The Pennsylvania State University

The Graduate School

College of Energy and Mineral Engineering

ANALYTICAL MODELING AND THERMODYNAMIC ANALYSIS OF

THERMAL RESPONSES IN HORIZONTAL WELLBORES

A Thesis in

Energy and Mineral Engineering

by

Muhamad Hadi Zakaria

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

May 2012

ii

The thesis of Muhamad Hadi Zakaria was reviewed and approved by the following:

Luis F. Ayala

Associate Professor of Petroleum and Natural Gas Engineering

Thesis Advisor

Yaw D. Yeboah

Professor and Department Head of Energy and Mineral Engineering

Li Li

Assistant Professor of Petroleum and Natural Gas Engineering

R. Larry Grayson

Professor of Energy and Mineral Engineering

Graduate Program Officer of Energy and Mineral Engineering

iii

ABSTRACT

A comprehensive thermodynamic analysis is conducted to aid the interpretation of

thermal responses from horizontal wells. The analysis shows that using values of Joule

Thomson (JT) coefficient of the flowing fluid as a tool to predict heating and/or cooling

effects in a horizontal wellbores can lead to significant misinterpretations. It is shown that

fluid thermal behavior cannot be solely tied to JT behavior given that horizontal wellbore

flow can be far from isenthalpic. The thermal response model proposed by this study

starts with Euler‘s fluid flow governing equations in their one dimensional, single-phase

form. By utilizing all three mass, momentum and energy balance equations, and

thermodynamic considerations, pressure and temperature responses are coupled. A

solution procedure involving a semi-analytical approach is proposed for the prediction of

temperature and pressure traces during fluid flow in a horizontal wellbore. In this study,

steady-state, single-phase flow is considered during parametric studies. It is observed that

fluid temperature response mimics the pressure profile in the wellbore regardless of the

fluid type and sign of the JT coefficient. Water flowing temperature is shown to be the

least sensitive to fluid flow conditions, closely followed by oil. Gas flowing temperature

exhibits the largest sensitivity to flow conditions. Overall behavior is a strong function to

the isentropic thermal coefficient of the fluid. This finding is used to clarify the common

misconception that employs the JT effect to explain horizontal wellbore thermal

responses. Parametric studies are also conducted to assess how the temperature and

pressure profile change in response to changes in well flowrate, inclination, completion,

radius and roughness.

iv

TABLE OF CONTENTS

LIST OF FIGURES ........................................................................................................... vi

LIST OF TABLES ............................................................................................................. ix

NOMENCLATURE ........................................................................................................... x

ACKNOWLEDGEMENT ................................................................................................ xii

CHAPTER 1 INTRODUCTION ........................................................................................ 1

CHAPTER 2 LITERATURE REVIEW ............................................................................. 4

CHAPTER 3 PROBLEM STATEMENT ........................................................................... 7

CHAPTER 4 THERMODYNAMIC PROCESS IN WELLBORE SYSTEM ................. 11

4.1 Governing Equations in Wellbore ........................................................................... 11

4.2 Steady state evaluation of thermal response equation ............................................. 16

4.3 Solution Procedure .................................................................................................. 18

4.2 Thermal Response Equation in Pipeline Flow ........................................................ 23

CHAPTER 5 RESULTS AND DISCUSSIONS............................................................... 25

5.1 Single Phase Non-Isothermal Flow Problem .......................................................... 25

5.2 Correlation between Thermal Coefficient and Joule Thomson Coefficient. .......... 34

5.3 Flowrate Effect ........................................................................................................ 39

5.4 Inclination Effect ..................................................................................................... 41

5.5 Type of Well Completion Effect ............................................................................. 43

v

5.6 Wellbore Radius Effect ........................................................................................... 46

5.6 Wellbore Roughness Effect ..................................................................................... 48

CHAPTER 6 CONCLUSIONS AND SUGGESTIONS .................................................. 50

REFERENCES ................................................................................................................. 53

APPENDIX A DERIVATION OF THERMODYNAMICS EQUATION IN

WELLBORE ..................................................................................................................... 55

APPENDIX B THERMODYNAMIC FLUID MODEL .................................................. 61

APPENDIX C DETERMINATION OF FRICTION FACTOR FOR MOMENTUM

EQUATION ...................................................................................................................... 72

APPENDIX D VALIDATION PROCESS ....................................................................... 74

D.1 Properties validation for Gas .................................................................................. 74

D.2 Properties validation for Oil ................................................................................... 77

APPENDIX E SAMPLE CALCULATION ..................................................................... 80

vi

LIST OF FIGURES

Figure 3-01 Predicted Temperature profile for different fluid types

(Yoshioka, 2005) ………………………………………………………..08

Figure 3-02 Thermal Response for Safah C producing well (Brown et al. 2003) ……09

Figure 3-03 Temperature profile along horizontal well on heavy oil production

(Foucault et al. 2004)…………………………………………………….09

Figure 4-01 Differential volume element for a wellbore in solution procedure with toe

condition…………………………………………………………………21

Figure 4-02 Solution Procedure Flow Chart..………………………………………....22

Figure 5-01 Phase envelope for oil generated by PVTsim20…………………………27

Figure 5-02 Phase envelope for oil generated by PVTsim20…………………………27

Figure 5-03 Flowrate of all fluids for baseline case…………………………………..29

Figure 5-04 Velocity of all fluids for baseline case…………………………………...30

Figure 5-05 Pressure profile of all fluids for baseline case…………………………...30

Figure 5-06 Temperature profile of all fluids for baseline case ……………………...31

Figure 5-07 The contribution of temperature gradient at the last iteration …………...33

Figure 5-08 Joule Thomson inversion curve of pure methane (Ayala, 2012) ………..34

Figure 5-09 Joule Thomson of pure methane (for Pr =0.5 to Pr=15) ………………...35

Figure 5-10 Joule Thomson of pure methane (for Pr = 0.5 to Pr=15) ………………..35

Figure 5-11 Isentropic Thermal Coefficient of pure methane (for Pr=0.5 to Pr=15)…36

vii

Figure 5-12 Difference between Joule Thomson coefficient and isentropic thermal

coefficient of pure methane (for Pr=0.5 to Pr=15) ……………………...37

Figure 5-13 Accumulated flowrate for each case …………………………………….39

Figure 5-14 Velocity for each case for flowrate study ……………………………….40

Figure 5-15 Pressure for each case for flowrate study ……………………………….40

Figure 5-16 Temperature profile for each case for flowrate study …………………...41

Figure 5-17 The pressure profile due to the wellbore inclination ……………………42

Figure 5-18 The temperature profile due to inclination (fixed toe temperature) ……..42

Figure 5-19 Flowrate for well completion case ……………………………………....44

Figure 5-20 Velocity profile for well completion case ……………………………….44

Figure 5-21 Pressure profile for well completion case ……………………………….45

Figure 5-22 Temperature profile for well completion case …………………………..45

Figure 5-23 Velocity profile for wellbore radius case study ………………………....47

Figure 5-24 Pressure profile for wellbore radius case study ………………………....47

Figure 5-25 Temperature profile for wellbore radius case study …………………….48

Figure 5-26 Pressure profile for wellbore roughness case study …………………….49

Figure 5-27 Pressure profile for wellbore roughness case study …………………….49

Figure D-01 Gas density value comparison between PVTsim 20 and model

calculation……………………………………………………………......75

Figure D-02 Gas enthalpy comparison between PVTsim 20 and model calculation.....75

Figure D-03 Gas thermal compressibility comparison between PVTsim 20 and model

calculation …….…………………………………………………………76

viii

Figure D-04 Gas thermal compressibility comparison between PVTsim 20 and model

calculation…….………………………………………………………….76

Figure D-05 Oil enthalpy comparison between PVTsim 20 and model calculation......78

Figure D-06 Oil thermal compressibility comparison between PVTsim 20 and model

calculation …….…………………………………………………………78

Figure D-07 Oil thermal compressibility comparison between PVTsim 20 and model

calculation…….………………………………………………………….79

Figure E-01 Radial influx for gas production ………………………………………...80

Figure E-02 Radial influx for liquid production ……………………………………...81

Figure E-03 Density of fluid at the last iteration ……………………………………...81

Figure E-04 Viscosity of fluid at the last iteration ……………………………………82

Figure E-05 Moody friction of fluid at the last iteration ……………………………...82

Figure E-06 Enthalpy of fluid at the last iteration …………………………………….83

ix

LIST OF TABLES

Table 5-01 Case Study Description (Baseline Case) ……………………………….26

Table 5-02 Mole Fraction of Component for Oil and Gas ………………………….26

Table 5-03 Isentropic thermal coefficient and Joule Thomson values for wellbore

fluids …………………………………………………………………….32

Table B-01 Passut and Danner (1972) coefficients ………………………………….65

Table B-02 Properties for pure components …………………………………………70

Table B-03 Peng-Robinson (1976) binary interaction coefficients ………………….71

x

NOMENCLATURE

A : Area, ft2

: Isobaric Heat Capacity, BTU/lbm R

: Isochoric Heat Capacity, BTU/lbm R

d : Wellbore Diameter, ft

e : Internal Energy per Unit Mass, BTU/lbm

F : Force, lbf

: Friction factor [-]

g : Acceleration factor, ft/s2

h : Fluid Specific Enthalpy at Wellbore conditions , BTU/lbm

: Fluid Specific Enthalpy at Reservoir conditions, BTU/lbm

: Energy unit conversion (psia ft3/BTU)

: Heat Capacity Ratio [-]

L : Length of Wellbore, ft

M : Mass Rate, lbm/s

MW : Molecular Weight, lbm/lb-mole

: Pressure, psia

Re : Reynolds number, [-]

S : Specific Entropy, BTU/lbm

T : Temperature, R

: Time, s

xi

U : Overall Heat Transfer Coefficient, BTU/ ft2 hr F

: Velocity, ft/s

W :Mass rate, lbm/s

GREEK:

: Coefficient of thermal expansion, 1/F

: Radial mass influx, lbm/ft3 s

: Joule Thomson coefficient F/psi

: Isentropic thermal coefficient F/psi

: Isothermal compressibility, 1/psi

ρ : Density lbm/ft3

: Viscosity, cp

: Shear stress tensor, lbm/ft-s2

: Coefficient of the cubic Peng-Robinson EOS in terms of Z

: Mixture viscosity parameter (Lohrenz et al. correlation) (ft-s/lbm)

SUBSCRIPT

a : phase ‗a‘

g : Gas phase

: Segment number

L : Liquid phase

m : Mixture

R : Reduced property

c : Critical point

xii

ACKNOWLEDGEMENT

I am grateful to the Lord Almighty, who has bestow me such a wonderful life and

guided me throughout my life in becoming who I am today.

I would like to express my sincere gratitude and appreciation to Dr Luis F.

Ayala, whose immense knowledge, encouragement, patient, supervision and support

from the preliminary to the concluding level enabled me to develop an understanding of

the subject. His guidance helped me in all the time of research and writing of this thesis. I

could not have imagined having a better advisor and mentor for this thesis. I also extend

my gratitude to Dr. Yaw D. Yeboah and Dr. Li Li for their interest in serving as

committee members.

I dedicate this thesis to my beloved parents Zakaria Hashim, Saadiah Ali and

family in Malaysia and Saudi Arabia. To my father and mother, who never stop believing

and supporting me since the day I was born and also to my siblings who constantly

reminding me about the value of hard work with endless kindness and support. I also

dedicated this thesis to Iiffa Nabilah, who I forever grateful for her never-ending love.

To all my fellow friends in Penn State, my deepest gratitude for the amazing

friendship we have along my path though my academic experience. Finally would like to

give my appreciation to Petroliam Nasional Berhad – PETRONAS – for sponsoring my

studies here in Penn State. Thank you all.

CHAPTER 1

INTRODUCTION

Horizontal wells have progressively become the most commonly used well

architecture in the oil and gas industry due to their ability to provide enhanced

productivity as compared to vertical wells. Early analysis techniques for horizontal

wellbore flow analysis embraced the commonly used isothermal flow assumption, which

has now proven to be a severe limitation as the availability of modern interpretation tools

such as Distributed Temperature Sensing (DTS) has become available. The application of

Distributed Temperature Sensing in modern well operation enables us to monitor the

temperature profile in real time data. This useful tool uses optical fiber optic sensor to

detect pressure, temperature and flow rate distribution along the wellbore and transmit

the data in at instant. Several studies have shown that the DTS can be very beneficial as it

can be used to detect water breakthrough along the wellbore (Wang et al., 2008) and

proven to be cost effective (Salim et al., 2011). As the use of DTS technology becomes

more widespread, the understanding of thermal response of fluid due to its withdrawal

from the reservoir also becomes crucial. The typically embraced isothermal fluid flow

assumption can no longer be used as it completely removes the complexity and influence

of thermodynamic principles in the process. Several studies have attempted to describe

the how DTS surveys should be interpreted–with many of them using Joule Thomson

cooling or heating effects to describe expected fluid behavior–but there only few that

2

have provided a complete analytical model for this thermal response (Zhuoyi et al., 2009,

Pourafshary et al., 2007, Livescu et al., 2007).

Chapter 2 of this thesis – Literature Review – provides a complete description of

some of the most prominent approaches currently available to describe the thermal

response of fluid in the wellbore. Many of these studies explicitly or implicitly conclude

that wellbore thermal responses can be directly explained using the value of fluid‘s Joule

Thomson Coefficient. Chapter 3 further explains the motivation of this study and details

the problem that needs to be addressed. Chapter 4 details the thermodynamic and

hydrodynamic development that shows that wellbore responses do not follow isenthalpic

paths. A mathematical predictive formulation is also proposed in this chapter, including

governing equations and proposed solution procedure. The model uses the Peng-

Robinson Equation of State (PR-EOS), an equation of state for hydrocarbon

multicomponent fluids originally based on the Soave-Redlich-Kwong (1974) equation,

for the prediction of fluid properties. The Peng Robinson EOS is popular in the petroleum

industry because it was originally derived for hydrocarbon mixtures. The use of a modern

EOS also allows to fully capture the thermodynamic process and thermal-pressure

response inside and in the vicinity of the wellbore region through a rigorous enthalpy

calculation should be implemented. In previous reservoir and wellbore flow models, the

pressure dependency of fluid properties such as enthalpy is commonly neglected for

simplification. Some reservoir fluids, however, can have significant enthalpy dependence

on pressure and temperature changes in the system. Chapter Five analyzes the coupling

of the thermal and pressure responses for several single-phase systems and discussions

3

are provided to explain the physical significance of these behaviors. Finally, Chapter Six

presents the main conclusions that have been drawn from this study and suggestions for

future work are also thoroughly discussed.

4

CHAPTER 2

LITERATURE REVIEW

One of the earliest models available for the interpretation of well temperature

solutions is that of Ramey (1962). Ramey (1962) presented a method of predicting

temperature distribution of a vertical well that incorporated the heat transfer between

wellbore and reservoir. The model is based on several assumptions: energy accumulation

is neglected, the effects of heat loss from the wellbore are ignored and working fluids

were limited to ideal gases and incompressible liquids. Using thermodynamic

considerations, Coulter and Bardon (1979) proposed an explicit equation for thermal

behavior prediction for flowing temperature in surface pipeline that can be used for both

liquid and natural gas flow. The proposed non-isothermal fluid flow equation in pipeline

flow integrated the concept of Joule Thomson Coefficient (JTC) into pipe analysis and it

can be used for the analysis of single phase or two phase flow. These studies successfully

provided the general framework for flowing temperature prediction in vertical wells and

surface pipes, respectively, and are often cited by other authors in the matter.

Hassan and Kabir (1994) presented analytical expressions for computing a time

dependent fluid temperature at any point in vertical well during fluid withdrawal both in

drawdown and buildup test Their method distinguished between two models: the

generalized and the flow-pattern based model. Both models incorporated the concept of

temperature diffusivity of the reservoir and also included the heat transfer by the tabular

5

and cement sheaths in the wellbore by introducing the thermal storage parameter. Hassan

and Kabir (2002) also presented many models of wellbore flow for predicting fluids‘

pressure and temperature behavior. This includes models for single/multi-phase flow,

multiple strings wellbore and production/drilling operation.

Yoshioka et al. (2005) developed a fully coupled reservoir/horizontal wellbore

model with linear radial-flow accounting both pressure and temperature behavior. In this

model, the temperature change of fluid is reported to be mainly depended on the

production rate and the fluid properties. A heavy dependency of thermal response on

Joule-Thomson effect is discussed and presented as one of the main factor driving

thermal changes. By applying Yoshioka et al. (2005) work, Zhuoyi and Ding (2009)

presented a prediction model to predict the flow profile of horizontal well by downhole

pressure and Distributed Temperature Sensing (DTS) data for water reservoir. In this

attempt, the model was shown to partially match the thermal response of the DTS data for

oil production

S.Livescu et al. (2009) also proposed a fully-coupled thermal in multilateral well

model for reservoir simulation. The study suggested that conductive heat transfer should

be ignored and the Joule Thomson effect is used throughout the formulation to

incorporate the enthalpy changes of fluid in the wellbore system.

Shirdel and Sepehrnoori (2009) present a steady-state, non-isothermal, fully-

coupled compositional wellbore/reservoir simulator to stimulate fluid flow in horizontal

well. The study suggested that since the time steps in reservoir model are in the order of a

day, the wellbore system reaches steady state at the end of the simulation time.

6

Spindler (2011) developed an analytical model for wellbore temperature

distribution. The model analyzed Hassan and Kabir (2005) transient in different

perspective to compare term quantitatively through nondimensionalization. Thus, an

improved model that uses explicit solution and also includes all the fluid interactions in

the wellbore is presented.

Radespiel (2010) developed a robust numerical model that can be used for

thermal response prediction in horizontal wellbore which is one of the first attempt

provide thermal interpretation in term of mass radial influx and velocity gradient. An

analytical-simplified theoretical model in order to explain the temperature behavior in the

horizontal wellbore for a single phase fluid was also developed. The study suggested that

a thermal coefficient (dependent on fluid‘s speed of sound) can be a major factor in the

prediction of temperature profile of the wellbore fluid. One of the interesting conclusions

mentioned is that temperature was always expected to drop in the same direction as

pressure, in spite of what is believed by many in the literature in terms of Joule-Thomson

cooling effects.

7

CHAPTER 3

PROBLEM STATEMENT

One of the potential pitfalls of currently available models for non-isothermal flow

analysis in horizontal wellbore systems is the misuse of common assumptions made

during the development of other fluid models. For example, almost all temperature

prediction equations for this case are basically derived from pipeline equations. On the

surface, this approach appears to be a valid choice because horizontal wellbore systems

can be roughly visualized as a pipeline placed in the middle of the reservoir. However,

there are few significant differences between the thermodynamically processes

experienced by the fluid in the environment. A careful review of these differences should

be undertaken to avoid potentially misleading conclusions.

Since many currently available wellbore models are mainly based on pipeline

modeling equations, the resulting temperature models tends to inherit their heavy

thermal-response dependency on the value of the Joule Thomson coefficient of the fluid.

Figure 3-1, taken from one wellbore study, shows the differences in predicted thermal

response of a horizontal wellbore when the flowing fluid changes from oil, to water, and

natural gas. The prediction follows a similar pattern as one would predict for fluid flow in

pipelines: fluids with negative Joule Thomson coefficients (water, oil) are expected to

heat with pressure drop, but fluids with positive Joule Thomson coefficients would be

expected to cool with pressure drop (natural gases). The Joule Thomson cooling/heating

8

argument has also been used by other authors to explain field-observed thermal response

of horizontal wellbore and pipelines.

Figure 3- 1 Predicted Temperature profile for different fluid types (Yoshioka, 2005)

Based on pipeline observations, gases are thus always expected to experience cooling

upon expansion while liquids would experience thermal heating upon expansion due to

the opposite signs of their Joule Thomson coefficient values. However, reported field

DTS data from horizontal wellbores in oil reservoirs seem to contradict these predictions:

field measurements show oil horizontal wellbores cooling as pressure decreases. Brown

et al. (2003) presented the temperature responses on producing well in Oman using DTS.

The temperature resposes of one of the wells - Safah C - can be shown in Figure 3-2.

Foucault et al. (2004) also show a similar thermal response in one of horizontal wells

operating in extra heavy oil field in Orinoco Belt of Venezuela (Figure 3-3).

9

Figure 3- 2 Thermal Response for Safah C producing well (Brown et al. 2003)

Figure 3- 3 Temperature profile along horizontal well on heavy oil production

(Foucault et al. 2004)

10

Both of these independent field studies show that the fluid inside the wellbore can

experience a significant temperature drop during oil production, which seems to

contradict predictions from currently available wellbore models and the expected heating

that oils should experience during expansion due to their negative Joule Thomson

coefficients (JTC). Brown et al. (2003) attempted to explain the wellbore cooling of

Figure 3-2 using JTC arguments by indicating that the produced oil contained significant

amounts of natural gas.

Unlike pipeline systems, an open wellbore system has a radial mass influx/efflux

coming in to (producer) or out of (injector) the wellbore (―porous pipe‖). The exchange

between the two systems is basically an exchange of mass and enthalpy, which should be

considered by the models and might influence wellbore responses to move away from an

isenthalpic path. Additional areas for potential improvement for currently available

wellbore models can be found in the calculation of mass and energy exchange between

the wellbore and reservoir system. The energetic exchange should be rigorously

calculated via equation-of-state-based enthalpy calculations that consider actual reservoir

and wellbore pressure and temperature conditions. Few models have used, instead, an

overall heat transfer coefficient U to quantify this energetic exchange (Shirdel and

Sepehrnoori, 2009, Livescu et al., 2009). This can be consider as another legacy item

borrowed from pipeline modeling, where the system-surrounding energetic exchange

involves heat lost due to convection and conduction through the system materials – pipe,

tubing, reservoir and space between the materials. In this study, a semi-analytical

approach is explored to address the concerns outlined above.

11

CHAPTER 4

THERMODYNAMIC PROCESS IN WELLBORE SYSTEM

4.1 Governing Equations in Wellbore

In this chapter, the governing equations for the wellbore system are considered. One-

dimensional governing equations for single phase fluid flow in wellbore which

incorporating the mass coming in or out radially from or to the reservoir are

implemented. Appendix A shows a detailed derivation of the equations discussed here,

while an abridged derivation is presented in the chapter. The development starts by

stating the Euler‘s conservation equations for mass, momentum and energy which can be

written as follows:

(4.1)

(4.2)

(

)

(

) (4.3)

where is the wellbore fluid velocity, is the wellbore pressure, is fluid enthalpy, e is

the internal energy, and the radial mass influx denoted as . These three equations are

arranged so that the right hand side of the equations represent the source terms of

equations. These include radial mass influx/efflux in or out the wellbore ( ), momentum

change due to the action of surface (shear stress) and body (gravity) forces (F),

incoming/outgoing energy due to mass exchange ( ), and work done by surface and

12

body forces (uF). Equations (4.1) to (4.3) are written in their conservative forms; i.e., in

terms of the conservative variables and . These equations are routinely

implemented as the governing equations for pipeline flow (Ayala and Adewumi, 2003;

Stadke, 2004), for which case there is no radial influx/efflux . In addition, the

radial heat exchange of conductive and convective between fluid and surrounding; which

normally expressed by the overall heat transfer coefficient is being replaced by the

enthalpy change experienced by the fluid . The energy equation (4.3) neglects heat

conduction effects along the wellbore as it would be largely overwhelmed by energy

convection within the wellbore.

By expanding all three conservation equations, they can be re-expressed in terms of their

non-conservative forms as shown below:

(4.4)

(4.5)

*

+

(4.6)

In fluid dynamics applications, fluid motion can be expressed in terms of material

derivatives. Material derivatives are used to describe the evolution of a property of a

given fluid parcel in time, as it moves along its pathline. Hence, in term of material

derivatives, and substituting relations in (4.1) and (4.5), (4.6) can be expressed as:

*

+

(4.7)

Equation (4.7) describes how the relation of the energy change experienced by the fluid

in the wellbore system. Equation (4.7) shows that wellbore fluids should not be expected

13

to experience iso-energetic process as the radial influx/efflux can never be zero in

wellbore system ). In order to see the entropy equation in the horizontal wellbore,

thermodynamic identity of

can be used in equation (4.7). This would

yield:

*

+ (4.8)

By observing equation (4.8), the wellbore system cannot be expected to experience truly

isentropic changes because there will always radial mass influx/efflux ).

Furthermore, entropy change in wellbore system is seen not to be directly dependent on

the velocity gradient.

Expanding equation (4.6) and substituting enthalpy definition ( ), one can

obtain relation of the enthalpy change as:

*

+

(4.9)

Equation (4.9) shows that wellbore flow cannot be truly isenthalpic because and

there is an additional contribution of the pressure gradients in this equation. The

relationship between changes of enthalpy with changes of pressure and temperature can

be expressed by the following:

(4.10)

where is the Joule-Thomson coefficient (JTC). By combining thermodynamics

identity from (4.9) with enthalpy expression in (4.10), one can re-express the thermal

response equation as:

14

*

+

(4.11)

Equation (4.11) is the model proposed by this study for the interpretation of horizontal

wellbore thermal responses. It is important to note that although they do not explicitly

appear in this equation, elevation effects are still being considered by the model. This is

because the force due to elevation (inclination of horizontal wellbore) is accounted when

in application of the momentum equation to calculate the pressure profile (forcing

function ―F‖).

The nature of the thermodynamic coefficient in front of the pressure material derivative

in Equation (4.11) can be illustrated by analyzing its relationship to isentropic processes.

The dependency of temperature and pressure drop for any entropy changes can be written

as:

(

) (

)

(4.12)

In order to find substitution of (

) , the thermodynamic identity

is

expressed to be at constant specific volume and dividing it by to obtain:

(

)

(4.13)

Furthermore, by the recognizing thermodynamic identity below from thermal volumetric

expansion term definition:

(

)

(

)

(4.14)

15

Substituting (5.1.3) and (5.1.4) into (5.1.2), the change of temperature due to pressure

change at constant entropy (

) can be expressed as the following:

(

)

(4.15)

where the following thermodynamic identity has been used:

(4.16)

Therefore, Equation (4.11) can be alternatively written in terms of the isentropic thermal

coefficient , which will result into:

*

+

(4.17)

or,

*

+

(4.18)

Equations (4.11), (4.17) and (4.18) show that the wellbore temperature behavior is not

directly influenced by the sign of Joule Thomson Coefficient of the fluid. Note that in a

case of production where and

are positive value whereas both have negative value

for injection.

Contrasting with most approaches in developing non-isothermal fluid flow in

wellbore system, the energy exchange between the surroundings and the wellbore is not

characterized through the use of an overall heat transfer coefficient U. In general, the heat

transfer exchange with the environment for a pipeline system is typically expressed using

U as follows:

(4.19)

16

where is the surrounding temperature and W is the total mass flowrate in the system.

Based on the equation, U can simply defined as the characteristic of a system that take

into account the heat interaction between the fluid system and the environment around it

– pipe or tubing material and reservoir. There are several ways to calculate or estimated

U. For example, Hasan and Kabir (1994) use the expression that involves conductivity of

all materials in the pipes and tubings. However the best way of estimating U-coefficient

is thru the actual flow test data. Although this approach is correct for a pipeline system,

the same thing cannot be said for open sections in a wellbore system. This is because the

energy exchange due to radial convection of mass influx/efflux is much more dominant

than conductive heat interaction between fluid and the environment. Thus the application

of a U coefficient would not reliably represent the energetic exchange with the wellbore

system as U never considers any radial mass influx contribution in the wellbore system.

4.2 Steady state evaluation of thermal response equation

In typical production and injection cases, and after the initial transients have

dissipated, the accumulation terms for mass, momentum, and energy provide an overall

small contribution to the overall pressure, velocity, and temperature response along the

wellbore. Thus a steady state condition inside a wellbore can be considered as a valid

approximation unless there is a perturbation of flow inside the wellbore. Thus equation

(4.11) can be reduced into:

*

+

(4.20)

17

Conventionally, the wellbore model always has positive direction from heel to toe. Thus

for both production and injection cases, the velocity always be has opposite sign

compared to and

. If , and because every other coefficient is bound to be

positive, the sign of the RHS of this equation is solely dependent on the signs of the

pressure derivative since the contribution of kinetic energy in the mass term is proven to

be very small compared to other terms. This means that the temperature profile is bound

to mimic the direction of pressure response under these conditions. However, if there is a

significant change between fluid enthalpy from environment and wellbore ( ), the

temperature profile will be strongly influenced by both terms in the equation.

At steady state, the one-dimensional governing equations for fluid flow in wellbore

subjected to incoming mass (production) or outgoing mass (injection) exchange can be

written as:

(4.21)

Similarly, the momentum balance of a wellbore can be expressed as:

(4.22)

All analytical equations (4.20) to (4.21) can be solved by finite differences. The

procedure can prove iterative since the equations are nonlinear and fluid thermodynamics

properties are dependent of both temperature and pressure. At the first iteration, the

model assumes pressure and temperature for every segment. Then, using a fixed radial

flow rate for every segment fluid velocity is calculated using mass balance equation.

Momentum balance and thermal response equations then can be applied to calculate the

18

pressure and temperature profile in the wellbore. Velocity, pressure and temperature will

be updated until these parameters are converged. At the boundary (toe), we assume there

is no heat or flow transfer. Section 4.3 explains in further detail about this solution

procedure.

These equations of the thermal and pressure response are also applicable to

inclined and vertical well as well. This is because applying the equation for these well

orientations does not break any assumption in the model development. However, one

should consider the temperature change due to the depth. By knowing the temperature at

datum, the formation temperature can be expressed as:

(4.23)

where G is the geothermal temperature gradient whereas Z is the vertical distance from

the reference. For closed wellbore section (no mass influx), the contribution of heat

exchange with the environment via overall heat coefficients (U) must be considered.

4.3 Solution Procedure

This section explains the numerical procedure and the proposed equations for steady state

modeling of single phase flow in horizontal wellbores. From equation (4.4), the steady

state equation for mass balance equation can be written as:

(4.24)

Above equation can be expressed in forward finite difference method. This step would

yield:

19

(4.25)

Thus, making the velocity of the investigated segment as subject, equation (4.25) can be

solved using the following expression:

(4.26a)

where:

(4.26b)

(4-26c)

Using the same principle, the momentum balance equation (4.22) at steady state can be

represented as following:

(4.27)

Note that the forces (F) involve in momentum balance are contributed by the shear stress

between the fluid and wall and the elevation (Fw) and the potential energy changes in the

system (Fg). These two forces can be calculated using the formulation below;

(4.28a)

(4.28b)

where:

(4.28c)

Note that the is the friction Moody‘s factor and calculated using Chen‘s friction factor

equation (Chen, 1979). In order to use Chen‘s friction factor equation, the Reynolds

number must be determined first. For this paper, the viscosity of oil is calculated using by

20

Lohrenz, Bray and Clark (1964) while viscosity of gas is determined by Lee, Gonzalez

and Eakin (1966) method. Therefore, by discretizing steady state momentum equation,

one can solve for pressure of the fluid at each segment by the following equation:

(

) (4.29a)

or,

(4.29b)

where:

–

(4.29c)

Finally, thermal response equation at steady state condition (4.20) is rewritten as using

forward finite difference as:

[

]

(4.30a)

Thus

(4.30b)

[

] (4.30c)

(4.30d)

Where is the energy unit conversion ( . All other fluid

properties in this paper are calculated using Peng-Robinson Equation of State (PR-EOS)

which will be described in detail in Appendix B. In order solve these equations

simultaneously; a flow rate entering the wellbore per segment of the wellbore is

generated first. The temperature and pressure profile are assumed first to obtain the

21

velocity profile from equation (4.17). Then the pressure profile is obtained using

equation (4.29) and both procedures will be repeated until the pressure is converged.

After that we calculate the temperature equation using (4.30) and the mentioned

procedure will be repeated until the temperature reach convergence. The flow chart of

this procedure can be shown of Figure 4-2.

𝒒𝒕𝒐𝒆 𝟎; 𝑼𝒕𝒐𝒆 𝟎; 𝑻𝒕𝒐𝒆 𝑻𝑹; 𝑷𝒕𝒐𝒆 𝑷𝑹

Figure 4- 1 Differential volume element for a wellbore in solution procedure with conditions at the toe segment

22

Test temperature

convergence

Yes

Write results

No

Calculate pressure using

momentum balance

Test pressure

convergence

Calculate temperature

using energy balance

Yes

No

Assume pressure for

each segment

Calculate fluid properties

using EOS

Calculate velocity using

mass balance

Assume temperature for

each segment

Input Data

Figure 4- 2 Solution Procedure Flow Chart

23

4.2 Thermal Response Equation in Pipeline Flow

In this section, the thermal responses equation for pipeline flow is presented for

comparison purposes. This derivation is crucial to see how the same energy equation can

be collapse differently for pipeline case compared to wellbore system. At steady state,

Euler‘s equation (4.1) to (4.3) can be written for pipeline flow as:

(4.31)

(4.32)

(

) (4.33)

Note that in pipeline system, there no radial mass influx/efflux coming in or out. The

source term in the equation (4.32) still contributed by the shear stress and the gravity-

similar to wellbore equation. The q in equation (4.33) takes into account the heat

exchange between fluid and the system environment and uses the concept of overall heat

coefficients (U). Expanding equation (4.33) and applying equation (4.31) results in:

*

+ (4.34)

In pipeline systems, the shear stress contribution in enthalpy is small and normally

neglected. Thus the enthalpy equation can be written as:

(4.35)

or,

(

) (

) (

)

(4.36)

24

These equations shows that enthalpy changes in the pipeline equation is contributed by

the heat transfer exchange with the environment

, the gravity or elevation effect (

)

and the acceleration contribution of the fluid, (

) . In equation (4.36), the pipeline can

only be in isenthalpic condition if heat transfer, acceleration and elevation are zero

simultaneously. This condition can be realized in insulated pipeline systems. However,

wellbore system presents completely different condition as equation (4.9) show that the

mass influx/efflux contribution is never zero; making the system to be never isenthalpic.

Note that equation (4.36) can also be change into thermal response equation using

relation in (4.10):

( )

(4.37)

where the explicit dependency of pipeline temperature on Joule-Thomson coefficient of

the fluid ( ) is highlighted. Ayala (2012) presented the derivation on how equation

(4.36) can be collapsed in to Coulter and Bardon equation (1979) (Equation 4.37).

25

CHAPTER 5

RESULTS AND DISCUSSIONS

In this chapter, wellbore response model in the previous chapter is used to predict

the pressure and temperature behavior for a number of cases. The chapter starts with a

comparison between different types fluid in single phase flow. Parametric studies are

then conducted to access how the temperature and pressure profile change in response to

changes in well flowrate, inclination, completion, radius and roughness.

5.1 Single Phase Non-Isothermal Flow Problem

As the main objective of this thesis is to clarify and identify the main reason

underlying the thermal behavior of fluid radially coming into the wellbore, the

comparison of case between different types of fluids is important. For this section, three

type of fluid will be used in our model: oil, water and gas. In order to present a

comparison solely based on thermodynamic behavior of these fluids, all other variables

and parameters are assumed the same for all cases. Table 5.1 presents the base case

description that will be used throughout this chapter.

26

Table 5- 1 Case Study Description (Base Case)

Initial Reservoir Pressure (psia) 3900

Initial Reservoir Temperature (F) 190

Length of Wellbore (ft) 4000

Number of Wellbore Segments 50

Wellbore Diameter (in) 2.5

Relative Roughness of Wellbore 0.027

Wellbore Inclination (degree) 0

Geothermal Gradient (F/ft) 0.01

In order make sure that investigated fluids are always in single phase in the system at all

time, the compositions of oil and gas are selected carefully. Figure 5-1 and Figure 5-2

show the phase envelope for each phase whereas Table 5-2 stated the overall molecular

fraction of the fluid used. Based on the two phase envelopes, the initial pressure and

temperature are far away from their respective two phase region and on the respective

single phase region. For the case of water, since its critical point is found around 705 F

and 3198.8 psia, it is always in its aqueous state throughout the system as temperature

changes would not reach values beyond its critical temperature.

Table 5- 2 Mole Fraction of Component for Oil and Gas

Oil Gas

Component mol % Component mol %

C1 0.500 C1 0.886

C3 0.100 C2 0.049

C6 0.100 C3 0.025

C10 0.100 nC4 0.010

C15 0.050 nC5 0.010

C20 0.050 N2 0.020

27

Figure 5- 1 Phase envelope for oil generated by PVTsim20

Figure 5- 2 Phase envelope for gas generated by PVTsim20

28

In this thesis, the baseline case study for single flow fluid uses a total of 5800

bbl/d and 8.2 MMSCF for liquid and gas flow flowing at the heel, as shown in Figure 5-

3. This later will be used in the mass balance equation to solve for the velocity of the

fluid. Note that instead of generating the influx flowrate, one can couple the mass balance

with reservoir simulation. This will give more parameters such as skin factor,

permeability and porosity to be investigated in sensitivity analysis. However, this will

only affect the flowrate and not the pressure and temperature profile; which is the main

objective of this thesis. For the base case, we make the radial flowrate for both oil and

water phase to the same for better comparable purposes. For gas radial flow rate however,

the generated flowrate is designed so that it would give comparable pressure drops to the

liquid phase. Note that the distribution of radial influx ( ) at each segment is generated

so that there is no radial influx at the toe and it is increasing linearly as it goes to the heel.

The radial influx distribution ( ), and other parameters at each segment are shown in

appendix E.

29

Figure 5- 3 Cumulative flowrate of all fluids for baseline case

Figures 5-4, 5-5 and 5-6 display the corresponding velocity, pressure, and temperature

wellbore responses, respectively, for the fluids under study and for the production data of

Figure 5-3. As expected from having the highest density among all the fluid, water

experienced largest pressure drop followed by oil and gas phase. This pressure behavior

can be explained as fluid flow in a straight horizontal section – no inclination or gravity

contribution in momentum loss – because irreversibility work losses due to the shear

stress ( between the flowing fluid and the internal wall of the wellbore is a strong

function of fluid density and viscosity. Thus fluid like water will have larger pressure

drop while gas will experience otherwise.

30

Figure 5- 4 Velocity of all fluids for baseline case

Figure 5- 5 Pressure profile of all fluids for baseline case

31

Figure 5- 6 Temperature profile of all fluids for baseline case

An interesting observation is that even though gas has the smallest pressure

response, it experiences the largest temperature response compared to other fluids. Oil,

while having a pressure drop profile similar to water, displays a more responsive change

in temperature distribution. The different degree of responsiveness among the three fluids

can be explained by the significantly different values of their

thermal coefficients

(isentropic thermal coefficients in Table 5-3), which predominantly appears in the

thermal response model. Based on table below, water has the lowest value of

whereas

gas high thermal responsiveness can be explained by its high value of this property. One

can also see how the contribution of the ‗volumetric heat capacity‘ contribution (

) in

the thermal response, which appears in both terms in the developed temperature model.

32

By having highest volumetric heat capacity, water requires higher pressure drop to have

same temperature profile as other fluids. This information can also be used in the sensing

the water breakthrough in the system using DTS data. Since water is expected to have

least sensitive temperature response, there will be sudden temperature change in the

system. This behavior shown in the toe region in the Figure 3-3.

Table 5- 3 Isentropic thermal coefficient and Joule Thomson values for wellbore fluids

Fluid Isentropic Thermal Coefficient (F/psi) Joule Thomson Coefficient (F/psi)

Heel Toe Heel Toe

Gas 0.0350 0.0340 0.0108 0.0118

Oil 0.0048 0.0047 -0.0044 -0.0040

Water 0.0010 0.0010 -0.0024 -0.0024

More importantly, these results show that the producing wellbore is predicted to

always experience cooling upon expansion and never heating—irrespective of fluid type.

This finding supports reported field DTS responses for horizontal oil wells (see Chapter

3). It also shows that whether the value of the Joule Thomson coefficient (also reported in

Table 5-3) is positive or not does not lead to wellbore heating. Table 5-3 shows that

always positive for all fluids, while the Joule Thomson coefficient for liquids is negative.

This Joule Thomson inversion had been used by previous studies to predict wellbore

heating for liquids – which contradicts real measured data.

An analysis to see which term in equation (4.20) plays the most significant role in

determining the temperature profile of the fluid in the system should be considered.

Figure 5-7 displays the total temperature change along the wellbore ( along with

33

the contribution due to the mass influx (

*

+ and the isentropic thermal

coefficient contribution (

). This figure shows that the enthalpy difference ( )

and ( ) contribution – which reflect the convection energy contribution via mass

influx –are always lower than the value of (

) as fluid approaches the heel. Thus the

temperature behavior of all fluids tend to mimic the pressure drop profile –with values

dropping from toe to heel – and can be said to be heavily controlled by the isentropic

thermal coefficient contribution.

Figure 5- 7 The contribution of temperature gradient at the last iteration

34

Because gas has been shown to be the most sensitive fluid in terms of the

temperature change, and since all other fluids– water and oil–would tend to follow the

gas trend but with lower sensitivity, the rest of the parametric studies shown in this

chapter use gas as the baseline case study.

5.2 Correlation between Thermal Coefficient and Joule Thomson

Coefficient.

As mentioned previously, the study wants to address the potential misconception

of using Joule Thomson inversion to explain differences between wellbore thermal

behavior of liquids and gases. From equation (4.20), one can still express the developed

thermal response equation in term of Joule Thomson Coefficient as shown below:

*

+

(6.1)

Equation (6.1) explicitly shows that the Joule Thomson coefficient can be related to the

isentropic thermal coefficient values. Thus a detailed analysis between these two

properties is performed by observing their differences in behavior for a given fluid.

Figure 5.8 shows the volumetric behavior of methane, highlighting the JT energetic

inversion envelope, as presented by the ASTM handbook of petroleum and natural gas

refining and processing (Ayala, 2012). Figure 5.9 to Figure 5.12 show the JT and

isentropic thermal coefficient behaviors of the same fluid in terms of reduced temperature

and pressure.

35

Figure 5- 8 Joule Thomson inversion curve of pure methane (Ayala, 2012)

Figure 5- 9 Joule Thomson of pure methane (for Pr =0.5 to Pr=15)

36

Figure 5- 10 Joule Thomson of pure methane (for Pr = 0.5 to Pr=15)

Figure 5- 11 Isentropic Thermal Coefficient of pure methane (for Pr=0.5 to Pr=15)

37

Figure 5- 12 Difference between Joule Thomson coefficient and isentropic thermal coefficient of pure methane(for Pr=0.5 to Pr=15)

According the theorem of corresponding states, all fluid will have approximately the

same trend of thermodynamic behavior when compared at the same reduced pressure and

reduced temperature (Cengel, 2007). Hence, the graphs presented for pure methane

would provide a valuable guide to examine behavior of most hydrocarbon fluids in

general. Figure 5-10 and Figure 5-11, show that the JTC of a fluid reaches a constant

value as fluid becomes less compressible (as pr significantly increases). In the limit, the

fluid is nearly incompressible and:

(6.2)

which is the JT value prediction expression used by Alves (1992) for nearly

incompressible fluids. In this limit, JTC is expected to remain negative and such fluids

38

would experience heating upon isenthalpic expansion. However, Figure 5-11 also shows

that the isentropic thermal coefficient is always positive even for liquids, regardless of

pressure and temperature conditions. As a result, a fluid will always experience thermal

cooling in the direction of expansion in horizontal wellbore systems since this term has

been shown to control wellbore thermal response. This is in agreement with field

observations, and it is concluded that the isentropic thermal behavior would be much

better representation to explain wellbore thermal responses. Note that the use of Joule

Thomson coefficient in predicting wellbore thermal response can yield the same

conclusion for gases; since both JTC and isentropic thermal coefficient for gases remain

are positive. However, these two approaches will diverge during liquid fluid flow

predictions as both would have opposite sign. From Figure 5-12, it can be said that the

isentropic thermal coefficient would be equal to JTC only at the critical point (Pr=1, Tr =

1). This is due to the fact that the difference of the two coefficients is dependent to

reciprocal of isobaric heat capacity—which becomes infinity at critical conditions—thus

driving the difference between the coefficients to zero.

39

5.3 Flowrate Effect

In this case study, the dependency on pressure and temperature behavior on the

radial and the total flowrate of horizontal wellbore system is being analyzed. Thus the

study employs different total gas flowrate - including the baseline case study – while

maintaining other parameters constant. Figure 5-13 to Figure 5-16 show the model

sensitivity to different flowrate. As the system have larger amount of inflow fluid, there

will be pressure drop required to transfer the fluid from the toe to the heel. As the

flowrate increases, the pressure drop and temperature drop increases. Strong evidence

that pressure drop will have significant impact on the temperature response can also be

observed. As the system experience higher pressure drop the temperature drop will also

be larger.

Figure 5- 13 Accumulated flowrate for each case

40

Figure 5- 14 Velocity for each case for flowrate study

Figure 5- 15 Pressure for each case for flowrate study

41

Figure 5- 16 Temperature profile for each case for flowrate study

5.4 Inclination Effect

One of the variations in horizontal wellbore is having an inclination from the heel

to the toe. This orientation is normally used when there the reservoir formation is thin and

bounded by anticline of syncline structure. For this case study, the effect of inclination to

pressure and temperature profile of the flowing fluid is analyzed. Note that one

significant difference for this case study is that the when calculating ( ), the reference

temperature (reservoir condition) will change due to the geothermal gradient; obeying

equation (4.23). Figure 5-17 to Figure 5-18 below show the model sensitivity to different

inclination. Note that temperature gradient of the formation G used in this study is 0.01

F/psia.

42

Figure 5- 17 The pressure profile due to the wellbore inclination

Figure 5- 18 The temperature profile due to inclination (fixed toe temperature)

The pressure drop for upward flow will be larger than horizontal flow as the hydrostatic

pressure drop increases. As the wellbore inclines upward from toe to heel, upward

43

elevation change increases, making the pressure drop required to transport the fluid from

toe to heel is larger. The reservoir temperature also decreases as fluid move from the toe

to heel due to the geothermal gradient effect. Gas experiences larger cooling effect due to

contribution of both thermal coefficient and geothermal gradient effect. Geothermal

gradient factor affects the convection heat contribution due to the mass influx

( ; where is a strong function of the reservoir temperature. Based on the both

upward and downward orientation comparison, it can be observed that geothermal

gradient effect is significant in the inclination case.

5.5 Type of Well Completion Effect

In this section will show how different type of well completion will affect the

pressure and temperature of fluid in the reservoir. Therefore, two types of well

completion are being considered; open-hole (baseline case study) and perforated

wellbore. Note that in slotted of perforated wellbore, the radial influx coming to the

wellbore is alternated with the non-producing segment throughout the wellbore. For the

closed wellbore sections, the proposed model should be modified in order to account for

potential heat transfer via overall heat transfer coefficients (U). However, for the sake of

simplicity, we assume U is zero or negligible in order to isolate the effect of . Figure 5-

19 shows a flowrate profile for both types of well completion. Figure 5-20 to Figure 5-22

show how these completions affect the pressure and temperature change in the wellbore.

44

Figure 5-19 Flowrate for well completion case

Figure 5- 20 Velocity profile for well completion case

45

Figure 5- 21 Pressure profile for well completion case

Figure 5- 22 Temperature profile for well completion case

46

As observed, regardless of type of completion, as long as the total flowrate for the gas is

the same, the final velocity flowing in the horizontal wellbore is the same; obeying the

mass balance constraint. However, the pressures drop of is much higher for slotted

wellbore compared to open -hole completion. The pressure and temperature ‗pulses‘ on

the slotted completion on Figure 5-21 and Figure5-22 are due to the injection at selected

segments. Note that these effects are significant at high flow rates. Note that this

observation can be applied in the industry as one can detect which zone has higher

permeability or production by observing the ‗pulses‘ in the DTS data.

5.6 Wellbore Radius Effect

Model dependence on wellbore radius is investigated in this section. For this case

study, the radial flowrate for this wellbore is fixed for all cases so that the wellbore radius

effect can be observed. Figure 5-23 to Figure 5-25 show the result of this sensitivity

study. For larger wellbore radiuses, the velocity of the fluid inside the wellbore will

decrease. This then consequently reduce the amount of friction produce between wall and

the wellbore surface. Thus the pressure and temperature drop of the fluid inside wellbore

will also decreases as the ramification of the lower friction factor. Note that the toe

region, the flowrate and fluid velocity changes are insignificant. Based on equation (4.9),

the system can be expected to approach isenthalpic conditions at the toe. However, the

pressure change in this section is the lowest (near zero) and thus the isenthalpic

temperature drop would still remain close to zero.

47

Figure 5- 23 Velocity profile for wellbore radius case study

Figure 5- 24 Pressure profile for wellbore radius case study

48

Figure 5- 25 Temperature profile for wellbore radius case study

5.6 Wellbore Roughness Effect

The thermal and pressure response of flowing fluid due to changing wellbore

roughness is conducted in this section. Note that the diameter used in this case is 2.5in.

Since the roughness is only involved in friction calculations, the mass balance equation

calculation would remain unchanged regardless of roughness value. Thus, Figure 5-29

and Figure 5-30 below illustrate the changes of pressure and temperature profile due to

different wellbore roughness. As the roughness increases, the viscous losses of the fluid

inside the wellbore will increase due to higher friction factor. This then will increase the

pressure drop and consequently increase the thermal response as well.

49

Figure 5- 26 Pressure profile for wellbore roughness case study

Figure 5-27 Pressure profile for wellbore roughness case study

50

CHAPTER 6

CONCLUSIONS AND SUGGESTIONS

Based on the study of thermal response for single phase and one dimensional flow

at steady cases condition, the following are the conclusion that can be drawn:

1. The use of Joule-Thomson coefficients as a tool to predict horizontal wellbore

cooling or heating responses has been shown to potentially lead to

fundamentally erroneous wellbore performance expectations.

2. For a single case fluid flow, water is the least sensitive to temperature change,

followed by oil and gas. This behavior has been shown to be heavily related to

the value of the isentropic (and not isenthalpic) thermal coefficient of the

fluid. As the fluid becomes less compressible, the value of the fluid isentropic

thermal coefficient approaches to zero. The opposite can be said when fluids

become more compressible.

3. Since the isentropic thermal coefficient is showed to be always positive at any

pressure and temperature condition, single-phase fluid flow in horizontal

wellbores is predicted to always exhibit a temperature curve that mimics its

pressure profile. This also provides a better tool to predict potential fluid flow

responses in horizontal wellbores than the use of the Joule Thomson heating

and cooling effect of the same fluid.

51

4. The temperature curve response can be also said to be driven by the flowrate.

Higher flowrate in wellbore will led to higher pressure drawdown in wellbore

and, consequently higher temperature drops. This can be explained due to

higher pressure drop is required to transfer larger mass.

5. Inclination of the wellbore also plays significant role in temperature profile of

the fluid. For upward flow, higher pressure drop will be needed to flow the

same amount of fluid in the wellbore and opposite effect is applied to the

downward flow. In addition, the prevailing geothermal gradient does also

affect the thermal response.

6. The completion of well can also affect the temperature change of the fluid.

The slotted wellbore would have higher pressure and temperature drop than

the open hole completion for the same amount of fluid produced at the

surface.

7. As the wellbore radius decreases, resulting pressure drops increase. This is

because smaller wellbores lead to faster velocities and larger friction losses in

the pipe. As the pressure response of wellbores with smaller radiuses becomes

exacerbated, the temperature curves will follow the same trend.

One of the recommendations for this study is to also enable the transient flow

effect in the system. This would increase the usability of the developed equations for

other different cases involving flow at early time regime. At that regime, the

accumulation of mass, momentum or energy could change the resulting behavior of the

fluid as predicted in this thesis. Secondly, the model can be also coupled with working

52

reservoir model which incorporates reservoir pressure and temperature development as

fluid is drained to the wellbore. This will give better understanding of how the fluid

would react in changing reservoir pressure and temperature as well as the radial flowrate

change due to reservoir depletion. Next, one should also consider extending the

application of the model for multiphase flow. In order to accomplish this, one should

consider the holdup ratio of the phases and the contribution of each phase in the mass,

momentum and energy balance. The split factor of the fluid at each section of the

wellbore as pressure and temperature changes should be considered to get better result for

this application. Lastly, the model can be also made applicable to vertical wellbore. This

modification should include the usage of overall heat transfer coefficient and geothermal

gradient effect as fluid moves upwards through the tubing.

53

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Measured in Multilateral Wells Equipped with Intelligent Completion‖,SPE-94097, 2005.

Wang X., Lee J. Distributed Temperature Sensor (DTS) system Modeling and

Application, SPE Paper 120805, May 2008.

Zhuoyi Li and Ding Zhu., ―Predicting Flow Profile of Horizontal Well by Downhole

Pressure and DTS Data for Water-Drive Reservoir‖, SPE-124873, October 2009

55

APPENDIX A

DERIVATION OF THERMODYNAMICS EQUATION IN WELLBORE

In this appendix, the derivation of all equations discussed in chapter four will be

explained and derived in detail. The derivations still start by going to each balance

equation first and continue to the derivation for fluid energy, entropy and enthalpy.

Finally, the equation of thermal response also will be derived in this appendix.

A.1 Mass Balance Equation

The modified balance Euler‘s equations expressing mass influx/efflux into the

wellbore system for one-dimensional flow can be written as:

(A.1)

As discussed before, equation (A.1) is expressed in their conservative form; .

Expanding this equation into their respective non-conservative form will result to:

(A.2)

A.2 Momentum Balance Equation

The momentum balance equation can also be defined in its conservative form as

follow;

(A.3)

56

Expanding equation (A.3);

(A.4)

Dividing equation (A.4) with density and rearrange it will yield:

*

+

(A.5)

Or

*

+ (A.6)

A.3 Energy Balance Equation

One can write the energy balance as for one dimensional flow in non-conservative

term as:

(

)

(

) (A.7)

The definition of enthalpy of a real fluid can be described as:

(A.8)

Thus, by substituting the enthalpy term and expanding all the derivative terms, equation

(A.8) can be re-expressed as the following;

(

)

(

)

*

+ (A.9)

Equation (A.9) can be further simplified by grouping the *

+ and dividing all

terms with density. This would result to:

57

*

+ (

) *

+

(

)

(

)

*

+

(A.10)

Using chain rule, one can said that:

(

)

;

(

)

(A.11)

Thus, with relationship shown in (A.11) ,mass balance definition in (A.1) and

relationship of internal energy in equation (A.8), equation (A.10) can be written as:

*

+ *

+

*

+

(A.12)

Note that from expanded momentum balance equation in (A.6), substitution of *

+ value can be done using equation (A.12); which results in:

*

+ *

+

*

+

(A.13)

Or simply;

*

+

*

+

(A.14)

In fluid dynamics application, since the quantity interested for equation (A.14) is the

specific fluid energy, material derivative can be used to describe the energy evolution of

58

the fluid parcel in time, it moves along its pathline; from toe to heel of the wellbore.

Thus:

*

+

(A.15)

A.4 Entropy Equation

In addition to that, by utilizing thermodynamic identity of

, one can

rewrite it in material derivative form as:

;

*

+

(A.16)

Applying relationship in (16) and mass balance definition in equation (1) in equation

(15), the time evolution of entropy of a fixed amount of mass as it moves through the

flow can be written as:

*

+ (A.17)

A.5 Enthalpy Equation

From equation (A.7), we can substitute with relationship of and this

would result to:

(

)

(

) (A.18)

Expansion of equation (A.18) will yield:

59

(

)

(

)

(A.19)

By dividing entire equation (A.19) by fluid density and grouping the *

+ would

result to:

*

+ (

) *

+

(

)

(

)

(A.20)

Thus applying the mass balance equation, one can re-express equation (A.20) to be:

*

+

(

)

(

)

(A.21)

Based on similar concept in (A.11) and expanded momentum equation in (A.6), equation

(A.21) can be simplified into the following:

*

+ *

+

(A.22)

Or

*

+ *

+

(A.23)

60

Thus the substance derivative for enthalpy can be written as:

*

+

(A.24)

A.6 Thermal Response equation

Using the thermodynamic identity, or

equation (A.24) can be written as:

*

+

Or

*

+

(A.25)

Note that

, thus equation (A.25) can be reduced to:

*

+

(A.26)

61

APPENDIX B

THERMODYNAMIC FLUID MODEL

A multi-component fluid can be described using relationship proposed by Equation of

State (EOS). The EOS expresses the relationship between density, pressure and

temperature of investigated fluid. In typical wellbore and pipeline cases, the pressure and

temperature will be changing continuously throughout the production or injection. This

results to continuous change in thermodynamic properties such as enthalpy and heat

capacity of the fluid. Therefore, the EOS can be utilized as a monitoring device to see

these changes since volumetric, pressure and temperature data can be obtained. Once two

of mentioned variables are obtained, many thermodynamics properties calculation can be

derived and calculated. This section will discuss the thermodynamics model used to

calculate necessary variables in thermal response calculation. As there are many EOSs,

Peng-Robinson Equation of State (PR-EOS) is chosen for this paper.

B.1 Peng Robinson Equation of States

Peng and Robinson (1976) developed a multicomponent fluid equation of states based on

Soave-Redlich-Kwong works (1974). This is more preferable equation of states as it was

derived for hydrocarbon carbon components. The P-explicit equation for this equation

can be expressed as the following:

(B.1)

62

Where = specific molar volume (

[ (

)]

Note that equation (B.1) can also be expressed in term of compressible factor using the

real gas relationship ( ). This will result into a cubic equation of Z factor:

(B.2)

where

In order to obtain the compressible Z factor, equation (B.2) can be solved either using

Newton Raphson method or analytical solution of a cubic equation. Note as well that so

far, the equations are expressed for a single component fluid. For a multicomponent

mixture, mixing rule will be used and this will change the following variables:

∑ (B.3a)

63

∑ ∑ √ (B.3b)

Note that ― ‖ is the molar percentage for each -th component present in the fluid

whereas ― ‖ is the binary interaction of the i-th component with the other component in

the fluid. If a multiphase fluid present in a particular case, ― ‖ must be replace by the

molar fraction for each phase- ― ‖ for liquid and ― ‖ for gas phase.

B.2 Thermodynamics Properties

B.2.1 Molecular Weight

The molecular weight of a multicomponent fluid is defined as the summation of each

molar contribution of each component molecular weight. Since the molecular weight

( ) and the molar fraction ( ) of individual components are known, the total

molecular weight of the single phase fluid can be written as:

∑ (B.4a)

B.2.2 Density

In previous section, we can predict the Z factor for the fluid using Peng-Robinson

Equation of State. The density of ―a‖ phase fluid is expressed as:

(

) (B.5)

Note that phase ―a‖ can be both applicable for gas and liquid phase. However respective

molecular weight and Z factor must be used.

64

B.2.3 Enthalpy

One of the most important properties in the fluid transport application is enthalpy. From

Peng-Robinson work, an expression of enthalpy from departure relation function can be

expressed as:

√

(√ )

(√ ) (B.7)

Therefore the enthalpy of the fluid at pressure and temperature can be calculated by

adding the enthalpy of the departure – from equation (B.7) – with the enthalpy of ideal

condition obtained at the same condition. Note that several procedures of calculating the

enthalpy are proposed by different authors. For this paper, we will use the correlation

made by Passut and Danner (1972) to calculate the ideal enthalpy which is expressed as

follow:

(B.8)

Where Apd, Bpd, Cpd, Dpd, Epd and Fpd are the coefficients given by Passut and Danner

(1972). The summarized of those coefficients can be described as follow:

65

Table B- 1Passut and Danner (1972) coefficients

Component Apd Bpd Cpd(103) Dpd (106) Epd (1010) Fpd (1014)

H20 -2.46342 0.457392 -0.05251 0.064594 -0.202759 0.23631

N2 -0.68925 0.253664 -0.01455 0.012544 -0.017106 -0.008239

CO2 4.77805 0.114433 0.101132 -0.026494 0.034706 -0.01314

C1 -5.58114 0.564834 -0.28297 0.417399 -1.525576 1.958857

C2 -0.76005 0.273088 -0.04296 0.312815 -1.38989 2.007023

C3 -1.22301 0.179733 0.066458 0.250998 -1.247461 1.893509

iC4 13.2866 0.036637 0.349631 0.005361 -0.298111 0.538662

nC4 29.11502 0.00204 0.434879 -0.08181 0.072349 -0.01456

iC5 27.62342 -0.0315 0.469884 -0.098283 0.102985 -0.029485

nC5 27.17183 -0.0028 0.440073 -0.086288 0.081764 -0.019715

neoC5 11.77146 0.004372 0.406465 -0.027646 -0.217453 0.468503

C6 32.0356 -0.0231 0.461333 -0.097402 0.103368 -0.030643

C7 30.70117 -0.02314 0.460981 -0.098074 0.104752 -0.03134

C8 29.50114 -0.0224 0.459712 -0.098062 0.104754 -0.031355

C9 28.56645 -0.02165 0.458518 -0.097973 0.104654 -0.031318

C10 28.4899 -0.02384 0.461164 -0.099786 0.108353 -0.033074

C11 28.06989 -0.02384 0.460773 -0.099839 0.108415 -0.033122

C12 26.21126 -0.01852 0.453893 -0.096464 0.101393 -0.029665

C13 26.97706 -0.02293 0.459517 -0.099758 0.108351 -0.033091

C14 26.50692 -0.02205 0.458079 -0.099164 0.107126 -0.032538

Note that equation (B.8) is suitable for calculation for ideal enthalpy for each component.

However, in multicomponent fluid, mixing rule should be applied so that:

∑

(B.9)

For this correlation, the unit for enthalpy given is in BTU/lbm. Thus an appropriate

universal gas constant R should be used and the expression should be divided by the

molecular weight so that enthalpy in Btu/lbmole. Furthermore, equation (B.7) requires

derivative of with respect of temperature, which can be written as:

66

√ ∑ ∑ ( ) * ( ) (

) (

)

(

) (

)

+ (B.10)

B.2.4 Isobaric heat capacity, Cp

Basic definition of specific heat capacity is generally defined as:

(

)

(B.11)

In calculating isobaric for can be derived by differentiating (B.7) with respect to

temperature. Thus the analytical relationship for isobaric heat capacity can be written as

follow:

( (

)

)

√ *(

)

(

)

( (

) (

) )

+

√

(B.12)

Where is the ideal isobaric capacity, also given by Passut and Danner (1972) as:

(B.13)

Note that the second derivative of with respect to temperature can be derived

from expression in (B.12),

67

√ ∑ ∑ ( ) * ( ) (

) (

)

(

√

) (

) (

)

( ( )

√

) + (B.14)

At this point the derivative of compressibility factor with respect to temperature at

constant pressure is required. The expression below offers one way to calculate it:

(

)

( (

) (

) (

) )

(B.15)

where,

(

)

(

)

(

)

(

)

(

)

(

)

(

)

[ (

)

(

)

(

)

(

) ]

(

)

(

)

68

B.2.5 Viscosity of the Gas

For this calculation, Lee, Gonzalez and Eakin‘s method (1966) can be used to

calculate the viscosity of a natural gas:

( (

)

) (B.15a)

( )

(B.15b)

(B.15c)

(B.15d)

Note that the temperature should be in Rankin and density in lbm/ft3 and the

calculated viscosity is in centipoise (cp).

B.2.6 Viscosity of the Liquid

In calculating the viscosity of the liquid phase, Lorenz-Bray-Clark‘s empirical

correlation is used. This is one of the best methods as the correlation was derived

empirically to find liquid hydrocarbon mixture by taking into account each of its

components. The equation was originated from Jossi, Stiel and Thodos (1962) for

calculating viscosity of dense-gas mixture, as shown as equation (4.2.5 a) below:

(B.16)

69

where μi is the liquid viscosity (cp), μ* is viscosity at atmospheric pressure (cp), m is

the mixture viscosity parameter (cp-1

) and ρr is the reduced liquid density. The reduced

density can be calculated using the Kay‘s mixing rule:

(

) (B.16a)

∑ ; ∑ ; ∑ ;

(B.16b)

Note that the is given in 0R, in psia and in ft

3/lbmol. Table B-2 and Table B-3

show the hydrocarbon properties and binary interaction between them needed for

calculations in this appendix. For the mixture viscosity parameter, Lohrentz (1969)

proposed the following equation:

√

(B.16c)

For the viscosity at atmospheric pressure, Lohrentz suggested to use the Hernin &

Zipperer equation. This equation put a condition in calculation one of its variable

depending of the value of the component‘s reduced temperature, Tr. The correlation can

be shown as follow;

∑

∑

(B.16d)

Where

; (B.16e)

70

; (B.16.f)

With all the variables defined and calculated, the equation 4.2.5 a can be applied to

evaluate the viscosity of the liquid.

Table B- 2 Properties for pure components used in this study (Ahmed, 1989)

Component MWi Tci (oR) Pci (psia)

wi Vci (ft3/lb)

N2 28.013 227.49 493.1 0.0372 0.051

CO2 44.01 547.91 1071 0.2667 0.0344

C1 16.043 343.33 666.4 0.0104 0.0988

C2 30.07 549.92 706.5 0.0979 0.0783

C3 44.097 666.06 616 0.1522 0.0727

iC4 58.123 734.46 527.9 0.1852 0.0714

nC4 58.123 765.62 550.6 0.1995 0.0703

iC5 72.15 829.1 490.4 0.228 0.0679

nC5 72.15 845.8 488.6 0.2514 0.0675

neoC5 72.15 781.13 464 0.1963 0.0673

C6 86.178 913.6 436.9 0.2994 0.0688

C7 100.205 972.7 396.8 0.3494 0.0691

C8 114.2 1024.22 360.7 0.3977 0.069

C9 128.25 1070.68 331.8 0.4445 0.0685

C10 142.28 1112 305.2 0.4898 0.0679

71

Table B- 3 Peng-Robinson (1976) binary interaction coefficients (Danesh, 1998)

No Component 1 2 3 4 5 6 7 8 9 10 to 24

1 N2 0.0000 0.0000 0.0310 0.0500 0.5150 0.0600 0.0852 0.1000 0.0711

2 CO2 0.0000 0.0000 0.1070 0.1200 0.1322 0.1300 0.1241 0.1400 0.1333

3 C1 0.0310 0.1070 0.0000 0.2150 0.0026 0.0330 0.0140 0.0256 0.0133

4 Ethylene 0.0500 0.1200 0.2150 0.0000 0.0089 0.0000 0.0100 0.0200 0.0200

5 C2 0.5150 0.1322 0.0026 0.0089 0.0000 0.0089 0.0011 -0.0067 0.0960

6 Propylene 0.0600 0.1300 0.0330 0.0000 0.0089 0.0000 0.0100 0.0080 0.0080

7 C3 0.0852 0.1241 0.0140 0.0100 0.0011 0.0100 0.0000 -0.0078 0.0033

8 iC4 0.1000 0.1400 0.0256 0.0200 -0.0067 0.0080 -0.0078 0.0000 0.0000

9 nC4 0.0711 0.1333 0.0133 0.0200 0.0960 0.0080 0.0033 0.0000 0.0000

10 iC5 0.1000 0.1400 -0.0056 0.0250 0.0080 0.0080 0.0111 -0.0040 0.0170 0.0000

11 Neopenthane 0.1000 0.1400 -0.0056 0.0250 0.0080 0.0080 0.0111 -0.0040 0.0170 0.0000

12 nC5 0.1000 0.1400 0.0236 0.0250 0.0078 0.0100 0.0120 0.0020 0.0170 0.0000

13 nC6 0.1496 0.1450 0.0422 0.0300 0.0140 0.0110 0.0267 0.0240 0.0174 0.0000

14 Met Cyc Pent 0.1500 0.1450 0.0450 0.0310 0.0141 0.0120 0.0270 0.0242 0.0180 0.0000

15 Cyc Hex 0.1500 0.1450 0.0450 0.0310 0.0141 0.0120 0.0270 0.0242 0.0180 0.0000

16 nC7 0.1441 0.1450 0.3520 0.0300 0.0150 0.0140 0.0560 0.0250 0.0190 0.0000

17 Met Cyc Hex 0.1500 0.1450 0.0450 0.0300 0.0160 0.0150 0.0580 0.0250 0.0200 0.0000

18 Toluene 0.1500 0.1800 0.0600 0.0400 0.0200 0.0210 0.0600 0.0300 0.0110 0.0000

19 o-Xylene 0.1441 0.1400 0.0470 0.0300 0.0160 0.0150 0.0590 0.0260 0.0120 0.0000

20 nC8 0.1500 0.1400 0.0470 0.0300 0.0160 0.0150 0.0590 0.0260 0.0120 0.0000

21 nC9 0.1500 0.1450 0.0474 0.0400 0.0190 0.0200 0.0070 0.0060 0.0100 0.0000

22 nC10 - nC14 0.1500 0.1450 0.0500 0.0450 0.0300 0.0250 0.0200 0.0100 0.0010 0.0000

23 nC15 - nC19 0.1500 0.1450 0.0600 0.0500 0.0400 0.0300 0.0250 0.0150 0.0010 0.0000

24 nC20 - nC24 0.1500 0.1450 0.0700 0.0600 0.0500 0.0350 0.0300 0.0200 0.0015 0.0000

72

APPENDIX C

DETERMINATION OF FRICTION FACTOR FOR MOMENTUM

EQUATION

In this paper, calculating the friction factor is extremely crucial as the shear stress

between wellbore and the fluid will heavily determine the pressure change in the system.

Therefore, the frictional loses can be expressed by the sheer force – shear stress multiply

the pipe wall surface area) divided by cross-sectional area of the flow in the pipe. The

friction factor is defined as the ratio of shear stress to kinetic energy :

(C.1)

From the dimensional analysis in circular pipes, it is demonstrated that the frictional

factor is directly dependent on Reynolds number for fully developed flows (Bird et al,

2002). Reynolds a dimensionless number from the ration of the inertia forces to viscous

forces or:

(C.2)

Where is the dynamic viscosity. Conventionally, if Reynolds number is less than 2100,

the flow is said to be laminar flow as the viscous forces is dominating over the inertia

forces of the flow. Thus, the friction factor can be directly obtained using the analytical

solution for circular pipe equation:

(C.3)

73

for Fanning friction factor while Moody friction factor is calculated using the following

expression:

(C.4)

However, when Reynolds number is higher than 2100, it is said that flow is in turbulent

flow; where inertia force is dominating over the viscous force. In this case, the friction

factor has shown to be dependent on both Reynolds number and the relative pipe

roughness (Bird et al, 2002). At this point there are several correlations that can be

used to get the friction factor at turbulent flow. For this paper, we will use the correlation

proposed by Chen (1979). This correlation is choose instead of others due to the fact that

Chen‘s friction factor equation is expressed in the explicit form; making calculating

done without any iterative method. Chen‘s friction factor equation can be written as

follow:

√

(

(

)

) (C.5)

74

APPENDIX D

VALIDATION PROCESS

In this this section, we will try to show that the model is calculating

thermodynamics properties with high precision and can also be collapsed to have the

same prediction made by other authors. One of the purposes of this section is also to have

a high degree of confidence in our model especially for the readers as the calculation of

some of the terms in the model can be confusing. Thus, two type of validation will be

performed; properties and model validation.

D.1 Properties validation for Gas

For properties validation, we compare out calculated value for major

thermodynamics properties used in our model and compare them with PVTsim 20.

PVTsim 20 is a handy PVT simulation program developed for the petroleum industries

and widely used by leading oil producing and operating companies throughout the world.

Since PVTsim 20 has a tremendous database and reliable EOS prediction model for

almost 20 years, comparing the values with data generated by our model will be a nice

touch to boost the confidence in our prediction model.

In our model, the most important thermodynamics parameters are the density ( ),

enthalpy (h), isobaric heat capacity (CP) and thermal expansion coefficient ( ). Therefore

we will use the compositional of gas on Table 5-1 to calculate these properties and

compare it with the results from PVTsim 20.

75

Figure D- 5 Gas Density value comparison between PVTsim 20 and model calculation

Figure D- 6 Gas Enthalpy comparison between PVTsim 20 and model calculation

76

Figure D- 7 Gas Thermal compressibility comparison between PVTsim 20 and model

calculation

Figure D- 8 Gas Thermal compressibility comparison between PVTsim 20 and model calculation

77

Figure D-1 until Figure D-4 show the comparison of density ( ), enthalpy (h),

isobaric heat capacity (CP) and thermal expansion coefficient ( ) respectively when

compared to the PVTsim 20 result. Since enthalpy is reference-base properties, all the

first data for each case are brought down to zero so that it would be compared. In this

section, only gas is used as we believed it would be sufficient to should one type of fluid

for the validation. Based on all diagrams shown, the calculation of these values match

with the results obtained from PVTsim 20 with absolute error less than 1%. This

validations show that the thermodynamic model that we use in Appendix A will produce

an accurate data for any fluid.

D.2 Properties validation for Oil

In this section, the same procedure is repeated for oil. The composition used in

this validation is described in Table 5-1. Figure D-5 until Figure D-8 show the

comparison of density ( ), enthalpy (h), isobaric heat capacity (CP) and thermal

expansion coefficient ( ) respectively when compared to the PVTsim 20 result. Note that

there is higher error for properties such as and for liquid compared to gas. This is

because enthalpy and Cp are dependent to Passut and Danner (1972) coefficient; which is

not used in PVTsim 20. However, the error is less than 10%. Thus the value used is still

in acceptable range.

78

Figure D- 5 Oil Density value comparison between PVTsim 20 and model calculation

Figure D- 6 Oil Density value comparison between PVTsim 20 and model calculation

79

Figure D- 7 Oil enthalpy value comparison between PVTsim 20 and model calculation

Figure D- 8 Oil enthalpy value comparison between PVTsim 20 and model calculation

80

Appendix E

SAMPLE CALCULATION

In this appendix, the sample values are presented so that one can reach the same

result without using Peng Robinson EOS. Note that the sample calculation is presented in

base case (refer Chapter 5.1) at the last iteration process. Thus, values such as radial

influx ( ), density, viscosity, friction factors, thermal compressibility, isobaric heat

capacity and enthalpy change for all the fluids used (gas, oil and water). Thus Figure E-1

to Figure E-7 show the properties need for calculation:

Figure E- 2 Radial influx for gas production

81

Figure E- 2 Radial influx for liquid production

Figure E- 3 Density of fluid at the last iteration

82

Figure E- 4 Viscosity of fluid at the last iteration

Figure E- 5 Moody friction of fluid at the last iteration

83

Figure E- 6 Enthalpy of fluid at the last iteration

Note that Figure E-3 to Figure E-6 show values of the properties for respective

fluid at the last iteration. All the thermodynamic properties for all fluid are calculated

from Peng Robinson EOS. It can be observed that most of the values needed in the

calculation are almost constant despite the pressure and temperature variation along the

wellbore. Thus one can reproduce the data fully analytical using the average value of all

the fluid properties during the iteration.