Gillermo Michel. Modeling of multiphase flow in wells by franco sivila

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

______________________________ Faruk Civan, Chair

______________________________ Roy Knapp

______________________________ Robert Hubbard

â&#x20AC;&#x153;Science without religion is lame, religion without science is blindâ&#x20AC;? 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.

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

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 nonequilibrium 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.

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.

1.2. 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.

Gas

Oil

Water

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

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

(a)

Water

Oil

(b)

Gas

(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 nonequilibrium 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.

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 pseudocomponents 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 pseudocomponents 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.

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:

•

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.

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.

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.

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.

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

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.

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.

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)

DownarZapolsky et al. (1996)

Badur and Banaszkiewicz (1998)

Febuire et al. (1993)

Civan (2006)

Present Study

Multiphase effect consideration

Mixture

Multi-fluid

Mixture

Mixture and Multi-fluid

Mixture

Homogeneous mass flux along well

Yes

Nonequilibrium model consideration

Holdup

Relaxation

Holdup and Relaxation

Orientation angle

Near Vertical

Topography

Horizontal

Vertical

Fluid type

Reservoir

Water

Reservoir

Slip ratio consideration

Yes

Thermal effect consideration

---

Isothermal

Adiabatic

Nonadiabatic

Isothermal

Nonadiabatic

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

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 ( V&Gs , V&Os and V&Ws ), the overall mass flow rate1 ( m& ) for the system can be determined by: m& = γ G ρ asV&Gs + γ O ρWs V&Os + ρWs V&Ws …………………...….……….………(3-1)

The water and air density at standard conditions (ρas and ρws) 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.

It is necessary to define the volumetric flow rate of each phase1 ( V&g , V&o and V&w ) by using the volumetric flow rates of the pseudo-components.

V&g = (V&Gs − Rg / oV&Os − Rg / wV&Ws ) Bg ……...………………………..............(3-2) V&o = V&Os Bo …………………………………………………….………...(3-3) V&w = V&Ws Bw ………………………………………………….…………..(3-4)

The formation volume factors for each phase, i.e. gas, oil, and water, (Bg, Bo and B

Bw) can be estimated by the correlations given in Appendix B. The equations 3-2, B

3-3 and 3-4 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 rate1 ( V& ) and the gas, oil, and water phase fractional flows1 (Sg, So and Sw) can be calculated as the following:

V& = V&g + V&o + V&w ……………….……………………...………………..(3-5) Sg =

V&g ……………………………….………………….……...…….(3-6) V&

So =

V&o …………………………………...………….………………..(3-7) V&

Sw =

V&w ………………………………………………………..………(3-8) V&

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:

S g + S o + S w = 1 ……………………………………………………….(3-9) S L = 1 − S g …………………………………………………………...(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:

x st =

(V&Gs − R g / oV&Os − R g / wV&Ws )γ G ρ as

……….…………………..……..(3-11)

The gas-in-solution ratios (Rg/o and Rg/w) can be estimated with the correlations presented in Appendix B.

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.

ρ st =

m& ……………………...……………………………………....(3-12) V&

Knowing the volumetric flow rate and the internal pipe diameter (D), the volumetric flux 16 (u) in a circular pipe can be obtained as: u=

4V& …………………………...…………………………….…...(3-13) πD 2

For a homogeneous fluid, the viscosity14 (μ) can be estimated by weighting the phase viscosities by their own fractional flow.

μ = S g μ g + So μo + Sw μw ………………………………………………(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.

η=−

1 cp

⎡ ⎛ ∂υ ⎞ ⎤ ⎢υ − T ⎜ ⎟ ⎥ …………..………..……….………………..(3-15) ⎝ ∂T ⎠ P ⎦ ⎣

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:

1 ⎛ ∂ρ ⎞ ⎜ ⎟ …………………………………………………………(3-16) ρ ⎝ ∂P ⎠ T

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. Re =

ρ vD ………………………………………………………..….(3-17) μ

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:

τw =

1 f M ρv 2 …………..……………………..…...………………...(3-18) 8

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. Qf =

4U (T − Ts ) ……………..……..……………..………………..(3-19) D

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. Ts = Ts 0 + α s l sin ϕ ………………...………..…………....…………...(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:

ρ = ρ g H g + ρ L H L ………………………………………...……….(3-21) H g + H L = 1 ………………..………………………………………(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).

ρ g = ρ st

x st ………………………………………………………..(3-23) Sg

ρ L = ρ st

(1 − x st ) …………………………………………………...(3-24) SL

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. vg =

uS g

vL =

uS L ………………………………………………….…………(3-26) HL

…………………………………………………….………(3-25)

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. v=

ρ g H g vg + ρ L H L vL ……………………….…………………….(3-27) ρg H g + ρLH L

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.

λ=

vg vL

………………………………………………………….……(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. HL = Hg =

λS L λS L + S g Sg

λS L + S g

………………………………………………..…….(3-29)

…………………………………………….……….(3-30)

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:

ρg ………………………………………………..………….(3-31) ρ

x = Hg

In this study, a new constitutive equation is proposed for modeling of the slip ratio and the slippage as the follows:

λ=

(ρ (ρ − ρ )(ρ − ρ ) (ρ (ρ − ρ )(ρ − ρ ) + (ρ − ρ )(ρ − ρ )

(ρ st − ρ st 0 )(ρ st − ρ L ) g

st 0

− ρ g )(ρ st − ρ L )

− ρ g )(ρ st 0 − ρ L )

λ0 ……………..…(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.

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. x st = x + θ

D x ………………………………………………………(3-33) Dt

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

∂x ⎞ ⎛ ∂x + v ⎟ ……………………………..……………..(3-34) ∂l ⎠ ⎝ ∂t

θ = (x st − x )⎜

For a steady-state flow regime, the relaxation time is approximated by disregarding the change of quality in time.

θ=

x st − x …………………………………………………….……..(3-35) dx v dl

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.

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.

Qf mout

Pout

τw min

Δl ϕ

(a)

(b)

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 Ď&#x2022;.

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 crosssectional 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).

(a)

(b)

T (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 crosssectional 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: ∂ ( Aρ ) + ∂ (vAρ ) = 0 …………………………………………..…(4-1) ∂t ∂l

After expanding the derivatives and rearranging the terms, the next expression is obtained.

ρ ⎛ ∂A ∂ρ ∂v ∂ρ ∂A ⎞ +ρ +v = − ⎜⎜ + v ⎟⎟ …………….………………….(4-2) ∂l ⎠ ∂t ∂l ∂l A ⎝ ∂t

Assuming a pipe with a constant and circular cross-sectional area, the final form for the law of mass balance is shown below: ∂ρ ∂v ∂ρ +ρ +v = 0 ……….....…………………..………………….(4-3) ∂t ∂l ∂l

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.

∂ ( Aρ v ) + ∂ (vAρuv ) = −C Aτ w − Aρ g sin ϕ − ∂ ( AP ) ...….……......(4-4) ∂l ∂t ∂l

After expanding the derivatives and rearranging the terms, the next expression is obtained. Aρ

⎡∂ ⎤ ∂v ∂v ∂ + vAρ + v ⎢ ( Aρ ) + (vAρ )⎥ ∂t ∂l ∂l ⎣ ∂t ⎦ ……...……………………...(4-5) ∂ = −C Aτ w − Aρ g sin ϕ − ( AP ) ∂l

By replacing the mass balance equation (equation 4-1) and expanding the derivatives, equation 4-5 reduces to:

C ∂v ∂v ∂P P ∂A + vρ = − A τ − ρ g sin ϕ − − ………………………(4-6) ∂t ∂l A ∂l A ∂l

For a pipe with constant and circular cross-sectional area, the momentum balance equation takes the final form:

4τ ∂v ∂v ∂P + vρ + = − w − ρ g sin ϕ ….……...…...………………..(4-7) ∂t ∂l ∂l D

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

loss or gain of energy. Therefore, the frictional term is not considered in the energy balance equation initially18. ∂ ( Aρ eT ) + ∂ (vAρ eT ) = − ∂ ( vAP ) − AQ f ……..………...………..(4-8) ∂t ∂l ∂l

By expressing the total energy in terms of enthalpy, kinetic energy, and potential energy, the equation 4-8 takes the form:

∂ ⎡ ⎛ Aρ ⎜ h − P + 12 v 2 + gz ⎞⎟⎤⎥ ⎢ ρ ⎠⎦ ∂t ⎣ ⎝ …...……..(4-9) ∂ ∂ ⎡ ⎤ ⎞ ⎛ 1 2 P + ⎢vAρ ⎜ h − ρ + 2 v + gz ⎟⎠⎥⎦ = − ∂ l ( uAP ) − AQ f ⎝ ∂l ⎣

After expanding the derivatives and rearranging the terms, the energy balance is set in the following convenient expression. Aρ

∂ ∂ h + 12 v 2 + gz + vAρ h + 12 v 2 + gz ∂l ∂t

(

)

(

)

⎡∂ ⎤ ∂ (vAρ )⎥ = ∂ ( AP ) − AQ f + h + v + gz ⎢ ( Aρ ) + ∂ t ∂ l ⎣ ⎦ ∂t

(

1 2

)

……....……..(4-10)

By replacing the mass balance equation (equation 4-1) and expanding the derivatives, the equation 4-10 is conveniently expressed as: Aρ

⎛ ∂h ∂h ∂v ∂v ⎞ + vAρ + v⎜⎜ Aρ + vAρ ⎟⎟ ∂t ∂l ∂t ∂l ⎠ ⎝ ∂ = ( AP ) − vAρ g sin ϕ − AQ f ∂t

…………………..………….(4-11)

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.

4τ P ∂A P ∂A ∂h ∂h ∂P ∂P + vρ − −v =v w + +v − Q f …...……...(4-12) ∂t ∂ l ∂t ∂l D A ∂t A ∂l

By expressing the enthalpy in terms of temperature as recommended by Brill and Mukherjee9, the energy balance equation takes the form:

ρ cp

∂P ∂P ∂T ∂T + vρ c p + (ρ ηc p + 1) − v(ρ ηc p + 1) ∂t ∂l ∂t ∂l ……….…….(4-13) 4τ w P ∂A P ∂A =v + +v − Qf D A ∂t A ∂l

Because it is assumed that the pipe has constant and circular cross-sectional area, the energy balance takes the final form:

ρ cp

∂P ∂P ∂T ∂T + vρ c p − (ρ ηc p + 1) − v(ρ ηc p + 1) ∂l ∂t ∂l ∂t 4τ = v w − Qf D

…………….…(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. ∂ ( Aρ x ) + ∂ (vAρ x ) = AΓ …………………………………………(4-15) ∂t ∂l

After expanding the derivatives and rearranging the terms, the mass balance for the gas phase is set conveniently as:

Aρ

⎡∂ ⎤ ∂x ∂x ∂ + vAρ + x ⎢ ( Aρ ) + (vAρ )⎥ = AΓ ……….……………(4-16) ∂t ∂l ∂l ⎣ ∂t ⎦

By replacing the mass balance equation (equation 4-1) and expanding the derivatives, the equation 4-16 is takes the form: ∂x ∂x Γ +v = ………..........………………………………………...(4-17) ∂t ∂l ρ

Bilicki and Kestin6 replaced the interface transfer term

Γ /ρ by using the

definition of the relaxation time stated in equation 3-33.

x − x st ∂x ∂x +v = − ………………..…...……………………….......(4-18) θ ∂l ∂t 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.

dv dρ +v = 0 ………………………………………………...…....(4-19) dl dl

vρ

4τ dv dP + = − w − ρ sin ϕ ………………...………….…………..(4-20) dl dl D

vρ c p

4τ dT dP − v(ρ ηc p + 1) = v w − Q f …….……………………….(4-21) dl dl D

x − x st dx =− ………..……………...……….…………..………...(4-22) dl θ

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.

ρv =

m& 4m& ……...……..…………………...……..……………...(4-23) = A πD 2

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:

dP v2 1 2 dρ −v = − fM ρ − ρ g sin ϕ ……………………………….(4-24) dl dl D 2

The momentum equation takes its final form by rearranging the kinetic momentum term and replacing the isothermal compressibility defined in equation 3-16.

v2 1 − fM ρ − ρ g sin ϕ dP D ………..…..…………….……..……...(4-25) = 2 dl ⎛ ∂ρ ⎞ 2 1− ⎜ ⎟ v ⎝ ∂P ⎠ T

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.

The energy equation is rearranged as the following:

dT dP dP 4τ w Q f ………………...…………….…(4-26) = c Pη +υ + − dl dl dl ρD ρv

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. dT T ⎛ ∂υ ⎞ dP 1 v2 UπD = + fM − (T − Ts ) …...………..…...…..(4-27) ⎜ ⎟ dl c P ⎝ ∂T ⎠ P dl 2 c P D c p m&

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:

x − x st dx =− ………...…………………………………………….(4-28) dl vθ

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 RungeKutta method (RK4) is selected.

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 multiphasefluid along the pipe occurring under this scheme is depicted in figure 5-1.

Consequently, The rate of change in pressure ( quality (

Δx Δl i

ΔP Δl i

), temperature (

ΔT Δl i

), and

) 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.

Pi +1 = Pi +

ΔP Δl ………………………………………………....….(5-1) Δl i

Ti +1 = Ti +

ΔT Δl ………………………………………………….....(5-2) Δl i

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.

OUTLET i=0

ϕ1

i=1 i=2 i=3 i=4

ϕ2

ϕ3

i = N-1 i=N INLET

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

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. ΔP 1 ⎛ ΔP = ⎜⎜ Δl i 6 ⎝ Δl

ΔT 1 ⎛ ΔT = ⎜⎜ Δl i 6 ⎝ Δl

+2 i ,1

ΔP Δl

ΔT Δl

+2 i,2

ΔP Δl

ΔT Δl

ΔP Δl

⎞ ⎟ …………………..….(5-3) ⎟ i,4 ⎠

ΔT Δl

⎞ ⎟ ……………………..(5-4) ⎟ i,4 ⎠

i ,3

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.

ΔP Δl

i ,1

( v i )2 1 − fM i ρ i − ρ i g sin ϕ i D = 2 …………...……...……….(5-5) ∂ρ 2 1− (v ) ∂P i i

ΔT Δl

= i ,1

T cp

i i

( v i ) 2 UπD ∂υ ΔP 1 + fM i − (T − Ts i ) …………...(5-6) ∂T i Δl i 2 c p D c p m& i i

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.

Pi ,1 = Pi + Ti ,1 = Ti +

1 ΔP 2 Δl

i ,1

1 ΔT 2 Δl

i ,1

Δl ………………………………………...………(5-7) Δl ………………………………………….……..(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.

ΔP Δl i , 2 ΔT Δl

1 − fM = 2

= i,2

T cp

i ,1 i ,1

i ,1

∂υ ∂T

ρ i ,1

( v i ,1 ) 2 D

− ρ i ,1 g sin ϕ i

∂ρ 1− ( v i ,1 ) 2 ∂P i ,1

i ,1

ΔP Δl

i ,1

1 + fM 2

( v i ,1 ) 2 i ,1

i ,1

−

…………….…..……(5-9)

UπD c p m&

(T − Ts i ) .…..(5-10) i ,1

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. Pi , 2 = Pi +

1 ΔP Δl …………………………………………….….(5-11) 2 Δl i , 2

Ti , 2 = Ti +

1 ΔT 2 Δl

Δl …………………………………………..……(5-12) i,2

In the same manner for estimating the previous sets, the third set of rates is formulated by the equations 5-13 and 5-14.

ΔP Δl i ,3 ΔT Δl

1 − fM = 2

= i ,3

i,2

ρ i,2

( v i,2 ) 2

∂ρ 1− ∂P

∂υ ∂T

i,2

ΔP Δl

i,2

− ρ i , 2 g sin ϕ i

( v i,2 )

…......……………...(5-13)

i,2

1 + fM 2

( v i,2 ) 2 i,2

i,2

−

UπD c p m&

(T − Ts i ) .....(5-14) i,2

For the fourth set of rates, the intermediate pressure and temperature are obtained by taking a full increment of the length Δl. Pi ,3 = Pi +

ΔP Δl ………………………………….……………….(5-15) Δl i ,3

Ti ,3 = Ti +

ΔT Δl

Δl ………………….……………………………….(5-16) i ,3

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.

ΔP Δl

ΔT Δl

1 − fM = 2

i ,3

i,4

= i,4

T cp

i ,3 i ,3

∂υ ∂T

ρ i ,3

( v i ,3 ) 2

∂ρ 1− ∂P

i ,3

ΔP Δl

i ,3

− ρ i ,3 g sin ϕ i

( v i ,3 )

……..…..………….(5-17)

i ,3

1 + fM 2

( v i ,3 ) 2 i ,3

i ,3

−

UπD c p m&

(T − Ts i ) ..…(5-18) i ,3

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:

•

Volumetric rate of production for the gas, oil and water pseudocomponents.

•

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.

αs =

TN − T0 …………………………………………..…………….(5-19) L

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

initial external temperature are changed simultaneously until the desired pressure and temperature drop are obtained. BEGIN

INPUT System’s Constants V&Gs , V&Os , V&Ws , γG, γO, ξ

INPUT System’s Constants L, D, ε, ϕ, Δl, TN, PN

INPUT Surface Data P0, T0

CALCULATE αs

GUESS λ 0, Ts0

CALCULATE Properties below bubble-point pressure

SET l = 0

Yes

V&gs ≥ 0

SOLVE the change in state with liquid hold up

SOLVE the change in state

CALCULATE Properties below bubblepoint pressure

CALCULATE Properties above bubble-point pressure

SET l = l -Δ l

l ≥ -L Yes

P = PN and T = T N

Yes END

Figure 5-2 : Flowchart for simulating multiphase flow in wells

A good initial guess is based on assuming an equilibrium for the initial slip ratio (Îť0=1) and a perfect insulation for the initial external temperature (Ts0=T0).

The presence of the gas phase ( V&gs â&#x2030;Ľ 0 ) 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 ( V&gs â&#x2030;¤ 0 ), the fluid is not flashing and it is considered to be at equilibrium.

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 γo

Nº 7 8 10 11 12 13 22 23 24 25

0.9826 0.9826 0.9516 0.9390 0.9390 0.9390 0.8236 0.8236 0.8236 0.8236

V&Gs

V&Ws

[Mscf/d]

[stb/d]

160.2 231.8 2104.5 1954.7 2154.8 3501.6 180.7 311.7 452.7 1060.7

0.761 1.101 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

V&Os

[stb/d]

[psia]

[in.]

[ºF]

761.1 1100.7 4128.0 3834.3 4226.8 6868.5 191.8 330.8 480.5 1125.9

569.4 589.3 769.6 813.7 661.7 428.8 1370.3 1341.9 1306.4 1171.5

5.000 5.000 5.000 2.875 2.875 2.875 2.875 2.875 2.875 2.875

104.0 107.6 124.9 118.9 132.8 138.9 92.1 89.6 89.4 98.6

222.8 226.4 188.6 186.8 187.9 187.9 167.0 167.0 167.0 167.0

γg 1.268 1.268 0.708 0.708 0.708 0.708 0.750 0.750 0.750 0.750

[wt%]

[ft.]

[psia]

3.0 10171 3.0 10171 0.0 7648 0.0 7579 0.0 7579 0.0 7579 0.0 8038 0.0 8038 0.0 8038 0.0 8038

4514.2 4546.8 3322.8 3439.2 3288.7 3244.7 3568.5 3503.1 3430.7 3222.0

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.

â&#x20AC;˘

Case 2 includes samples 22, 23, 24 and 25 for representing two-phase flow of light crude oil.

â&#x20AC;˘

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. Hg =

1 ⎛ 1 − x st 1 + C ⎜⎜ ⎝ x st

⎞ ⎟⎟ ⎠

⎛ ρg ⎜⎜ ⎝ ρL

⎞ ⎟⎟ ⎠

⎛ μL ⎜ ⎜μ ⎝ g

⎞ ⎟ ⎟ ⎠

…………..….…………………..(6-1)

The equation 6-1 is rearranged in order to perform a power law correlation as indicated in the following. ⎛ 1 − x st 1 − 1 = C ⎜⎜ Hg ⎝ x st

⎞ ⎟⎟ ⎠

⎛ ρg ⎜⎜ ⎝ ρL

⎞ ⎟⎟ ⎠

⎛ μL ⎜ ⎜μ ⎝ g

⎞ ⎟ ……………………….……….….(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.

54 Sample 10 Sample 11 Sample 12 Sample 13

1/Hg-1

0 0

12 16 (1/xst-1)q(ρ g/ρ L)r

Figure 6-1: Correlation for Case 1

These plots show a direct relationship between the expression

expression

(

− 1)

( )

q ρg

xst

− 1 and the

ρL

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 Sample 22 Sample 23 Sample 24 Sample 25

1/Hg-1

0 0

30 40 (1/xst-1)q(ρ g/ρ L)r

150

180

Figure 6-2 : Correlation for Case 2

Sample 8 50

Sample 7

1/Hg-1

0 0

(1/xst-1)q(ρ g/ρ L)r

120

Figure 6-3 : Correlation for Case 3

Table 6-2 : Adjustable parameters and Coefficient of Determination Nº 7 8 10 11 12 13 22 23 24 25

λ0 1.800 3.800 2.945 1.817 2.247 1.024 1.672 1.345 1.175 1.152

T s0 [ºF] 77.74 76.01 109.84 98.44 117.79 117.27 81.61 73.10 67.36 65.22

C 0.4915 0.2882 12.7908 2.1830 9.9014 1.1154 0.8377 0.8656 0.9269 0.9877

q 0.9823 0.9317 0.4939 0.8182 0.5787 0.9804 0.9391 0.9685 0.9828 0.9818

r 0.4528 -0.1661 1.0161 0.8734 1.0897 1.0085 0.5482 0.7247 0.8542 0.9031

R 0.99954 0.99937 0.99971 0.99848 0.99907 0.99795 0.99812 0.99772 0.99754 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.

[

( )]

⎧ H − 1 + c(1− xst )q ρ g x st ρL ∑ g i =0 ⎨ ⎩ R2 = 1− N ∑i=0 (H g − S g )2 N

r −1

⎫ ⎬ ⎭ ………….…………….(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.

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. 3500 Sample 10 Sample 11 Sample 12 Sample 13

3190 2880

Pressure [psia]

2570 2260 1950 1640 1330 1020 710 400 0

770 1540 2310 3080 3850 4620 5390 6160 6930 7700 Length [feet]

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.

0.8 Sample 10 Sample 11 Sample 12 Sample 13

0.7

Void Fraction [fraction]

0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.1

0.2 0.3 0.4 0.5 0.6 Gaseous Fractional Flow [fraction]

0.7

0.8

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.

189 Sample 10

182

Sample 11 Sample 12

175

Sample 13

Temperature [ยบF]

168 161 154 147 140 133 126 119 0

770

1540 2310 3080 3850 4620 5390 6160 6930 7700 Length [feet]

Figure 6-6 : Temperature drop for Case 1

22 Sample 10 Sample 11 Sample 12 Sample 13

19.8

Temperature Difference [ยบF]

17.6 15.4 13.2 11 8.8 6.6 4.4 2.2 0 0

770

1540 2310 3080 3850 4620 5390 6160 6930 7700 Length [feet]

Figure 6-7 : Temperature difference for Case 1

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. 7.0% Sample 10 6.3%

Sample 11 Sample 12

5.6%

Sample 13

Dryness [fraction]

4.9% 4.2% 3.5% 2.8% 2.1% 1.4% 0.7% 0.0% 0

770 1540 2310 3080 3850 4620 5390 6160 6930 7700 Length [feet]

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.

10000

Relaxation Time [s]

1000

100

Sample 10 Sample 11 Sample 12 Sample 13

0.1

0.2 0.3 0.4 0.5 0.6 Gaseous Fractional Flow [fraction]

0.7

0.8

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 twophase 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

3600 Sample 22 Sample 23 Sample 24 Sample 25

3350 3100

Pressure [psia]

2850 2600 2350 2100 1850 1600 1350 1100 0

810 1620 2430 3240 4050 4860 5670 6480 7290 8100 Length [feet]

Figure 6-10 : Pressure drop for Case 2

0.5 Sample 22 Sample 23 Sample 24 Sample 25

0.45

Void Fraction [fraction]

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0

0.05

0.1

0.15 0.2 0.25 0.3 0.35 0.4 Gaseous Fractional Flow [fraction]

Figure 6-11 : Void fraction for Case 2

0.45

0.5

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. 168 Sample 22

160

Sample 23 Sample 24

152

Sample 25

Temperature [ยบF]

144 136 128 120 112 104 96 88 0

810

1620 2430 3240 4050 4860 5670 6480 7290 8100 Length [feet]

Figure 6-12 : Temperature drop for Case 2

However, this asymptotical trend is not observed in the difference between the fluid temperature and the surroundings temperature as shown in figure 6-13. Conversely, the heat transfer is increasing non-monotonically.

36 Sample 22 Sample 23 Sample 24 Sample 25

32.4

Temperature Difference [ยบF]

28.8 25.2 21.6 18 14.4 10.8 7.2 3.6 0 0

810

1620 2430 3240 4050 4860 5670 6480 7290 8100 Length [feet]

Figure 6-13 : Temperature difference for Case 2

9.0% Sample 22 8.1%

Sample 23 Sample 24

7.2%

Sample 25

Dryness [fraction]

6.3% 5.4% 4.5% 3.6% 2.7% 1.8% 0.9% 0.0% 0

810 1620 2430 3240 4050 4860 5670 6480 7290 8100 Length [feet]

Figure 6-14 : Dryness gradient for Case 2

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.

10000

Relaxation Time [s]

1000

100

Sample 22 Sample 23 Sample 24 Sample 25

0.1

0.2 0.3 0.4 Gaseous Fractional Flow [fraction]

Figure 6-15 : Relaxation time for Case 2

0.5

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. 4600

Sample 8

4190

Sample 7

3780

Pressure [psia]

3370 2960 2550 2140 1730 1320 910 500 0

1020 2040 3060 4080 5100 6120 7140 8160 9180 10200

Length [feet]

Figure 6-16 : Pressure drop for Case 3

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. 0.3 Sample 8

0.27

Sample 7

Void Fraction [fraction]

0.24 0.21 0.18 0.15 0.12 0.09 0.06 0.03 0 0

0.03

0.06

0.09 0.12 0.15 0.18 0.21 0.24 Gaseous Fractional Flow [fraction]

0.27

0.3

Figure 6-17 : Void fraction for Case 3

The temperature drop in this case is clearly nonlinear for both samples. The temperature change is mainly accountable to the heat transfer magnitude. As the fluid departs from the bottom-hole, the heat transfer seems to increase asymptotically as described in figure 6-18. However, this asymptotical trend is not observed in the difference between the fluid temperature and the surroundings temperature as shown in figure 6-19. This break in the tendency is due to the change from two-phase to three-phase flow.

230 217

Sample 8

204

Sample 7

Temperature [ยบF]

191 178 165 152 139 126 113 100 0

1020

2040

3060

4080

5100

6120

7140

8160

9180 10200

Length [feet]

Figure 6-18 : Temperature drop for Case 3

Temperature Difference [ยบF]

33 29.7

Sample 8

26.4

Sample 7

23.1 19.8 16.5 13.2 9.9 6.6 3.3 0 0

1020 2040

3060 4080

5100 6120

7140 8160

9180 10200

Length [feet]

Figure 6-19 : Temperature difference for Case 3

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.

2.0% 1.8%

Sample 8

1.6%

Sample 7

Dryness [fraction]

1.4% 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% 0.0% 0

1020 2040 3060 4080 5100 6120 7140 8160 9180 10200

Length [feet]

Figure 6-20 : Dryness gradient for Case 3

100000

Relaxation Time [s]

10000

1000

100

Sample 8

Sample 7 0.1

0.05

0.1 0.15 0.2 0.25 Gaseous Fractional Flow [fraction]

Figure 6-21 : Relaxation time for Case 3

0.3

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.

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 Lagrangeâ&#x20AC;&#x2122;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 Butterworthâ&#x20AC;&#x2122;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.

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 relaxationgradient 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 TwoPhase 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.

[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 Nonequilibrium 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

[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.

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:

ρ = ρ g H g + ρ L H L …………………………………………….……..(A-1) v=

ρ g H g vg + ρ L H L vL ………………………………………………(A-2) ρg H g + ρLH L

V& ………………………………………………………………....(A-3) A

ρg H g ……………………………………………………………(A-4) ρ

The mass flow rate can be written as: m& = ρ g AH g vg + ρ L AH L vL …………………………………………...(A-5)

By rearranging the equation A-5 and combining the equations A-1 and A-2, the next relationships are obtained:

ρv = ρ g H g v g + ρ L H L v L ………….……………………..………..…..(A-6)

ρv =

m& …………………………………………………………...…..(A-7) A

The volumetric flow rate of the multiphase fluid system, the gas phase and the liquid phases can be defined as:

V& = V&g + V&L …………………………………………………………...(A-8) V&g = AH g v g ..………………………………….……………………...(A-9) V&L = AH L v L …..……………………………….………………….....(A-10)

By replacing the equations A-3, A-9 and A-10, the equation A-8 becomes: u = H g v g + H L v L …………………………….………...……....……(A-11)

Assuming that vL=v* and vg= λv* where λ is the slip ratio, the equations A-6 and A-11 take the form:

ρv = (λρ g H g + ρ L H L )v * ……………..……………………………(A-11) u = (λH g + H L )v * …………………….……..……………………..(A-12)

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: v = [1 + x (λ − 1)]v * ……………...……....…………………………..(A-13)

u = [1 + H g (λ − 1)]v * ………………...….…………………………..(A-14)

By combining the equations A-13 and A-14, the flowing or actual velocity takes the form:

v=u

1 + x (λ − 1) …………………………….……………………(A-15) 1 + H g (λ − 1)

The equilibrium density can be defined as:

ρ st =

m& …………………………………………………………….(A-16) Au

By replacing the equations A-7 and A-15 into the equation A-16, the flowing or actual density can be expressed:

ρ = ρ st

1 + H g (λ − 1) 1 + x (λ − 1)

……………..…………………………………(A-17)

The liquid hold up and the void fraction can be defined as a function of the fractional flow of the phases:

HL =

Hg =

λS L

λS L + S g Sg

λS L + S g

…………………………………………………..…(A-18)

……………………………………………………..(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):

v = u ………………………………………………………………...(A-20)

ρ = ρ st …………………………………….………………………...(A-21) H L = S L ……………………………………..………………………(A-22) H g = S g ……………………………………..………………………(A-23)

Note that the slip ratio is equal to one if the phases are flowing at the same velocity.

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.

Bg : [ft3/scf]

Bo : [bbl/stb]

Bw : [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.

T pc = 169.2 + 349.5γ g − 74.0γ g2 ……………………………………….……...(B-1) Ppc = 756.8 − 131.0γ g − 3.6γ g2 ………………………………….…………….(B-2)

Pseudo-reduced Temperature and Pressure. These properties are defined as

follows:

T pr =

T ……………………………………..……………………………....(B-3) T pc

Ppr =

P ………………………………………...……………………..…….(B-4) Ppc 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.

⎛ A A ⎞ A A z = ⎜ A1 + 2 + 33 + 44 + 55 ⎟ ρ pr ⎜ T pr T pr T pr T pr ⎟⎠ ⎝

⎛ A A ⎞ 2 + ⎜ A6 + 7 + 28 ⎟ ρ pr ⎜ T pr T pr ⎟⎠ ⎝

⎛A A ⎞ − A9 ⎜ 7 + 28 ⎟ ρ 5pr ⎜ T pr T ⎟ pr ⎠ ⎝

(

+ A10 1 + A11 ρ

2 pr

)

ρ pr2 T pr3

− A11 ρ 2pr

+ 1 ………………………………………(B-5)

The pseudo-reduced density is given by:

ρ pr = 0.27

Ppr zT pr

………………………………………………………………(B-6)

The eleven constants (A1 to A11) for equation B-5 are defined as follows:

A1 = 0.3265

A7 = −0.7361

A2 = −1.0700

A8 = 0.1844

A3 = −0.5339

A9 = 0.1056

A4 = 0.01569

A10 = 0.6134

A5 = −0.05165

A11 = 0.7210 91

A6 = 0.5475 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: zT ………………………………………….…………………...(B-7) P

B g = 0.0283

Gas density. Equation B-8 states the density of a gaseous hydrocarbon:

ρ g = 10.736

Mw P …………………………………..…………...…………...(B-8) zT

Mw = 28.9625γ g ……….………………………….…………………….…...(B-9)

Gas viscosity. The Lee et al. correlation is used for estimating the gas viscosity. ⎛ ρg ⎞ 1 ⎟⎟ X 1 ⎜⎜ ⎝ 62.36 ⎠ Y

μ g = 10−4 K1e K1 =

…………………………...……………..…………(B-10)

(9.379 + 0.01607Mw)T 1.5 ……………………………….……….…...(B-11) (209.2 + 19.26Mw + T )

X 1 = 3.448 +

986.4 + 0.01009 Mw ……………………………..…....…..….(B-12) T

Y1 = 2.447 − 0.2224Mw …………………………..………….……………..(B-13)

Gas solubility of saturated oil. The gas solubility (Rb) is estimated at bubble-

point conditions using: V&Gs Rb = s …………………………………………………………….……..(B-14) V&O

API gravity. The API gravity is defined as:

γ API =

141.5

γo

− 131.5 ………………………………..…………………...…..(B-15)

Oil compressibility. The oil compressibility at pressures above the saturation

pressure is estimated using the Vasquez-Beggs correlation. co =

5 Rb + 17.2(T − 460 ) + 12.61γ API − 1433 ....................................................(B-16) 10 5 P

Gas solubility in oil. The Standing correlation states that:

Rg / o

⎡⎛ P ⎤ ⎞ = γ g ⎢⎜ + 1.4 ⎟10 X 2 ⎥ ⎠ ⎣⎝ 18.2 ⎦

1.2048

………………………………………....(B-17)

X 2 = 0.0125γ API − 0.00091(T − 460) ………………………………....….....(B-18)

Saturation Pressure. The bubble-point pressure (Pb) is obtained by solving for

pressure in the Standing correlation.

⎡⎛ R Pb = 18.2 ⎢⎜ b ⎢⎜⎝ γ g ⎣

⎞ ⎟ ⎟ ⎠

0.83

⎤ 10 X 2 − 1.4⎥ …………………………………….………(B-19) ⎥ ⎦ 93

Oil formation-volume-factor. The Standing correlation for saturated oils is used:

⎡ ⎛γ Bo = 0.0012⎢ R g / o ⎜ o ⎜γ ⎢ ⎝ g ⎣

⎞ ⎟ ⎟ ⎠

0.5

⎤ + 1.25(T − 460)⎥ ⎥ ⎦

1.2

+ 0.9759 ………………………………...………………………………..(B-20)

The oil formation-volume-factor at above-bubble-point pressures is computed as follows:

Bo = Bo,b e − co ( P − Pb ) ……………………………….……………………...(B-21) The oil formation-volume-factor at the bubble-point pressure (Bo,b) is estimated B

by replacing the gas solubility at bubble-point conditions in Eq. B-20.

Oil Viscosity. The Beggs-Robinson correlation for saturated oils is used:

μ o = X 3 μ od Y

………………………………………..……………………(B-22)

log(log(μ od + 1)) = 3.0324 − 0.02023γ API − 1.163 log(T − 460) ……….………………………………..……………(B-23)

X 3 = 10.715( R g / o + 100) −0.515 ……………………………………………….(B-24) Y3 = 5.44( R g / o + 150) −0.338 …………………………...…………...…………(B-25) The Vasquez-Beggs correlation for under-saturated oils is used: ⎛P μ o = μ o ,b ⎜⎜ ⎝ Pb

⎞ ⎟⎟ ⎠

………...…………………………….…………………….(B-26)

X 4 = 2.6 P 1.187 e −11.513−0.0000898 P …………………………………..…………(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

(

)

Rw / o = K 5 + X 5 P + Y5 P 2 Z 5 …………………………………….……….…(B-28) K 5 = 2.12 + 3.45 × 10 −3 (T − 460)

+ 3.59 × 10 −5 (T − 460) 2 ……………….…………………………………..(B-29) X 5 = 0.0107 − 5.26 × 10 −5 (T − 460)

+ 1.48 × 10 −7 (T − 460) 2 ……………………………………………….…..(B-30) Y5 = −8.75 × 10 −7 + 3.9 × 10 −9 (T − 460)

+ 1.02 × 10 −11 (T − 460) 2 ..............................................................................(B-31) Z 5 = 1 − [0.0753 − 0.000173(T − 460)]ξ ………………………………….....(B-32)

Water viscosity. The water phase is considered to have some level of salinity.

Therefore, the McCain correlation is applied.

μ w = X 6 (T − 460) Y K 6 ……………………………………....…….... …....(B-32) 6

X 6 = 109.574 − 8.40564ξ + 0.313314ξ 2

+ 8.72213 × 10 −3 ξ 3 …………………………………….…………...……(B-33)

Y6 = −1.12166 + 2.63951 × 10 −2 ξ − 6.79461 × 10 −4 ζ

− 5.47119 × 10 −5 ξ 3 + 1.55586 × 10 −6 ξ 4 .......................................................(B-34) K 6 = 0.9994 + 4.0295 × 10 −5 P

+ 3.1062 × 10 −9 P 2 ……….…………………………………..……………(B-35)

Water

formation-volume-factor.

The

water

formation-volume-factor

computed using the McCain correlation. Bw = (1 + X 7 )(1 + Y7 ) …………………………………..……………………(B-36) X 7 = −1.00010 × 10 −2 + 1.33391 × 10 −4 (T − 460)

+ 5.50654 × 10 −7 (T − 460) 2 ………………………………………….……(B-37) Y7 = −1.95301 × 10 −9 P (T − 460)

− 1.72834 × 10 −13 P 2 (T − 460) − 3.58922 × 10 −7 P − 2.25341 × 10 −10 P 2 …………………………………..(B-38)

Specific heat. The Gambill correlation is used for an estimation of the specific

heat for hydrocarbon mixtures. Cp =

0.338 + 0.00045(T − 460)

γo

…………………………………..………...(B-39)

Overall heat transfer. The Shiu and Beggs correlation is used for computing the

overall heat transfer for producing pipes.

C p m& UπD

= 0.0149m& 0.5253 (12 D) −0.2904 γ API

0.2608

γ g 4.4146 ρ 2..9303 …………………………………………….………………..(B-40)

Friction factor. The explicit approximation for the Colebrook equation

developed by Zigrang and Sylvester is used. 1 fM

⎡ 2ε 5.02 ⎛ 2ε 13 ⎞⎤ = −2 log ⎢ − log⎜ + ⎟⎥ ………………………………....(B-41) ⎝ 3.7 D Re ⎠⎦ ⎣ 3.7 D Re

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

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