg VVV &&& +=
ggg vAHV =& ..………………………………….……………………...(A-9)
…..……………………………….………………….....(A-10) LLL vAHV =&
By replacing the equations A-3, A-9 and A-10, the equation A-8 becomes:
…………………………….………...……....……(A-11) LLgg vHvHu +=
Assuming that vL=v* and vg= λv* where λ is the slip ratio, the equations A-6 and
A-11 take the form:
*)( vHHv LLgg ρλρρ += ……………..……………………………(A-11)
*)( vHHu Lg += λ …………………….……..……………………..(A-12)
87
By introducing the equations A-1 and A-4 into the equation A-11 and the
expression Hg+HL=1 into the equation A-12, the following relationships are
obtained:
*)]1(1[ vxv −+= λ ……………...……....…………………………..(A-13)
*)]1(1[ vHu g −+= λ ………………...….…………………………..(A-14)
By combining the equations A-13 and A-14, the flowing or actual velocity takes
the form:
)1(1)1(1
−+−+
=λλ
gHxuv …………………………….……………………(A-15)
The equilibrium density can be defined as:
Aum
st&
=ρ …………………………………………………………….(A-16)
By replacing the equations A-7 and A-15 into the equation A-16, the flowing or
actual density can be expressed:
)1(1)1(1
−+
−+=
λλ
ρρx
H gst ……………..…………………………………(A-17)
The liquid hold up and the void fraction can be defined as a function of the
fractional flow of the phases:
88
gL
LL SS
SH+
=λλ …………………………………………………..…(A-18)
gL
gg SS
SH
+=λ
……………………………………………………..(A-19)
Now, the equations A-15, A-17, A-18 and A-19 are simplified for case when the
slip ratio is equal to one (λ=1):
………………………………………………………………...(A-20) uv =
stρρ = …………………………………….………………………...(A-21)
……………………………………..………………………(A-22) LL SH =
……………………………………..………………………(A-23) gg SH =
Note that the slip ratio is equal to one if the phases are flowing at the same
velocity.
89
APPENDIX B: CORRELATIONS AND BASIC RELATIONSHIPS
The correlations used in this study have been obtained from Lee and
Wattenbarger (2004), and Brill and Mukherjee (1999). These correlations are
summarized in the following, involving the units, given below.
BBg : [ft /scf] Bo3
B : [bbl/stb] BBw : [bbl/stb]
c : [psia-1] Cp : [BTU/lbm-ºR] D : [ft]
Mw : [lbm/lbmol] P : [psia] T : [ºR]
R: [scf/stb] U : [BTU/s-ft2-ºR] ρ : [lbm/ft3]
μ : [cp] ε : [ft] ξ : [wt%]
Pseudo-critical Temperature and Pressure. The gas phase is assumed to be free
of contaminants. Therefore, the Sutton correlations can be applied.
20.745.3492.169 ggpcT γγ −+= ……………………………………….……...(B-1)
26.30.1318.756 ggpcP γγ −−= ………………………………….…………….(B-2)
Pseudo-reduced Temperature and Pressure. These properties are defined as
follows:
pcpr T
TT = ……………………………………..……………………………....(B-3)
pcpr P
PP = ………………………………………...……………………..…….(B-4)
90
Gas compressibility factor. The Dranchuk and Abu-Kassem correlation is used
to compute an approximation of the Standing and Katz chart for gas
compressibility factor.
prprprprpr T
ATA
TA
TAAz ρ⎟
⎟⎠
⎞⎜⎜⎝
⎛++++= 5
544
332
1
2287
6 prprpr T
ATA
A ρ⎟⎟⎠
⎞⎜⎜⎝
⎛+++
5287
9 prprpr T
ATA
A ρ⎟⎟⎠
⎞⎜⎜⎝
⎛+−
( ) 112
11
3
22
1110 +++ − prA
pr
prpr e
TAA ρρ
ρ ………………………………………(B-5)
The pseudo-reduced density is given by:
pr
prpr zT
P27.0=ρ ………………………………………………………………(B-6)
The eleven constants (A1 to A11) for equation B-5 are defined as follows:
3265.01 =A 7361.07 −=A
0700.12 −=A 1844.08 =A
5339.03 −=A 1056.09 =A
01569.04 =A 6134.010 =A
05165.05 −=A 7210.011 =A
91
5475.06 =A
Note that equation B-5 formulates the gas compressibility factor as an implicit
equation. The evaluation of this factor has been done by the Newton-Raphson
iteration technique.
Gas formation-volume-factor. The gas formation-volume factor is known as:
PzTBg 0283.0= ………………………………………….…………………...(B-7)
Gas density. Equation B-8 states the density of a gaseous hydrocarbon:
zTPMw
g 736.10=ρ …………………………………..…………...…………...(B-8)
gMw γ9625.28= ……….………………………….…………………….…...(B-9)
Gas viscosity. The Lee et al. correlation is used for estimating the gas viscosity.
1
1 36.621
410
YgX
g eK⎟⎟⎠
⎞⎜⎜⎝
⎛
−=ρ
μ …………………………...……………..…………(B-10)
)26.192.209()01607.0379.9( 5.1
1 TMwTMwK
+++
= ……………………………….……….…...(B-11)
MwT
X 01009.04.986448.31 ++= ……………………………..…....…..….(B-12)
MwY 2224.0447.21 −= …………………………..………….……………..(B-13)
92
Gas solubility of saturated oil. The gas solubility (Rb) is estimated at bubble-
point conditions using:
sO
sG
b VV
R&
&= …………………………………………………………….……..(B-14)
API gravity. The API gravity is defined as:
5.1315.141−=
oAPI γ
γ ………………………………..…………………...…..(B-15)
Oil compressibility. The oil compressibility at pressures above the saturation
pressure is estimated using the Vasquez-Beggs correlation.
PTR
c APIbo 510
143361.12)460(2.175 −+−+=
γ ....................................................(B-16)
Gas solubility in oil. The Standing correlation states that:
2048.1
/2104.1
2.18 ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ += X
gogPR γ ………………………………………....(B-17)
)460(00091.00125.02 −−= TX APIγ ………………………………....….....(B-18)
Saturation Pressure. The bubble-point pressure (Pb) is obtained by solving for
pressure in the Standing correlation.
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡−⎟
⎟⎠
⎞⎜⎜⎝
⎛= 4.1102.18 2
83.0
X
g
bb
RP
γ…………………………………….………(B-19)
93
Oil formation-volume-factor. The Standing correlation for saturated oils is used:
2.15.0
/ )460(25.10012.0⎥⎥
⎦
⎤
⎢⎢
⎣
⎡−+⎟
⎟⎠
⎞⎜⎜⎝
⎛= TRB
g
oogo γγ
9759.0+ ………………………………...………………………………..(B-20)
The oil formation-volume-factor at above-bubble-point pressures is computed as
follows:
)(,
bo PPcboo eBB −−= ……………………………….……………………...(B-21)
The oil formation-volume-factor at the bubble-point pressure (BBo,b) is estimated
by replacing the gas solubility at bubble-point conditions in Eq. B-20.
Oil Viscosity. The Beggs-Robinson correlation for saturated oils is used:
33
Yodo X μμ = ………………………………………..……………………(B-22)
APIod γμ 02023.00324.3))1log(log( −=+
)460log(163.1 −− T ……….………………………………..……………(B-23)
515.0/3 )100(715.10 −+= ogRX ……………………………………………….(B-24)
338.0/3 )150(44.5 −+= ogRY …………………………...…………...…………(B-25)
The Vasquez-Beggs correlation for under-saturated oils is used:
4
,
X
bboo P
P⎟⎟⎠
⎞⎜⎜⎝
⎛= μμ ………...…………………………….…………………….(B-26)
PePX 0000898.0513.11187.14 6.2 −−= …………………………………..…………(B-27)
94
The oil viscosity at the bubble-point pressure (μo,b) is estimated by replacing the
gas solubility at bubble-point conditions in Eq. B-22.
Gas solubility in water. The Ahmed correlation is used for the gas/water
solubility
( ) 52
555/ ZPYPXKR ow ++= …………………………………….……….…(B-28)
)460(1045.312.2 35 −×+= − TK
25 )460(1059.3 −×+ − T ……………….…………………………………..(B-29)
)460(1026.50107.0 55 −×−= − TX
27 )460(1048.1 −×+ − T ……………………………………………….…..(B-30)
)460(109.31075.8 975 −×+×−= −− TY
211 )460(1002.1 −×+ − T ..............................................................................(B-31)
[ ]ξ)460(000173.00753.015 −−−= TZ ………………………………….....(B-32)
Water viscosity. The water phase is considered to have some level of salinity.
Therefore, the McCain correlation is applied.
666)460( KTX Y
w −=μ ……………………………………....…….... …....(B-32)
26 313314.040564.8574.109 ξξ +−=X
331072213.8 ξ−×+ …………………………………….…………...……(B-33)
95
2426 1079461.61063951.212166.1 ζξ −− ×−×+−=Y
4635 1055586.11047119.5 ξξ −− ×+×− .......................................................(B-34)
PK 56 100295.49994.0 −×+=
29101062.3 P−×+ ……….…………………………………..……………(B-35)
Water formation-volume-factor. The water formation-volume-factor is
computed using the McCain correlation.
)1)(1( 77 YXBw ++= …………………………………..……………………(B-36)
)460(1033391.11000010.1 427 −×+×−= −− TX
27 )460(1050654.5 −×+ − T ………………………………………….……(B-37)
)460(1095301.1 97 −×−= − TPY
)460(1072834.1 213 −×− − TP
2107 1025341.21058922.3 PP −− ×−×− …………………………………..(B-38)
Specific heat. The Gambill correlation is used for an estimation of the specific
heat for hydrocarbon mixtures.
op
TCγ
)460(00045.0338.0 −+= …………………………………..………...(B-39)
Overall heat transfer. The Shiu and Beggs correlation is used for computing the
overall heat transfer for producing pipes.
96
2608.02904.05253.0 )12(0149.0 APIp DmDUmC
γπ
−= &&
9303..24146.4 ργ g …………………………………………….………………..(B-40)
Friction factor. The explicit approximation for the Colebrook equation
developed by Zigrang and Sylvester is used.
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ +−−=
Re13
7.32log
Re02.5
7.32log21
DDfM
εε ………………………………....(B-41)
97
APPENDIX C: NOMENCLATURE
Symbols
A = cross-sectional area of the producing pipe, L2
B = formation value factor, dimensionless
c = compressibility, M-1Lθ2
C = flow parameter, dimensionless
cp = specific heat, L2θ-2T-1
CA = Perimeter if the cross-sectional area, L
D = pipe diameter, L
ƒ = friction factor, dimensionless
g = gravitational acceleration, Lθ-2
h = specific enthalpy, L2θ-2
H = volumetric fraction, dimensionless
l = distance measured from the surface, L
L = length of the producing pipe, L
m& = mass rate, Mθ-1
Mw = molecular weight, dimensionless
P = pressure, ML-1θ-2
q = flow parameter, dimensionless
Q = heat-flux rate, ML-1θ-3
r = flow parameter, dimensionless
R = solubility ratio, dimensionless
Re = Reynolds number, dimensionless
s = flow parameter, dimensionless
S = volumetric rate ratio, dimensionless
t = time, θ
T = Temperature, T
98
u = volumetric flux, Lθ-1
U = overall heat transfer coefficient, Mθ-3T-1
v = velocity, Lθ-1
V& = volume rate, L3θ-1
x = gas mass fraction, quality or dryness, dimensionless
α = thermal gradient, L-1T
γ = specific gravity
Γ = interface mass-transfer rate, ML-3θ-1
ε = roughness, L
η = Joule-Thompson coefficient, M-1Lθ2T
ϕ = pipe angle from the azimuth, degrees
λ = slip ratio, dimensionless
μ = viscosity, ML-1θ-1
π = trigonometric constant, dimensionless
ρ = density, ML-3
τw = wall shear stress, ML-1θ-2
υ = specific volume, M-1 L3
ξ = salinity, ML-3
Subscripts
a = air
b = bubble-point
g = gas phase
G = gas pseudo-component
L = liquid phases
M = Moody
o = oil phase
O = oil pseudo-component
99
od =dead-oil
pc = Pseudo-critical
pr = Pseudo-reduced
R = reservoir
s = surrounding or external
st = static or equilibrium
w = water phase
W = water pseudo-component
Superscripts
i = initial
s =standard
100