Gillermo Michel. Modeling of multiphase flow in wells

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)


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)


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


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]


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]


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]


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]


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]


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

]


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]



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]



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]



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]



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]



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

]



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


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


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


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


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


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


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

Gillermo Michel. Modeling of multiphase flow in wells

Documents

graduate studies

nonequilibrium conditions

graduate faculty

faruk civan

roy knapp

present work

present study

list of figures