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Microwave heating for wax precipitation prevention and near wellbore conformance control

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Microwave Heating for Wax Precipitation
Prevention and Near Wellbore Conformance
Control
A Thesis
Submitted to the Faculty of Graduate Studies and Research
in Partial Fulfillment o f the Requirements for the
Degree o f Master o f Applied Science in
Petroleum Systems Engineering
University o f Regina
By
Mohammad Saeid Sheidaei
Regina, Saskatchewan
MARCH 2005
© Copyright 2005: Mohammad S. Sheidaei
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UNIVERSITY OF REGINA
FACULTY OF GRADUATE STUDIES AND RESEARCH
SUPERVISORY AND EXAMINING COMMITTEE
Mohammad Saeid Sheidaei, candidate for the degree of Master of Applied Science, has
presented a thesis titled, Microwave Heating fo r Wax Precipitation Prevention and
Near Wellbore Conformance Control, in an oral examination held on October 4, 2004.
The following committee members have found the thesis acceptable in form and content,
and that the candidate demonstrated satisfactory knowledge of the subject material.
External Examiner:
Dr. Dena McMartin, Faculty of Engineering
Co-Supervisor:
Dr. Koorosh Asghari, Faculty of Engineering
Co-Supervisor:
Dr. Raman Paranjape, Faculty of Engineering
Committee Member:
Dr. Gang Zhao, Faculty of Engineering
Committee Member:
Dr. Mingzhe Dong, Faculty of Engineering
Chair of Defense:
Dr. Yee-Chung Jin, Faculty of Engineering
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ABSTRACT
Production from heavy paraffinic oil reservoirs is not a new challenge for
scientists and engineers. The wax precipitation in and around wellbores is quite a
common phenomenon in such reservoirs. If precipitation occurs it will reduce the
productivity of the well and this reduction will diminish the total profit.
Application of microwave energy is a new technique among near wellbore
conformance control methods. This method can be used in oil reservoirs to prevent and/or
remove wax depositions in near the wellbore region. This will increase the production
rate and at the same time decrease the pressure gradient needed for production.
In order to have a better understanding of the e ffects of microwave heating on
porous media, a mathematical model in the form of a partial differential equation is
developed. The model is solved analytically by two methods, and the solution is used to
simulate the heating effects of microwaves in porous media. The effects of frequency,
fractional flow, injection fluid salinity, microwave power, injection temperature, and
injection rate are investigated with the mathematical model. The results of the
mathematical model are later used to design the experiments, in order to examine and
validate the results of mathematical model study. Based on the results obtained from the
mathematical model, incident powers between 15 and 30 watts and injection rates of 5 to
15 ml/min with 0 to 1 wt% salinity were chosen for the laboratory experiments.
The experimental part of the investigation confirmed the validity of the
mathematical simulation results.
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ACKNOWLEDGMENTS
I would like to express my sincere gratitude to my academic supervisors, Dr.
Koorosh Asghari and Dr. Raman Paranjape, whose expertise, understanding, and
encouragement added considerably to my graduate study. I deeply appreciate their vast
knowledge and skills in many research areas, Their sound advice, careful guidance, and
excellent supervision on this thesis research project. Also, I would like to thank Dr.
Asghari for his great assistance in preparing this thesis and writing papers during my
Master’s degree program. Furthermore, his valuable guidance, inspiring discussion and
constructive criticism motivated me, both academically and professionally.
I would like to sincerely thank Dr. Sam Fluang, Dr. Norman Freitag, Dr. Selim
Sayegh and Mr. Keith Hutchence at the Saskatchewan Research Council for their
insightful advice and guidance.
My sincere appreciation is for Petroleum Technology Research Center and
Faculty of Graduate Studies and Research for providing funding and other support.
Finally, I would like to express my deepest gratitude to my dear parents Mr.
Ahmad Sheidaei and Mrs. Dr. Barbra Nouraldini, and to my wife Mrs. Marsha Battiste,
for their unconditional support, understanding, and encouragement; I would like to
acknowledge my son Mikayle and my daughter Monique, the former was born in Regina
during my Master’s degree program.
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TABLE OF CONTENTS
ABSTRACT...............................................................................................................................ii
LIST OF TABLES..................................................................................................................vii
LIST OF FIGURES................................................................................................................ vii
NOMENCLUTURE................................................................................................................ ix
CHAPTER 1 INTRODUCTION............................................................................................ 1
1.1 The Recovery Stages of Oil and Enhanced Oil Recovery...........................................2
1.2 Wax Deposition.............................................................................................................. 4
1.3 Near Wellbore Conformance Control and Effect of Wax Deposition........................5
1.4 Microwaves and Their Application in Enhanced Oil Recovery and Near Wellbore
Conformance Control............................................................................................................ 6
CHAPTER 2 LITERATURE REVEIW................................................................................9
2.1. Heat Transfer in Porous M edia..................................................................................... 9
2.2. Heating by Microwave Irradiation.............................................................................10
2.3 Fluid and Rock Dielectric Properties...........................................................................12
2.3.1 Dielectric Properties o f Rock and Reservoir Fluids...........................................12
2.3 2 Thermal Proprieties o f Reservoir Rock and Fluids............................................13
2.3.3 Previous Laboratory Studies on Microwave Pleating o f Reservoirs.................16
2.3.4. Field Test Results..................................................................................................16
2.3.5. Modeling Background.................................................
17
2.4 Wax Deposition............................................................................................................ 19
CHAPTER 3 EXPERIMENTAL SETUP FOR MICROWAVE HEATING OF
POROUS MEDIA................................................................................................................... 23
3.1 Introduction................................................................................................................... 23
3.2 Experimental Setup.......................................................................................................23
3.2.1 Data Acquisition System.......................................................................................23
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3.2.2 Data Storage......................................................................................................... 24
3.2.3 Sandpack................................................................................................................24
3.2.4 Wave Guide............................................................................................................24
3.2.5 Power Load........................................................................................................... 25
3.2.6 Microwave Power Source (Magnetron)...............................................................25
3.2.7 Crystal Sensors..................................................................................................... 27
3.2.8 Injection Pump...................................................................................................... 27
3.2.9 Pressure Transducer.............................................................................................27
3.2.10 Thermocouple..................................................................................................... 28
3.3 Procedure for Preparation of Sandpack for Experiments and Selection Procedure.28
3.4 Experimental Procedure for Microwave Heating...................................................... 29
3.5 Experimental Procedure for Wax Removal Using Microwave Heating...................30
4.1 Introduction................................................................................................................... 31
4.2 Mathematical Model for Microwave at the Inlet....................................................... 31
4.2.1 Conservation o f Mass in Cartesian Coordinates................................................32
4.2.2 Conservation o f Energy in Cartesian Coordinates.............................................35
4.3 Mathematical Model for Microwave at the O utlet.................................................... 38
4.3.1 Conservation o f Mass in Cartesian Coordinates................................................38
4.3.2 Conservation o f Energy in Cartesian Coordinates.............................................38
4.4 Microwave Energy.............................................................................................
42
4.4.1 Lambert Method.................................................................................................... 42
4.4.2 Maxwell Approach................................................................................................ 44
4.5 Modeling Two Phase Flow in Porous Media..............................................................45
4.5.1 Two Phase Model fo r Microwave at Inlet........................................................... 45
4.5.2 Two Phase Model fo r Microwave at Outlet........................................................ 48
4.6 Boundary Conditions for Microwave Heating............................................................49
4.7 Solution to Mathematical Models................................................................................51
4.7.1 Analytical Solution................................................................................................ 51
4.7.2 Second Analytical M ethod....................................................................................60
4.8.1 Analytical Solution to Model with Microwave at Outlet................................... 65
4.9 Fluid and Rock Property Data and Supporting Data for Dielectric Properties
68
4.9.1 Correlations Used for Physical Properties of W ater...........................................68
4.9.2 Oil Physical Properties..........................................................................................69
4.9.3 Correlations Used for Sand...................................................................................70
CHAPTER 5 RESULTS OF MATHEMATICAL MODEL STUDIES............................71
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5.1 Introduction
71
5.2 Penetration Depth......................................................................................................... 71
5.2 Effect of Microwave Power........................................................................................ 79
5.4 Effect o f Flow R ate...................................................................................................... 85
5.5 Effect o f Inlet Temperature......................................................................................... 92
5.6 Effect of Salinity.......................................................................................................... 96
CHAPTER 6 ........................................................................................................................... 98
Experimental Results and Discussions.................................................................................98
6.1 Introduction...................................................................................................................98
6.2 Primary Experiments...................................................................................................98
6.2.1 Material Selection Experiments and Procedures.............................................. 98
6.2.2 Primary Flow Experiments with Fresh Water..................................................103
6.3 Experiments with Brine.............................................................................................109
6.4 Experiments with Paraffin.........................................................................................112
CHAPTER 7 CONCLUSION AND RECOMMANDATIONS FOR FUTURE STUDIES 114
7.1 CONCLUSIONS..........................................................
114
7.2 RECOMMENDATIONS...........................................................................................114
CHAPTER 8 REFERENCES.............................................................................................116
APPENDIX A Pressure Transducer Calibration................................................................122
APPENDIX B ....................................................................................................................... 123
THE SIMULATION PROGRAM.......................................................................................123
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LIST OF TABLES
Table 2. 1 Physical properties of oil and sand used for mathematical modeling.............. 15
Table 6. IThe material under investigation for the sandpack and the results o f
100
LIST OF FIGURES
Figure 2. 1 Relative permittivity of pure water, 0.2% Brine solution, 1% brine solution
and 5% brine solution......................................................................................................14
Figure 3. 1 Schematic setup for Microwave heating experiment...................................... 26
Figure 4. 1 Element of volume in Cartesian coordinates used for conservation of mass. 34
Figure 4. 2 Element of volume in Cartesian coordinates used for conservation of energy
...........................................................................................................................................37
Figure 4. 3 Element o f volume in Cartesian coordinates used for conservation of mass
...........................................................................................................................................39
Figure 4. 4 Element of volume in Cartesian coordinates used for conservation of energy
...........................................................................................................................................41
Figure 5. 1Penetration depth versus length for sand, oil, water and different concentration
inlet. Phi=30% initial Temp=20 °C, Incident Power=22 watts................................... 72
Figure 5. 2 Penetration depth for sand versus length for different frequencies. Phi=30%
initial Temp=20 °C, Incident Power=22 watts..............................................................74
Figure 5. 3 Penetration depth for water saturated sand versus length for different
frequencies. Phi=30% initial Temp=20 °C, Incident Power=^2 watts........................75
Figure 5. 4 Absorbed power after 1000 sec. irradiation for sand versus length for different
frequencies. Phi=30% initial Temp=20 °C, Incident Power=22 watts........................77
Figure 5. 5 Power absorption for water saturated sand versus length after 1000 sec. for
different frequencies. Phi=30% initial Temp=20 °C, Incident Power=22 watts
78
Figure 5. 6 Effect o f microwave power on outlet temperature. Phi=30% initial Temp=20
°C, injection flow rate=10 ml/min..................................................................................80
Figure 5. 7 Water temperature profile at the outlet of sandpack. Injection rate:10 ml/min
and microwave and flow directions are co-current. Phi=30% initial Temp=20°C.. 82
Figure 5. 8 Temperature distribution in sandpack saturated with water and water flow at
10 ml/min after 6000 seconds. Phi=30% initial Temp=20 °C, Injection rate=10
ml/min...............................................................................................................................84
Figure 5. 9 The effect of fractional flow rate on outlet temperature profile. Phi-30%
initial Temp=20 °C, Incident Power=22 watts.............................................................86
Figure 5.10 Effect of flow rate on outlet temperature profile for the case microwave and
fluid flow are counter-current. Phi=30% initial Temp=20 °C, Incident Power=22
watts..................................................................................................................................87
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Figure 5.11 Outlet temperature profile versus time for various injection rates and
fractional flow rates, phi=30%, Incident power= 22 watts, Initial Temp=20 °C
89
Figure 5. 12 Temperature distribution profile for different water flow rates and
microwave at the outlet and comparison with microwave at the inlet. Phi=30%
initial Temp=20 °C, Incident Power=22 watts..............................................................91
Figure 5.13 Effect of inlet temperature on outlet temperature profile when microwave
and fluid flow are co-current. Phi=30%, Incident Power=22 watts............................93
Figure 5. 14 Temperature distribution profile for various inlet temperatures for co-current
microwave and fluid flow. Phi=30%, Incident Power=22 watts.................................95
Figure 5. 15 Outlet temperature profile versus time for different salinities, Phi=30%,
Incident Power=22 watts, initial temperature=20 °C................................................... 97
Figure 6. IThe setup for experiment with sand-filled waveguide.....................................102
Figure 6. 2 The outlet temperature profile for water injection of 10 ml/min................... 105
Figure 6. 3 The outlet temperature profile for water injection of 5 ml/min......................106
Figure 6. 4 The first experiment using thermocouple. 10 ml/min injection rate, fresh
water, initial temperature=20 C, incident power=22
..........................................108
Figure 6. 5 Outlet temperature profile for experiment with 1 wt% brine, injection flow
rate= 10 ml/min, incident power = 22 watts...............................................................110
Figure 6. 6 Outlet temperature profile for experiment with 0.2 wt% brine, injection flow
rate= 10 ml/min, incident power = 22 watts...............................................................I l l
Figure 6. 7 The experiment for wax removal. The experiment shows lower pressure
gradients for after wax removal procedure..................................................................113
vm
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NOMENCLATURE
p = Density, rn/s
cp = porosity
u = velocity, m/s
x ,y ,z = Coordinates
t =Time, s
T = Temperature, °C
q = Heat, Watts
E = Electrical field intensity (of Microwave), total microwave energy, Volt/m
H = Magnetic field intensity (of Microwave), Ampere-turn/m
B = Magnetic flux density, Weber ’s/m2
P = Microwave energy (inside the material under microwave rays), Watts
ks = Thermal conductivity of sand, W/m°C
kt = Thermal conductivity o f liquid (Oil+ Brine or Water), W/m°C
s ■Permittivity, F/m
e0 = Permittivity of vacuum , 8.854-10-12, F/m
s' — Relative dielectric loss factor of material
s"= Relative dielectric constant o f material
A0 = Wave length of microwave in free space, m
r = Relaxation time, s
co = Angular frequency, rad/s
a = Electric conductivity, Ohm
tan 8 = Dielectric loss coefficient
S = Solution concentration
Sw = Saturation of water
p = Permeability of a medium, is the measure of the amount of the electromagnetic
field passes through the medium, Weber ’s/m
ju0 = Permeability of vacuum
p e = Total charge density, the amount of charges present on the surface of medium,
Coulomb / m 2
J = Total conduction, Ohm
h = Thermal Convection, Watts/m2
Cp = Heat Capacity, W/m3oC
A = Area
erfc= Error Function
DP= Penetration Depth
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SUBSCRIPTS
r = Relative
b= Brine
w = Water
/ = Liquid
s = Sand, Solid Portion
0 = Incident distance, The Inlet
/ = Final
x
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CHAPTER 1 INTRODUCTION
The existence of large amounts of heavy paraffinic components in some oils and
the increasing demand to resolve the problem of wax scaling near the wellbore region
justifies the efforts made to examine various methods for reservoirs that suffer from the
wax scaling. Wax and/or asphaltene precipitation is a major problem encountered in
many oil reservoirs. The conventional remedies, such as solvent injection and hot water
injection, are usually short-term solutions and/or are uneconomical. The remedial
operations, including thermal and solvent injection methods, are the only methods that
will remove the heavy wax from the wellbore area and resolve deposition problem. The
needed heat for thermal methods can be supplied by hot fluid injection, electrical current
inside the reservoir, and electrical induction using coils and microwave heating. The
solvent injection and hot-fluid injection both will increase the mobility of wax and thus
facilitate its removal. Although these methods seem effective, they are expensive and
time consuming [1-4].
The above mentioned remedial operations are temporary solutions for wax
deposition. When the old problem recurs, it can permanently damage the producing
formation. In addition, these well simulation processes require a well shut down resulting
in production delays, and this may not be economical.
Microwave heating has been proposed as an alternative to these methods. The
concept behind microwave heating is that the microwaves penetrate the material and
vibrate the polar molecules at high frequencies, producing energy in the form of heat
within them. These characteristics and the potentially economical applications of
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microwave heating make it worthwhile to study for near wellbore heating to resolve wax
precipitation problems.
1.1 The Recovery Stages of Oil and Enhanced Oil Recovery
Subdivision of oil recovery into the three stages of Primary, Secondary, and
Tertiary is based on the chronological sequence of production from reservoirs. Primary
recovery is the result o f production by the natural energy of a reservoir without applying
any external force. The natural energy can be provided by an active aquifer, reservoir
depletion by gas drive, and solution gas drive. After decline of the primary energy, the
amount o f non-recovered oil is about 70% to 85% of Original Oil in Place (OOIP). After
this stage, production is continued by the aid of external energy provided by injection of
fluids such as water. The secondary method results from amplification of natural energy
through injection o f displacing fluid(s) to displace the oil and move it toward the
producing well(s) [1],
As mentioned, water is the most common injection fluid for secondary recovery
processes; hence, the term secondary recovery is synonymous with waterflooding.
Maximum recovery after the first two production procedures in individual reservoirs
might approach 35% to 50% of OOIP. The residual oil in the water-flooded sections of
reservoir consists of highly isolated and trapped droplets in the pores or as films around
the rock porous surface. The shape o f the trapped oil in pores is dependent on the rock
type, pore diameter and its wettability [ 1].
After the second production stage of a reservoir becomes uneconomical, the third
stage i s applied t o r ecover t he r emainder o f t he o il. T he p rocesses applied f or t ertiary
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recovery include miscible gas injections, chemicals/surfactants, and/or thermal energy.
Economics is the predominant reason to terminate any stage at any time [2,4],
The tertiary recovery process is responsible for producing the remaining 50 to
60% of OOEP. The “Enhanced Oil Recovery” or “Tertiary” methods include mobility
control and thermal and gas injection methods. It must be noted that some scientists have
classified certain gas injection methods as secondary recovery methods. This
classification is due to their immiscible behaviors [1].
Thermal processes are based on thermal energy injection or generation inside the
reservoir to improve r ecovery [2], The well-known heating methods include hot water
injection, steam injection, in-situ combustion, and electrical or electromagnetic heating
methods [6]. The hot-water injection method is used rarely. Steam injection consists of
two methods, namely cyclic steam injection and steam flooding or steam drive. Cyclic
steam injection is a single well process but the steam drive requires a series of wells for
injection and production. The new electrical/electromagnetic methods being studied for
tertiary production use electrical energy to heat the reservoir. However, the large size of
reservoir limits its application. Another limitation for such heating methods is the wiring
required
for transmission of electrical power.
Due to
these
limitations
the
electrical/electromagnetic heating methods are best for near well-bore heating only [1,2],
Near wellbore conformance control methods are helpful techniques for improving
oil recovery. These methods include near well-bore heating, gel injection in fractures and
high permeable zones, and sand production prevention [7], Microwave can be used as
one of the near wellbore heating methods for preventing wax precipitation. A variety of
methods for near wellbore and reservoir heating including induction heating by means of
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electricity and steam and hot water injection. The first is a method used only for near
well-bore heating. Although microwave has been available for more than four decades,
its applications have not yet been adopted commercially in the petroleum industry.
Since the chemical and physical properties of oils produced from different
reservoirs vary, some reservoirs are more susceptible to wax deposition than others.
Sometimes the crude oil produced from a single reservoir o r even a single well show
differing physical properties during the production. This variation occurs because lighter
hydrocarbons can be produced more easily than heavy fractions. The amount of heavy
fractions in crude oil usually determines its viscosity. In addition, the density of crude
oils with larger amounts o f heavy fractions is higher, sometimes even higher than water
[1-3]. These higher densities and viscosities reduce the productivity of wells and increase
the wax and asphaltene scaling near the wellbore during production. The precipitation of
heavy hydrocarbon molecules, such as wax and asphaltene in the near wellbore region
reduces the permeability of that zone and consequently the overall oil production from
the well.
1.2 Wax Deposition
Wax deposition is one of the major problems that can occur in the near wellbore
region of heavy paraffinic oil reservoirs. However, wax deposition also happens in crude
transmission lines and gas condensate reservoirs. Additionally it can reduce the quality of
oil due to precipitation o f paraffinic fractions [8]. The paraffinic deposits consist o f nparaffins (i.e. the linear Alkanes), small amounts of branched alkanes and aromatic
compounds. Investigations conducted by Manssori [10] showed that naphthenic and long-
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chain paraffins have a dominant effect on growth of microcrystalline waxes and
microcrystallines [9,10], Most researchers cite temperature reduction as the cause of wax
deposition [9-14]. However, others showed that velocity, molecular diffusion, shear
dispersion and concentration can cause wax deposition as well [11-14].
There are various methods, classified into two major categories, used to overcome
paraffin deposition. These are removal and inhibition/prevention methods. The removal
methods include mechanical, thermal and chemical methods to remove wax depositions
from scaled areas, such as the near wellbore region or pipeline. The inhibition/ prevention
methods include similar mechanical, chemical or thermal methods, but are based on
crystal dispersion or crystal modifiers. There is a potential for significant cost reduction
using inhibition methods [15].
Mechanical methods include pig running in pipelines and using agitators to
increase the washing ability o f oil. Thermal methods include near wellbore heating for
near wellbore wax deposition problems and using local heaters in or around pipeline for
pipeline deposition. In the chemical methods, a solvent is used to remove the wax.
1.3 Near Wellbore Conformance Control and Effect of Wax Deposition
Near wellbore conformance control is usually synonymous with gel treatment or
other techniques applied to reduce excessive water production and to accelerate oil
production. A lthough p reduction o f h igh a mounts o f w ater o ccurs i n m any reservoirs,
there are other issues such as wax or asphaltene deposition and sand production may also
occur.
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Any damage to the near wellbore region of a production well will adversely affect
oil production. Wax or asphaltene scaling in the near wellbore region leads to the
blocking o f f ormation a nd a r eduction i n p ermeability t hat requires h igher p ressure t o
produce crude. This reduction in permeability occurs because deposition plugs the pores.
Higher pressures may also be required for pumping due to altered wettability because the
deposited wax is oil-wet [15]. There are several methods to remedy wax deposition
including a pplying s olvent to w ash t he s caled z one o r u sing h ot water i nj ection. A cid
injection and hot water or hot diesel fuel injection are among the most common methods.
These methods require shutting in the well and may not be economical [15].
1.4 Microwaves and Their Application in Enhanced Oil Recovery and Near
Wellbore Conformance Control
Microwaves are electromagnetic fields with very high frequencies (300 MHz to
300 GHz). Under microwave irradiation, the molecules of most materials oscillate with
respect to the microwave frequency and their polarity that shows itself as dielectric
properties of material. The dipole moments of molecules try to align themselves with the
alternate nature of the electromagnetic field creating oscillation. These oscillatory
movements produce heat due to the higher internal energy and friction of molecules. The
optimization of this process at resonance frequency requires much consideration and
experimentation. Another source of heat is the conduction of ionic charges induced by the
field in ionic solutions. Even small amounts of current can produce considerable amounts
of heat within ionic solutions. A small electric current in oil-bearing formation can
produce 3.413 B.T.U. per kilowatt/hr [6,16], Rajagopal and Tao in their study show that
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electrolytic solutions can be dried faster than non-electrolytic liquids under microwave
irradiations [17].
Research on microwaves and their heating effect has a history of five decades.
The early work also identified the heating and decomposing effects. The fundamental
rule behind the heating and decomposing effects is related to the polar characteristic of
atoms and molecules. The frequency of microwave and the electrical properties of
material have considerable effects o n the heating and decompositionprocesses, as has
been shown for carbohydrates such as proteins [18-20]. Other studies show the
effectiveness of microwaves for the decomposition of heavy hydrocarbon chains and for
upgrading to lighter hydrocarbons [21]. The presence of some additives, such as water,
will increase the degradation effect due to increased polarity [21],
Many factors determine how an object will heat when subjected to microwave
radiation. These include the geometry of the cavity in which heating takes place, the
geometry and size o f the object and its electromagnetic and thermal properties. Prediction
of how a compound will heat is a preferred option to building prototypes and observing
individual heating patterns, as this is expensive and time consuming. Prediction of
heating patterns can be effectively accomplished through mathematical modeling and
numerical simulation. Any of the aforementioned factors that influence the heating can be
easily changed in a computer program to quickly determine the optimal configuration that
is then used for experimental design.
In order to study the applicability of microwave heating techniques to increase
production from oil reservoirs and to reduce asphaltene and wax scaling, experimental
and theoretical analysis are required. The theoretical analysis consists of two steps. First,
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a mathematical model o f fluid flow behavior is developed based on the foundations of
heat transfer in porous media. Second, the developed model is tested in real reservoirs.
The experimental part of this study is used to verify the mathematical model.
In this work the heat transfer equation in porous media wais developed from the
fundamental equations of continuity and energy balance. Then, this mathematical model
was solved analytically. In order to present the solutions, a computer program was
written to include the analytical solutions and compare them to the solutions from
experimental work.
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CHAPTER 2 LITERATURE REVEIW
2.1. Heat Transfer in Porous Media
Compared to microwave literature for modeling in material science, and the food
industry, and such, the published studies on the heating of oil reservoirs and porous
media are sparse. The thermal recovery processes and present knowledge about reservoir
rock and fluid properties as well as dielectric properties, will be reviewed in this section.
Fourier and Poisson derived an equation for heat transfer in the presence o f fluid
flow in 1840 [6]. In 1856, Darcy published his work on the flow of water through sand
beds. Darcy’s, Carman’s and Leveret’s experiments and investigations directly contribute
to current understanding o f transport in porous media [22], Carman [23] provided a
permeability equation based on specific area, while Leveret’s [24] work was based on
idealized reduced capillary pressure function. Leveret introduced the J function for
correlation of capillary pressure data. The work by Hagen on equation for flow in pipes,
Knudsen’s work on slip flow and experiments with rarefied gases, and Taylor’s work on
hydrodynamic dispersion inside tubes, had indirect contributions, as their work is not
specifically about porous media [22], Lauwerier [25] completed one o f the earliest
investigations in the field of hot fluid injection. This work inspired many other studies
[26-31],
Heat transfer in porous media has three basic components. The first is forced
convection that happens by fluid flow. The second component is conduction that takes
place between sand grains. The third component is the convection between fluid and
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solid. The third component o f heat transfer takes place through the film layer of liquid
that covers the solid matrix. The literature shows the existence of two schools of thought
related to heat transfer. The first postulated by Klinkenberg [29] and Peterson et. al. [30]
assumes that the predominant factors in heat transfer are the forced conduction and liquid
film layer transfer, described mathematically as:
f :=V (7>-7d +v-(v-:o
The second school of thought, proposed by French mathematicians Fourier and
Poisson in the mid nineteenth century, takes forced convection and conduction as the
major sources of heat transfer in porous media [6,32], expressed as:
[(1 - *X p C ,), + * G o C ,) ,] |f + (pCf ),[V.(vT)] = ( # , + (1 - t ) k , ) V 2T
(2. 2)
Preston and Hazen [30] show that equation 2.1 is not realistic since it does not
account for conductive heat transfer in porous media. Therefore, the second model is a
more realistic model for the heat transfer with or without flow in oil reservoirs.
2. 2. Heating by Microwave Irradiation
Microwave heating is based on the capability of the electromagnetic field to
polarize charges that are carried by molecules and the resistance of molecules to
changing polarity.
Since the publications by Copson [33], Puschner [34] and Okress [35] in the field
of heating by microwaves and microwave devices, researchers have tried to solve the
Maxwell equations (Equations 2.3 to 2.6) for heating of material under microwave
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irradiation. These equations govern electromagnetism (i.e., the behavior of electric and
magnetic fields) and were first written in complete form by physicist James C. Maxwell,
who added the so-called displacement current term to the final equation.
For time-varying fields, the differential equations (SI) are
(2.3)
(2.4)
dt
V.B = 0
(2.5)
dE
dt
VxB=/u0J +£0{i0—
(2 .6 )
Generally, investigation of microwave heating includes the Maxwell equation
coupled with a heat equation. All magnetic, electrical, and thermal properties of material
are non-linearly temperature dependent [18]. The non-linearity of properties and the need
for coupling heat transfer and Maxwell equations make the governing equations non­
linear. In order to make the governing equations solvable, some researchers have tried to
simplify the coupled equations to solve them analytically or numerically [36]. Others
have focused on solving the heat equation using a value for microwave heating term in
which they neglect the electromagnetic effects [36].
The assumed term for microwave power dissipation is usually in the form of
power law, exponential, or a combination [37]. In addition to frequency, dielectric
properties play a large role in bulk heating by microwaves. Further, there are direct
conduction effects, such as redistribution of charges under the electromagnetic field.
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Debye, Frolich, Daniel, Hill et al., and Hasted have broadly studied the polarization and
conduction effects [18].
2.3 Fluid and Rock Dielectric Properties
2.3.1 Dielectric Properties o f Rock and Reservoir Fluids
The literature is almost devoid of extensive theoretical and experimental
reports related to the dielectric properties of reservoir rock and fluids. However, there is
extensive data for other materials that can be used to predict the required values within
reasonable limits. Metaxas and Meredith [18] provided a table o f different properties for
approximately 50 different materials for frequencies of 107 Hz, 109 Hz and 3xl09 Hz at
22 °C. Dielectric properties of pure water and brine solutions are both frequency and
temperature dependent. Debye [18] provided the following correlations for water:
_ _
rw
i
rw H
tan
. 8 n*L
.
rv/H
( £ rwL ~ £ rwH
£•
')
2
l + a>5 Tw
;
£ rwL
nr
S nvH ®
(2.8)
Tv
Stogryn [38] provides with two correlations similar to Debye’s equations for
dielectric properties of brine solutions.
£
—
*
£
H—
—
S ')
------------ — —
J+
< - 9)
( 2 .
[ £n a .+ E n M < ° h
<°£ o \
Where w and b stand for water and brine respectively.
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1 0 )
Ratanadecho et al. [39] developed a detailed diagram of dielectric properties of
water and brine at different concentrations (Figure 2.1). Klein and Swift [40] improved
the Stogryn model to predict salinity effects on microwave heating, as discussed further
in Chapter 4.
Few data on dielectric properties for oils are available, although data exists for
bitumen, hexane and some plastics or petrochemical products such as melamine. The data
for hexane was chosen to represent the oil dielectric properties for this study.
For rock properties, data from Grandjean et al. [41] and Nguyen [42] were used.
Both of these studies were conducted on radar penetration in sands and soils.
2.3 2 Thermal Proprieties o f Reservoir Rock and Fluids
The literature is abundant with data for thermal recovery calculations and
reservoir rock and fluids thermal properties. The thermodynamic properties of water were
selected from Keenan and Keyes [43], Farouq Ali [44] also provided a series of data for
water and steam properties. The Gros [45] correlation is used to calculate oil density in
this investigation.
Data from Green and Willhite [1] and Boberg [46] were used for other required
data, such as rock density and for determining the physical properties of reservoir rock
and fluids. This data include consolidated and unconsolidated sands, and correlations for
viscosity for different types of oil such as heavy, medium heavy, medium and light. Table
2.1 displays the properties of oil and sand used in the present investigation.
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y = 1E-10xe - 5E-08xs + 6E-06x4 - 0.0003x3 + 0.0065x2 - 0.3266x + 67.575
pure water
y = -7E-12xs + 9E-09x - 2E-06x4 + 0.0002x3 - 0.0067X2 - 0.2651x + 82.567
Relative Permittivity
0.2% Brine
1% Bnne
y = 6E-12x6 + 4E-09x5 - 1E-06x4 + O.OOOIx3 - 0.0055x2 - 0.2674x + 80.303
5% Brine
= -1E-11x6 + 1E-08x5 - 2E-06x4 + 0.0002x3 - 0.007x2 - 0.2646x + 83.133
40
410
20
30
40
50
60
70
80
90
100
Tem perature
Figure 2.1 Relative permittivity of pure water, 0.2% Brine solution, 1% brine solution and
5% brine solution.
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'^ '^ - - ^ M a t e r i a l
Property
—
Density
Sand
Oil
110 lbm/ff*
p o =
p or
+ C 2(T
-
-
c ,(r
-
6 0 )
6 0 ) 2
(lb m / f t 3)
Thermal
= 0 . 7 3 5 - 1.3* + 7 ^ 7
(T
(B tu/hr-ft-°.F )
Conductivity
Relative Dielectric
loss Factor
Relative Dielectric
constant
k h0
= [ 1 . 6 2 (1 - 0 . 0 0 0 3
-
32))] / y 0
( B t u / h r - f t - “F )
3.87 ( F/ m)
1.9 ( F / m )
0.000387
0.0005
Table 2.1 Physical properties o f oil and sand used for mathematical modeling
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2.3.3 Previous Laboratory Studies on Microwave Heating o f Reservoirs
There are very rare reported laboratory tests directly related to microwave heating
of reservoirs. However, Jackson [21] studied its applications for upgrading heavy oils.
Kislitsin and Fadeev [47] experimentally determined the dielectric properties of highviscosity and high-paraffinaceous oils; and Wei et al. [5] studied the heat and mass
transfer in sandstone. Further, Chute e t al. [48] determined the electrical properties o f
Athabasca tar sands and Bosisio et al. [49] published experimental results on microwave
heating of Athabasca tar sands. In addition to the above, Abemethy and McPherson et a.l
[50], have conducted experiments that were related to the microwave heating of
reservoirs.
2.3.4. Field Test Results
Abemethy [50] reported a series of field test results as well as a model for
electromagnetic heating. A microwave radiating device w as inserted in the production
zone of a reservoir. Conditions of flow and no flow from reservoir to the well bore were
studied with microwave device and a combined heat transfer and microwave heating
model was used. The model is solved using a dimensionless analytical method. Equation
2.11 represents Abemethy’s proposed model for electromagnetic heating of reservoir.
dT _
dt
1
f aP0exp(-a(r - r0))
27vrhpoUS \
4.18
(2. 11)
This model is the representation of microwave heating o f reservoir in cylindrical
coordinates where: r is the radius of reservoir, T represents the temperature in the
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reservoir under microwave radiation, and a is the coefficient of electromagnetic
adsorption.
Davison [51], studied the stimulation of Lloydminster heavy oil with
electromagnetic heating. Two wells were selected, one for microwave and one as a
production well. The results study show an increase in production shortly after the
beginning of irradiation. Microwave failure due to electrical power loss resulted in
reduced production. In addition to electromagnetic reservoir heating, Davison applied this
method to increase the tubing temperature in order to study the effect of tubing
temperature on near well bore conformance control. It was concluded that the positive
production response was due to the electromagnetic heating effects. Davidson attributed
the electrical power disruption and electricity line for field test failure and the results
support this reasoning. The investigations showed that the reservoirs studied, even at low
power input levels, responded quickly to heating.
In the mid 1980s, McPherson and co-workers attempted to recover Athabasca tar
sands by applying electromagnetic processes [52]. The presence of Alberta and
Saskatchewan tar sands provides a unique opportunity to investigate electromagnetic
irradiations for deeper reservoirs. McPherson et al. [52] used a high frequency
electromagnetic device to produce heat in net pay. The produced heat caused gravity
drainage of tar which is produced from production well.
2.3.5. Modeling Background
The Maxwell Equation [18] is the fundamental governing equation for microwave
heating. The differential form of the Maxwell equation is expressed in terms of electric
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and magnetic field densities. In these equations, E and H represent electric and magnetic
field densities respectively.
v,
s
dB
V x £ = ------dt
(2.4)
V x H = J + -----
(2.12)
dt
V.Z) = q
(2.13)
V.B = 0
(2.5)
There are three constitutive relationships between J (total conduction), D (electric
field flux), and B (magnetic flux density) and E and H. Note that due to the three
dimensional nature of electromagnetic fields, all equations are in vector format.
J = <JE
(2.14)
B - juH
(2.15)
D = eE
(2.16)
Substituting equations 2.14 to 2.16 are substituted intoequations 2.4, 2.5and 2.13 to 2.14
results in equations 2.17 to 2.20 forelectric and magneticfields, respectively.
pjTT
V x £ = - / / ----dt
(2.17)
dE
V x H = crE + s —
dt
(2.18)
V.H = 0
(2.19)
V J =-
(2.20)
These are combined to give the microwave heating in material:
Qmicrowave
= 55‘61 X10 ^ f l ^ s ' t g d
[18]
(2.21)
Wadadar et. al. [53] provided a model for electromagnetic heating of Alaskan tar
sand as well as a numerical simulation of their proposed model. Their model, which is
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given in equation 2.22, is a three dimensional three-phase model. The Darcian velocity
model is used to represent the phase velocity.
~^{0{SaiPaieOiI + ^GasPGaseGas + ^wPwew) + 0-~^KP^'p)r'^) = V(AVT) +
J
+PGJ ) + M A » ( v P a + PmZ) + B M
V
Pgos
Pm
l
(VP w w
) ]
(2. 22)
Pw
Where, e is the internal energy, h is the convective heat transfer coefficient and k
is the absolute permeability.
2.4 Wax Deposition
Jessen and H owell [ 11] s howed t hat an i ncrease o f v elocity i ncreases t he w ax
deposition and since velocity is higher in the near wellbore region than in the reservoir
wax deposition here. Jessen and Howell [11] proposed that the presence of a higher mass
transfer coefficient at higher flow rates and a sloughing effect due to viscous drag
exceeding the shear stresses within the deposited wax may explain this phenomenon.
Increased deposit hardness at increasing flow rates was also observed. Two mechanisms
of “paraffin deposition-deposition of paraffin” from crude oil at the pipe walls (molecular
diffusion) and particle transport to the wall were proposed, with the observation that
molecular diffusion is controlling process.
Hunt [12] conducted a paraffin deposition study using the flow and cold-spot
apparatus. The observed the mechanism of deposit growth by diffusion of paraffin
molecules from solution was consistent with lab and field measurements. It was also
noted that as deposit thickness increases, the shearing force on it increases and may
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become sufficient to tear it loose (sloughing). Hunt also noticed that it was not possible to
form deposits under constant temperature.
Bott and Gudmundsson [13] observed that flow rate and temperature decrease the
number of particles available for deposit while greater wax concentration increases the
member of particles. It was shown that during the deposition process heat flux decreases
(due to insulation and decrease in bulk oil temperature) and shear stress increases (due to
reduced diameter).
Burger et al. [14] emphasized on molecular diffusion and shear dispersion as the
two main deposition mechanisms and at the same time discounted Brownian diffusion
and gravity settling. For turbulent flow, the region of activity was identified to be
localized in the laminar sub-layer. The model predictions matched field data relatively
well. From calculations at low heat fluxes, shear dispersion was found to be the dominant
factor in wax deposition. At high heat fluxes, the critical role parameter appears to be
molecular diffusion. Like Hunt [12], there is no deposition at zero heat flux. Here, the
wax deposit was considered as a porous media filled with oil. The measured average wax
content of the deposit was between 14 to 17%.
Weingarten et al. [54] measured the deposition rates by diffusion and also studied
the effect o f shear o n d eposit v olume using a set o f a d hoc flow setups. A s p er their
observations, both shear transport and diffusion play definitive roles in wax deposition.
Also, upon normalizing the temperature effect (diffusion), the deposition rates were
found to increase linearly with shear rate, as proposed earlier by Burger et al. [14].
However, at a critical shear rate, the rate of total deposition was found to approach zero.
The sloughing rate was n ot quantified. However, it was observed in their experiments
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that sloughing had no relation to the turbulence of the flow. As long as the wall shear
stress exceeded the deposit strength, waxes would be sloughed off even if the flow was
laminar. This contradicts Jessen and Howell [11], which placed an emphasis on the
transition from laminar to turbulent for the onset of sloughing. Additionally, in the
absence of a temperature gradient, despite the oil temperature being below the
crystallization temperature, they also observed no deposition [54].
Hsu et al. [55] analyzed wax deposition of waxy live crudes in a high pressure
turbulent flow loop. Here too it was observed that wax deposition is negligibly small
when the heat flux across the wall is small or negative. These experiments showed that
wax deposition occurs only when the temperature of the deposition surface is lower than
the cloud oil point o f the average oil temperature. Hsu et al. [55] also found the sloughing
effect to be significant enough for turbulent flow to be ignored in modeling. It was
demonstrated that as retention time increases, the carbon number of the deposits and wax
hardness also increase.
Hamouda and Davidsen [56] experimentally noted that, though gravity settling
contributes to the total deposition mechanism, it does so to a lesser extent than shear
dispersion and molecular diffusion. O f the latter two, molecular diffusion appears to be
the controlling factor, while shear dispersion becomes important only at low temperature
gradients. Hamouda and Davidsen [56] found wax deposition to be negligible under near
zero heat flux conditions. They also observed initial sloughing at 3500 seconds; no
deposition occurred at 5500 seconds. To account for this behavior in their model, the
“paraffin adhesion constant” was introduced which multiplies the overall deposition rate
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(the sxim of molecular diffusion and shear dispersion). The value of this constant was set
at one at 3500 seconds and zero at 5500 seconds for the paraffin investigated.
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CHAPTER 3 EXPERIMENTAL SETUP FOR MICROWAVE HEATING
OF POROUS MEDIA
3.1 Introduction
A series of experiments was conducted to understand the phenomenon of
microwave heating of porous media. Each experiment was designed to address the effect
of the specific parameter under study. This chapter provides a description of the
experimental apparatus used.
3.2 Experimental Setup
A schematic setup o f the experimental apparatus is shown in Figure 3.1.
3.2.1 Data Acquisition System
The data acquisition system used was a UPC608 Validyne with 16 ports that can
be used to collect data from 32 sensors simultaneously. For the microwave experiments
two ports were used for temperature and pressure data collection. Further the UPC608
provides 14 bit A/D resolution and 11 stages of programmable gain amplification to
allow accurate measurement from various signal sources. A special two-step A/D
conversion process allows correction of zero offset errors from low-level measurements.
The device provides data with a frequency as rapid as 0.1 second.
A thermocouple reading requires reference junction compensation. Hence, two
channels of Analog output (±10Vdc or 4-20 mA) were also included with 16 bits of
digital I/O and two frequency channels.
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3.2.2 Data Storage
The data collecting device is a personal computer (PC) that receives information
from data acquisition device and stores it in the desired spreadsheet or data file. The data
file is used for data handling and analysis. The ISA Validyne card was installed on an
ISA slot of PC to read data more accurately and in the desired frequency.
3.2.3 Sandpack
In order to provide a porous medium that can handle high temperature and can be
used for flow experiments, a cylindrical sandpack was used. The sandpack was designed
to simulate a one-dimensional flow condition.
A series of experiments was conducted to test the endurance o f available materials
such as Plexiglas and polyurethane, but showed that these materials either bum or deform
under microwave radiation. On the other hand, experiments with Teflon® showed it to be
a reliable material for building the sandpack.
3.2.4 Wave Guide
Microwave radiations are extremely harmful to humans. Radiations with power
levels higher than 0.01 watts can break protein molecules and lead to the death of living
cells. In order to prevent damage to human tissues it is necessary to use wave guides for
lab experiments and industrial handling of microwaves. The waves are guided and the
junctions are sealed, preventing harm to humans.
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Three types of waveguides are available including, circular, rectangular with
square cross section, and rectangular. F or these experiments, a waveguide with square
cross section was chosen because of its well developed and known physical and
mathematical model. Modeling of microwave heating in circular waveguides is more
complicated due to semi-circular pattern of wave flux produced. The chosen WR-380
waveguide has dimensions of 600 mm length, and 89 mm in height and depth.
3.2.5 Power Load
In order to absorb the microwave radiations passed through the sand pack and in
order to reduce the heating of waveguides, a water cooled power load is required. The
proposed flow rate of water was determined by the flow control of the microwave power
generator. The power load is similar to a waveguide in shape; with the exception of a
Teflon® barrier built inside the power load. The cavity is filled with water to absorb the
microwave radiations since it is one of the best absorbers available.
3.2.6 Microwave Power Source (Magnetron)
The microwave power source, or magnetron used was a Curling-Moore TEio
magnetron (1975). It can produce continuous variable power between 0 to 3000 watts.
The TEio mode produces electromagnetic radiations with zero magnetic fields in z
direction of coordinates. This mode is required to ensure the crystal sensors measure the
correct amount of microwave power and to facilitate the mathematical modeling.
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1- Data AcquiMLiu
2- Computer
3- SandPack
4- Data Line
5-Wave-guide
6- Power Load
7- Microwave
Power Source
8- Liquid Inlet
9- Liquid Outlet
10-Crystal sensors
Figure 3 . 1 Schematic
setup for Microwave heating experiment
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3.2.7 Crystal Sensors
Two sets of two Hewlett-Packard crystal sensors were utilized in order to
measure the incident, reflected, passed, and reflected from power load. Each set of
sensors were connected to watt meters to measure the power. One set of sensors was
installed on the inlet of the waveguide, and the other at the end of the same waveguide.
3.2.8 Injection Pump
To simulate fluid flow in porous media, fluids such as water, brine, or oil in a are
injected in the sandpack utilizing an injection pump. Here, an Omega injection pump
with a capacity of up to 500 cc/min was used at injection rates of 5, 10, and 15 cc/min are
selected for experiments.
3.2.9 Pressure Transducer
The pressure transducer used was a variable reluctance pressure transducer
made of magnetically permeable stainless steel and held between two stainless steel
blocks. Each block had an E-shaped core that clamps an inductance coil. The applied
pressure difference deflects the diaphragm toward the lower pressure cavity, changing the
inductance value o f each coil and produceing a voltage difference. The voltage difference
is then transformed into a pressure difference.
The Validyne pressure transducer was installed on the flow line to measure the
pressure gradient during the injection period. The measured pressure gradient is later
used for calculation of the permeability of the sandpack. The pressure transducer has a
membrane to separate the two flows and measures the gradient pressure. A 20 psi
diaphragm was used.
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3.2.10 Thermocouple
The temperature of effluent was measured using a type J thermocouple that was
attached to the data collection device. The type J thermocouple is composed of a positive
leg of iron and a negative leg of 45% nickel and 55% copper. Type J is useable from 0 to
816 °C, (32 °F to 1500 °F) but is susceptible to aging in the temperature range of 371 °C
to 538 °C, (700 to 1000 °F) in which the thermocouple losses sensitivity. This
thermocouple has benefits of low cost and stable calibration, and is used primarily with
96% pure MgO insulation and a stainless steel sheath.
The thermocouple was inserted into the effluent line to increase the
measurement accuracy. Temperature measurements were taken at a frequency o f 20 sec.
3.3 Procedure for Preparation of Sandpack for Experiments and Selection
Procedure
The Teflon® sandpack was filled with graded sand and packed to form a
homogeneous medium during injection. CO2 at a pressure of 10 Psia was used to displace
air from the sandpack. After vacuuming the CO 2 out, the sandpack was attached to a
pump to determine its pore volume using a solution of 1 wt% brine to saturate the
sandpack.
To determine an optimized flow rate and microwave power, a series of
experiments were required. The first series of experiments was conducted at 5 and 10
ml/min flow rates and with a dial power of 2 and 4, equal to 22 and 39 watts respectively.
Distilled water was used for injection. Formation of excess vapor bubbles indicated when
to stop the microwave power to reduce fire risk. Temperature measurements for the first
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five experiments were performed using a 5 cc glass vial and a digital thermometer. High
vapor levels reduced the accuracy of temperature measurements.
The incident and reflected microwave power were also monitored using Watt
meters attached to crystal sensors were used. The results showed that the amount of
microwave power exiting the sandpack is zero until the time at which the injected water
is vaporized. These results confirm previous investigations that indicate water as a good
absorber for microwave power.
3.4 Experimental Procedure for Microwave Heating
The general procedure that is used for the experiments is as follows.
1. Fill the Teflon® sandpack with graded sand and shake to ensure even distribution
of sand grains throughout.
2. Attach the sand filled sandpack to the CO2 storage tank and inject CO2 at 10 psig
from the bottom inlet while the sandpack is in a vertical position. After 30
minutes, seal the inlet and outlet with caps.
3. The sandpack is attached to vacuum pump to evacuate CO2 .
4. Saturate the CO2 flooded sandpack with brine to determine the pore volume.
5. Determine the porosity from pore volume and sandpack dimensions.
6. Inject brine and/or water at 5 and 10 ml/min or 10 and 15 ml/min to determine the
initial permeability of the sandpack to brine and/or water .
7. Place the sandpack in the waveguide and seal the waveguide.
8. Turn on the microwave and select the desired power.
9. Operate the experiment until a constant temperature is reached.
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10. Gather data and other encountered evidence.
11. Turn off the system.
3.5 Experimental Procedure for Wax Removal Using Microwave Heating
The procedure used for these experiments is identical to part 3.4 with the
exception that the graded sand is mixed properly with grounded paraffin. This set of
experiments were designed to simulate the removal of wax from plugged or reduced
permeability wellbore area. Therefore the sand/wax mixture should resemble the
reservoir.
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CHAPTER 4
MATHEMATICAL MODELING OF ELECTROMAGNETIC HEATING
OF MULTI-PHASE FLOW IN POROUS MEDIUM
4.1 Introduction
Investigations o f large-scale applications for microwave heating require either a
field test or a numerical simulation. To simulate the microwave heating effects, a
mathematical model is required. The mathematical model will enable researchers to
predict microwave-heating effects in reservoirs and/or at the lab scale.
The mathematical model was developed for Cartesian coordinates enabling
researchers to calculate the temperature profiles for one-dimensional problems similar to
the described experimental setup. The model was developed for use when the microwave
energy source is located either at the inlet or the outlet of the sandpack. If the microwave
energy is located at the inlet, the respective model will simulate the case in which
incident microwave energy is at the injector well. If the microwave source is located at
the outlet, it is analogous to locating the microwave source at the production well.
4.2 Mathematical Model for Microwave at the Inlet
One scenario studied to reduce wax deposition in and around the wellbore area,
applied the microwave radiation at the injection location. Here, the microwave radiation
and fluid flow are co-current and thus the microwave energy dissipation takes place in the
direction of fluid flow. The temperature gradient in this scenario increases gradually.
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The heat transfer equation in this circumstance is a coupled conductionconvection heat transfer equation. With microwave irradiation at the injector, it simply
heats the injection liquid. The model was developed by applying the basis for fluid and
heat transfer in porous media and the mass and energy conservation. The model is solved
for both co-current and counter-current fluid flow and microwave energy dissipation.
4.2.1 Conservation o f Mass in Cartesian Coordinates
Considering the one-dimensional fully developed flow from the left side of a
selected element (Figure 4.1), that is the control volume for mass balance, to the right
hand side, the mass transfer equation is developed as follows:
Mass flux in (x direction) = p u A
(4
Mass flux out {x direction) = {p + Ap)(u + Au)A
(4.2)
C h a n g* e
o Jf
ma s s rate
-
Ap
—-
AxA
At
Mass flux in - mass flux out = change o f mass rate
Ap
Ap
—A x A
= p u A - { p + A p ) { u
—A x A
= p u A - p u A
Ap
A t
— p A u A - A p u A - A p A u A
A x A — —p A u A
By dividing both sides by A x A
Ap
(A
^
(4 . 3 )
(4 . 4)
(4 5)
(4 6)
1
*
(4- 7)
gives:
Am
~ a T = ~ P
dp
+ Au)A
^
(4-8)
~ a 7
du
" ~ d f = ~P ~ ^
(4.9)
This is the continuity equation for one-dimensional flow.
32
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Assuming complete incompressibility and no change in density with temperature
for the liquid phase, the continuity equation is simplified as follows:
Ap
■ =
0u
(4- io)
Therefore,
Au
Ax
0
(4-11)
8u
l k =0
(4-12)
Note that Equation 4.12 is the derivative form of Equation 4.11 and can be derived
when Ax—» 0 .
33
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z
Az
>
Ax
Figure 4.1
Element of volume in Cartesian coordinates used for conservation of mass
34
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4.2.2 Conservation o f Energy in Cartesian Coordinates
Figure 4.2 was used to develop the conservation of energy equation in Cartesian
coordinates in our study. The mathematical model for conservation of energy includes
both solid and liquid components to address the total heat transfer in porous media. The
modeling is divided into three parts, conduction, convection, and total heat transfer.
For solid portion (i.e. sand):
(Heat conducted in-Heat conducted out)+ (Microwave energy in- Microwave energy out)
= rate o f change o f energy
(4.13)
Microwave energy out = Egut
Microwave energy out = Egut
I \
AT
Heat conducted in = \qx) A = - k —— A
V
Ax
. . ....
v**
' X
1
V
Heat conducted out ={qx)
A = (q) A +
AxA + Higher orders (Taylor expansion)
' Jx+Ax
' *x
Ax
=- k
AT
Ax
A 2T
A -k— r A
( 4
1
Ax
AT
AT
A2T
A - (-k ---- A - k — r~A)
Ax
Ax
Ax
at
at
a 2t
a 2t
a2r
= - k ----- A + k --- A + k — y A = k — z-A = k — r-A
Ax
Ax
Ax
Ax
ox
15}
'
(Heat conducted in - Heat conducted out) = ~k
dT_
Rate o f change o f energy = (pCp)s~gy
dt
(4.16)
(4-17)
Dealing with only the solid portion, the overall heat transfer equation is:
dT
(1 —(f) ) ( p C p ) J ^ —(1 - (j) )k s
d 2T
+ (1 _ $ ^)QMicrowavesS
For the liquid portion (i.e. water or brine and oil), the following apply:
35
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^
(Heat convected in-Heat convected out) + (Microwave energy in- Microwave energy out)
+ (Heat conducted in-Heat conducted out) = Rate o f change o f energy
(4 . 19)
H e a t c o n v e c t e d in - p C puTxA
(4 . 20 )
H e a t c o n v e c t e d o u t - p C pu A T x+AxA = p CpuTxA + p C pu A T xA
^ 21 )
For the convective heat transfer in the liquid phase, the following equation is used:
{Heat convected in - Heat convected out) = p CpuTxA - (p CpuTxA +p CuHTxA)
cT
pCpuTxA - p C puTxA -p C puKrxA= -pC puISTxA - - p C pu— A
(4 . 22)
The equation for heat transfer in the liquid phase is:
ST
8T
cfT
^^PCp),~dT+(PCd‘U‘H' =^ k'Hf +^
(4 . 23 )
Therefore, the total heat transfer in porous media is:
Total heat transfer in porous media— (heat transfer in solid part + heat transfer in liquid
part)
[ ( l - > ) { p c p )s +<!> (j>Cp ) l] ^ + ( j > C p )lul ^
= ( t kx +(1-(Z> ) k ] ^ - + Q mcrowave
(4 . 24)
36
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z
>•
Ax
Figure 4 . 2
Element of volume in Cartesian coordinates used for conservation of energy
37
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4.3 Mathematical Model for Microwave at the Outlet
This section provides the mathematical model when the microwave source is at
the outlet o f the sandpack. Investigation of microwave heating at the outlet simulates the
field application o f microwave heating for wax removal technology realistically.
4.3.1 Conservation o f Mass in Cartesian Coordinates
Similar to microwave irradiation at the inlet, if we consider one-dimensional fully
developed flow from the left side of the element of volume toward the right side of the
element block (Figure 4.3) is considered. The derivation method is as given in 4.2.2.
4.3.2 Conservation o f Energy in Cartesian Coordinates
Figure 4.4 represents the volumetric element of control volume for conservation of
energy in Cartesian coordinates. Again, assumption of solid and liquid parts is inevitable
and the modeling for heat transfer is divided into three parts, including conduction,
convection, and total heat transfer.
For the solid portion o f porous media (i.e. sand):
(Heat conducted in-Heat conducted out)+ (Microwave energy in- Microwave energy out)
= rate o f change o f energy (Heat)
(4.25)
For the liquid portion of media (i.e. oil and water or brine):
(Heat convected out - Heat convected in) + (Microwave energy in- Microwave energy
out) + (Heat conducted in-Heat conducted out) = Rate o f change o f energy
H e a t c on v e c t e d in = p C pu A T x+6xA = p C puT xA + p C pulsTxA
(4.26)
H e a t c o nv ec te d o u t = p C puTxA
(4.27)
38
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Az
z
u
Ay
Ax
Figure 4 .3
Element of volume in Cartesian coordinates used for conservation of mass
39
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The heat convection that takes place between the solid and liquid portion is:
{Heat convected out-Heat convected in ) = (pCpuTxA + pCpuATxA) - pCpuTxA
3T
= pCpuTxA + pCpuATxA - pCpuTxA = pCpuATxA = pCpu— A
(4.28)
The equation for heat transfer in the liquid portion of porous media is:
^ s ST
Cp)l
q
, ^ N
dT
~ { P C p ) l Ul ^
d 2T
— 9*1
■
+ rQirficrowave+l
(4.29)
Equation 4.35 represents the overall heat transfer in porous media.
Total heat transfer in porous mediae (heat transfer in solid part + heat transfer in liquid
part)
[(i - 1 X p c P) , + ^ p c p\ - \ H - { p c p)lUl A L = ( # , + (i - * )* ,) |
p
QMicrowave
(4.30)
40
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z
Ay
Ax
Figure 4 .4
Element of volume in Cartesian coordinates used for conservation of energy
41
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4.4 Microwave Energy
Two methods are available to calculate the amount of microwave energy that is
absorbed in the sandpack during the experiments. The first, Lambert Method combines
the effects of the electric and magnetic fields and the amount of energy applied in heating
can be used to develop the mathematical model. Using Lambert method, the microwave
energy can be presented in a one-dimensional format similar to the experimental setup
used here. The second approach, based on Maxwell’s equations for microwave
dissipation, takes into account both the effects of electric and magnetic fields in
developing the mathematical model. This method requires a three-dimensional solution
for regular types o f microwave devices and a two-dimensional solution for a TEio
microwave power source.
4.4.1 Lambert Method
In this approach, a natural logarithmic decay is used to predict microwave power
dissipation at any distance from the surface where radiation takes place. Lambert was the
first to introduce the decay equation for microwave dissipation [38,43]. This equation can
be used to provide energy dissipation in one dimension as well as cylindrical and
spherical coordinates. The crystal sensors measure both the electric and magnetic fields
and the output is the sum of both. Although it will reduce the accuracy for conventional
forms of microwave power source, the error for TEio models of microwave power source
is zero, because as indicated in Chapter 4, the magnetic field in z direction is zero.
The Lambert power decay model is represented as follows:
42
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Where:
P = power (Watts/m )
0 = the incident distance
x — the distance from incident
D p = penetration depth of microwave:
D =^~
p An
1+
's
(4.32)
-1
Dp is a function of temperature itself and is due to the temperature dependency of
s ' and s " , which represent the relative dielectric loss factor of material and relative
dielectric constant of material respectively.
The of microwave energy absorption is:
O M icrow ave = (VPx - P x + h x J^
(4. 33)
f
x + Ax
\
QMicrowave ~ P) eXP
D> /
f
QM
mu
icrowave = p0 exp
V
(
\
X
~zT
V
PJ
(
x
-e x p
V
+ Ax
(4.34)
\\
Z)
P J/
43
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(4.35)
(
O
- o Microwave
r
- P0 exp
/
\
X
-\
x
-ex p
f
exp
Du
r
c
\
X
O-Microwave
--------
h eXP
v
D
V
\
f
X
*
P0 exp -------\
Dp
j
v
(
1 -e x p
V
\\
D>>
Ax
--------
1 -e x p
pJ
r
Q Microwave
mu
(
Ax
(4. 36)
\\
Z>
p J/
Ax
(4. 37)
\ \
----------
\
Dn
p )
)
(4. 38)
4.4.2 Maxwell Approach
The effect o f the magnetic field in heating is accounted for by the Maxwell
equation. To calculate the contribution of electrical and magnetic field in heating, the
oscillation of the electric and magnetic fields are determined by magnetron. The
following equations represent the incident electric and magnetic fields used to determine
the power dissipation and flux [36,40]:
/
Em sin
nx
\
sin (2rtf')
T
(4.39)
K x>
Ein .
sm
Zh
f
\
TCX
UJ
sin (2;r/)
(4. 40)
(4. 41)
A h r
The Poynting Vector represents the power flux that is associated with the propagating
electromagnetic wave:
S = 0.5Re(ExH*)
(4. 42)
44
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Pm = J\ s d A = —
. j
A
E ml
( 4 . 43)
^ H
After calculating the electromagnetic energy, the following equation can be used for
determining the electromagnetic contribution to heating:
Qmicrowave =55.61x10_I4fE1s'tgS
(4. 44)
where:
£ = <j)Sl + (1 — (f>)£s
a n (i
( 4. 45)
tgS = <i>tg8l + (1 - <f>)tgSs
(4. 46)
4.5 Modeling Two Phase Flow in Porous Media
The co-production of water and oil in production wells requires models
representing the phenomenon. Therefore application of microwaves for near wellbore
conformance control will need mathematical models that can simulate the two-phase
flow, as well as the single phase.
4.5.1 Two Phase Model for Microwave at Inlet
The governing mathematical equations for fluid flow in reservoirs include flow
equations for the various phases present in the reservoir. Reservoirs generally contain two
or three different fluids including oil, brine and gas. The gas phase is found dissolved in
oil or free in the gas cap.
45
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When the gas is dissolved in the oil the governing equations for two phase
properties are sufficient although three phases are present. The equations are complex
and non-linear and require further assumptions for simplification.
The first assumption is that of uniform distribution of two liquids in the reservoir.
This means that in any particular volume of reservoir the oil and brine saturation is
constant giving the following governing equation for two-phase oil-water flow:
q = <lo+ <lw
(4.47)
In Equation 4.52, q represents the flow rate and indices o and w are for oil and
water, respectively. If the cross section is assumed constant, Equation 4.47 is transformed
to Equation 4.48 in which u is the Darcian:
u —u„+u'w
(4. 48)
By assuming uniform saturation throughout the sand pack, the capillary pressure is
constant throughout the sandpack. Therefore the capillary pressure gradient equals to zero
and results in the following equations.
dPo
8pw _ d p c _ 0
dx
dx
(4.49)
dx
OPo _ op,
dx
(4. 50)
dx
Where P is the pressure, and represents the capillary pressure gradient.
This will yield:
u0
u
(4. 51)
w
The substitution of Equation 4.51 into Equation 4.48 will yield Equation 4.52.
46
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Rewriting Equation 4.35 to consider two-phase flow and to replace oil velocity with
water or brine velocity, the following results:
d 2T
dT
(1 - <j>){p Cp ), + </>[jB {p Cp )w + (1 - fi ){p C p ) 0 \ dx2
______ P (P Cp) wuw + (1 - p ) { p Cp)0uo_______ d^T_
dt
)(P Cp)s +</> [P ( p C P)w + ( l - P ) ( p C p)0] dx
(1
x
Pa exp
v OpJ
+ ■
(i -
1
r
dx
1 - exp
\\
(4.53)
)(p c P)s +<
t>\p (P c p)w+ (i - p ){P c P)0]
and
dT
dt
(/?*. + ( i - / ? ) * . ) + ( W ) * , )
-
f t
(1 - t ) { p C P) , + t [ f i (,pCp)w + { \ - / 3 ) { p c P)0]
u*[P <J>Cp)w+ ( l - P ) ( p C p)0je-]
(1 - 4 ) ( p C p ) , + t [ P
dT
( p C P ) w + ( l - P ) ( p C p ) o] dx
r
P0exp
dx1
1 - exp
\\
dx
(4.54)
v
+ -
) { P C P ) S + </> [ f i ( P C p ) w + ( l - P
) ( p C p ) 0\
Finally replacing water velocity with overall velocity yields the overall heat transfer
equation:
47
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8T
(j* (fi k» + Q--P )k0) + Q.-4 )k,)
Q^T
~ d 7 = ( \ - 4 > ) { p C p ) , + t \ f l { p C P)w + { \ - f i ) { p C p )J
Sx2
u[fi ( p C p )w + ( \ - f i ) ( p C p ) 0 ^ - ]
dT
(1 + — )[(1 - t )( p c P\ + 0 [fi ( p c F )w + (1 - fi ) {p c p )0\\ dx
w
/
\\
f
> f
X
dx
* 1 - exp
D
\
pJ
\ Dp / j
+ (W ){pCP)s +0 [fi (p c p)w + ( l - f i ) { p c p)0]
4.5.2 Two Phase Model for Microwave at Outlet
The governing heat transfer equation is changed but the two-phase flow equations
and capillary pressure assumption remains the same. The governing heat transfer
equation (Equation 4.35) is transformed to:
(fi kw + { \ - f i )k0)+(\-<f> )ks)
dT
~dT = {!-</) ) { p C p ) s +</> [fi ( p C p ) w + ( l ~ f i ) ( P C p ) J
P ( p C p) wu w + ( I - f i ) ( p C p ) 0u0
+
dx2
dT
(1 - t ) ( p C P) , + t [ f i ( p C P) w + ( l - f i ) ( p C P) J dx
f
X
P0 exp
v
D,J
1 - exp
V
(4. 56)
dx
jj
+ -
(1 - <j> ) { p C P ), + <j> [fi ( p C p ) w + (1 - fi ) { p C p ) 0 ]
Which can be rewritten as:
48
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dT
( t i P K + Q- - P ) k 0) + (X-4>)k,)
d2r
~dT = ( \ - 0 ) ( p C P)s + 0[>8 ( p C p)w + ( l - f i ) ( p C P)0] dx2
+ -
uw[fi ( p C p)w+ ( \ - f i ) ( P Cp)0^ - ]
J
dT
) { p c p)s + </>[fi ( p C p)w + { \ - f i )(p C p) J dx
\
/
\ r
dx
X
* 1 - exp —
exp
D
\ np ' j
V
(1 - * )(p Cp)s + </>[fi (p Cp)w + (1 - fi ){p Cp)0]
(4.57)
f
\
+
P
J
For the total velocity the water velocity term must be replaced with the total velocity
term.
dT
dt
( H P kw + ( \ - p ) k o) + {i-<f>)ks]
a2r
{!-</> )(p Cp)s +</>[fi {p CP)W+ (1 - fi )(p Cp)a] 3v2
A.
> +
t ^ -]
U[fi ( p C, p)w
+ (]
( l -- f Bi ) V( pn C p)0
P '
dT
(i + Y ~ m - <f>) { p c P)s +<t> [fi (p c P)w + ( i - f i x p c P\ ] ] 8x
r
exp
v D,j
1 - exp
V
dx
\\
(4. 58)
Dp
+•
(1 - 1 )(p CP), + </>[fi (p Cp )w+ (1 - fi ){p CP\ ]
4.6 Boundary Conditions for Microwave Heating
The required boundary conditions to model microwave heating include those
essential f or m icrowave e nergy a dsorption a nd a ttenuation i n p orous m edia a s w ell a s
those for heat and fluid flow through the sandpack. For adsorption of microwave energy,
the rate of attenuation and properties of solid and liquid components must be considered.
The solid portion (sand) is generally can be considered as silica, which is almost
transparent t o microwaves [ 36]. The 1iquid p ortion i s w ater, b rine, o il or a m ixture o f
oil/brine, depending on the experimental conditions.
49
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The temperatures of liquid and solid portions are assumed equal at any element of
distance and time.
By the above assumptions, the boundary and initial conditions are written as
follows:
du
1- The velocity of liquid is constant in the sandpack
2- The velocity of liquid is constant during the experiment
3- Inlet temperature is equal to ambient temperature
8
u
= 0
(o, O =r 0
t
4- The temperature of solid and liquid phases are equal T s ( x , t ) = T L ( x , t )
5- The dissipation of microwave energy follows the Lambert decay law
„
6- There is no radial heat and/or mass transfer
_
or
r,
- u »
dT
n
- u
or
7- The sand pack has no radiative and/or convective heat transfer with surroundings.
8- All of the microwave energy is decayed inside the sandpack P ( L , t ) = 0
;
meaning that for the experiments the sandpack islonger than the microwave
penetration depth
9- The temperature before the commencement of the experiment is ambient
temperature r ( * , o ) =
t
0
10- The temperature in infinity is equal to ambient temperature
3T(0,t)
n -
0^
t
( °o , o ) =
T
0
„
= °
12-The convective heat transfer element of the mathematical model can be replaced
Therefore the convective heat transfer term in Equation 4.35 can be
represented by its equivalent that is the result of multiplication of a coefficient, h,
and the difference between the inlet and outlet temperature of the element.
50
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4.7 Solution to Mathematical Models
To verify that the derived mathematical model is correct, the model must be
solved using analytical or numerical methods. In the present investigation, an analytical
solution is provided. Two different solution methods, the Laplace Transformation
Method and the Transfer Method are given. Any soluble non-linear differential equation
has a general, particular and overall solution. The solutions to both analytical methods are
identical, but the Laplace Method has a solution for the particular part as well.
4.7.1 Analytical Solution
To obtain the analytical solution of the heat transfer equation the equation was
changed to:
(M +(W)^)
dT
d2T
{pCp)^
dT
{ l - t X p C r X + t i p C , , ) , dx2 ~ {\-<f> )(pCp)s +(/) {pCp)l ~dx
_________
Q-Microwave__________
(4 . 59)
(1 - ^ ) ( p C Pl + ^ ( p C r ),
Substituting the microwave heating term (Equation 4.43) into Equation 4.59:
dT
d2T
dt
( l - t X p C r l + t i p C ^ dx2
f
P0exp
1 - exp
dx
( p C p)[Ul
dT
(1 - f ) ( p C p)l + t ( p C p)l dx
^
(4 . 60)
V D P J)
+ -
The Laplace transformation of results in:
51
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ST(x,S)-T(x,0):
(1 - ^ { p C r l + H p C r l
■T"(x,S)
X
Pqexp
(p
c m
■T’(x,S)+-
(L -tX p M + tip M
/
\
f
*
V Dp
[(1 - ^ (
p
J
f
1-
Ax
exp
V
M + H
Dp
\
p
\\
yj
(4. 61)
C M
where ^(X O ) - r o is a constant.
Equation 4.61 is transformed into an ordinary non-homogenous second order
differential equation:
(1-?S)Go C Fl + ^ ( p C r ),
f
(P Cp);
■T'(x,S)~
(1 - * ) ( p C P)t + t ( p C P)l
\ '
f
S\
X
Ax
1 - exp
Pq exp
v D , j
V
JJ
-ST(x, S)++ 7(x, 0) = 0
t m - 4 y j > c p) , + 4 { p c p\ \
D
■T'(x,S)
p
(4. 62)
That can be rearranged to:
(P Cp)/ ut
T ”( x , S ) -
(\-(/>){pCP)s +<j)(joCP)l
(\-<i>)(pcP)5 +<j>{pcP\
f
\ c
f
x
Ax
1 - exp
Pqexp
v
-S T ( x ,S )
D P J
v
■T'(x,S)
(4 . 63)
Pp jj
■-T
X{\
m -M ip C rX + iip C M
and has two sets of solutions, the particular and the homogeneous solutions.
To obtain the homogeneous solution o f the differential equation, it is assumed that
T ( x , S ) = eXx
(4. 64)
Therefore, the differential equation transforms into:
52
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(M
+(W )£,)i____ %2e ,
^
(1-tXpC'X+tipC,,),
(pCpXut
,
/Le^ _*Se 1x ~ 0
{\-<t>){pCp)s +HpCP)l
“
.
____________
(4-65)
which becomes:
fc, + (1 -</> )ks
( i -<f>)(pCP)s +</> (p C Px
(P
-r
Cp )/
( i -(/>){pCP)s + t ( p C Px
-X-S = 0
(4. 66)
A second-degree equation has the solution:
ip CpX ul
^1,2
\2
i P C p) l u l
+
+
$ [(!-? > X p C „ ) , + </>( p C r ),]
( t k , + (l - t ) k , )
( P
a
2
c
p
) i u i
( k ,+ (\0
(4 . 67)
0
(4. 68)
)ks)
and
P =
ST(W X /?CP)s +^(/>CP),]
ip C pXui
2 ^ k, + (1 -<j>)*,)_
(4. 69)
\2
i p C p),u,
A=
2(0k,+(l-</>)ks)
B=
(4 . 70)
ji-tXpCpX +</>(pCPx
(4. 71)
k, + (1 -(/> )ks
The overall results depend on X. If the solution yields two solutions, the homogeneous
equation is:
53
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Th(x, S ) = C, (S)e*x + C2(S)e^
(4. 72)
If it yields one answer, the result is:
r4(x,5) = ^ (C 1+ C2x)
(4. 73)
Additionally if the answers are in the complex form:
Th{ x , S ) ^ C / a+lP] + C2e(a-lP)
(4 . 74)
The non-homogeneous differential equations have a homogeneous or general
solution, a particular solution and an overall solution that is the sum of the general and
particular solutions. The particular solution must have a form that is similar to the nonhomogeneous part of the differential equation. In this case, if we select the form that is
similar to the left hand side (exponential form) of the differential equation is selected, the
particular solution has the following form.
X
C3e Dp)
(4. 75)
The first and second derivatives with respect to S result in Equations 4.75 and
4.76.
X
(4. 76)
54
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C3ev DpJ
t; ( x , s
)=
SD]
(4. 77)
Substituting one of the constants, these equations become:
c3
k,+(\-(/>)ks)
(1 - t X p C J ' + t i p C r l
( p C )iu,
SDI
+ (1 - t X p C r l + t i p C r l
C3e\P, j
SDp
Ax
P0e{Dp 1 - e
-S
Ce
■+cA
(4. 78)
m - t x p c P) , + * ( p c P),]
-sc4=-T0
(4 . 79)
Using equation 4.1 results in:
C4=T0/ S
(4. 1)
and
Ax
Po 1-e
a
{<£K +(1 -(/) )ks) - D P{pCp)lul
Dt
- m - ^ ( p c P)s + ^ ( p c P)!]
(4. 80)
and the answer to the differential equation would be:
T ( x , S ) = Th( x ,S ) + Tp ( x , S )
(481)
Note that the first two constants, Ci and C2 , can be any two functions of S, and
the solution for the homogeneous part depends on the calculated values for Ai>2.
55
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To find the results in time domain the inverse Laplace of Equation 4.81 is
calculated:
T{x, t) = L~\T{x, S)}
(4. 82)
T(x, t ) = r 1[Tk (x, S ) + Tp (x, S)] = r 1[Tk (x, S)] + r 1[Tp (x, S)]
(4. 83)
r l [Th(x, S )] = IT1[Q ( 5 ) ^ ] + r 1[C2(5)^** ]
(4. 84)
= L~x[Cx(S)ea V x] + r 1[C 2(S )ea
-
x]
" r 1[Q ( S ) ^ x] + ea xL~l [C2( S ) e p x]
If Ci and C2 are independent of S, they are independent of time, which means that Ci and
C2 can be functions o f x or constant numbers. The result is as follows:
a x -B I 2
=C /
0.282095c
B.
, 0.25 Bx1/ _ A t / s
A
Zb >
(-0.25
■-C2eax~B'2 x
0.282095xcs~
B.
&
(4. 85)
'B*
If the constants are equal, the above equation is equal to zero. If the constants are
numbers, the equation is rewritten as:
a x -B /2
r \ T h(x,S)] = (Ci - C 2)e‘
0.282095c
(-0 .2 5 B x 2- A/ b )
X
B
(4 . 86)
B&x
1
1 -e
V
D P
IT1[C
] = L~][-----------------------------------------3 S
(0k, + ( l - 0 ) k s) - D P(pCp),u,
-]
Dl
(4. 87)
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A*
[(l-jit ) k s t - D P ( p C p ) i u , ] t
« £ p [(i-e> ) ( p C P )s +t
Z ) ; p 0[1 - e v * " ] [ ! - e
( p C P ),]
[{0 k j + (1 - ^ ) £ , ) and
ir1[c4] =r 1[r0/Js,]=r0
(4. 88)
When Ci and C2 are not constants the results must be added and the inverse
Laplace transform of the result after substitution of boundary and initial conditions must
be calculated. C 3 and C 4 are known. The initial and boundary conditions must to be used
to find the Ci(S) and C2 (S):
T ( x , S ) = Th( x , S ) + T ( x , S ) = C , 0 S > v + C 2( S)e^x + T J S +
&x
P0 ^ Dp' 1 - e
(4 . 89)
[0 k, + (\-(/> ) ks) ~ D p { p C p )l ul
S
s 2[ ( \ - ^ ) ( p C P)s + H p C P)l ]
Dt
where
{ p C p ) l ul
^ 1,2
+
k, + ( l - 0 ) k s)
(P C )/ u i
+
2(0 kl + { \ - 0 ) k s)
(4 . 90)
and
57
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T(0,S) = T0/ S
(4. 91)
T / S = C ,( S ) + C (S) + T J S +
0
2
Ax
P0[l - e ^
' 2]
U k , + ( l - 4 ) k s) - D p (P C p )l u l
(4.92)
------------------------- - S
^
Up
[ ( l - t ) ( p C P) s +<p (,o C P) , ]
2
' Ax
E0[l - ^ °PJ]
C l( S ) + C (S)
2
kj + (1 -tf> ) k \ - D p ( p C ),i
S' tO - ^ )(/> C, ) , +0 (/? CP),] - 5 - ------------
'
(4. 93)
Considering the second boundary condition yields:
dT (0,Q
ax
(4- 94)
The Laplace transformation will yield:
- ^ B Z ) = A C ,( 5 ') + ^ C ! (S) = 0
(4 . 9 5 )
Substitution of boundary conditions into Equation 4.95 results in:
^ 0[ l - e v ° " ]
1-1M ,
C, ( S) = ------------------------------------------------ l- f ------------------------------------------
S 2[(l-<f>)(pCP)s+ H p C P)l]
{$ k, + ( \ - < f > ) k \ - D p ( p C )l u,
-
S
±
------------*---Up
Axl
C2(5) =
E0[ l - e v
1-/L//L
Considering the third boundary condition results in:
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.96)
07X0,S') _ h(Tf - T 0)
(4. 98)
dx
A1Cl(S) + J7C2(S) =
KT f - T 0)
(4. 99)
S
Calculation of Ci, C2 yields:
C ,(S ) =
A [i - e
,;1
h ( T f - T0) x
1 - A t IA 2
S 2[(l-<z> ) { P C P) S +</> ( p C F ) , ] ~ S
d
:
(4. 100)
C ,(S ) =
A ll 1 - A2 !Ax
MA - A)*
S 2[ ( W
) { p C P) s +<!> ( p C P) , ] - S
k, + (1 -t/> ) * , ) - D P( p C p ) l u,
D~l
(4 . 101)
Substitution into Equation 4.84 yields:
f
a, Y\
Ax
v
JJ
- p-'p
D 2P0 1 - exp
1
T (x ,t)-T
Tf - T
0
0
{kfj> + (1
f
D 2R 1 - exp
P
0.5(T0 +
0
)k,) + D p i p C ^ U f
f
"A
Ax
■eDp +
V D p JJ
{kjf> + ( l - 0 ) k s) + Dp( p C p),ul
(4. 102)
Equation 4.102 is the governing equation for microwave heating under one-dimensional
conditions. This equation can be altered to consider cylindrical coordinates as well as
spherical coordinates. In this investigation, the one-dimensional solution to the heat
transfer equation in porous media was used due to similarity to the lab experiments.
59
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4.7.2 Second Analytical M ethod
A second method of analytical solution using a direct method of solution of nonhomogeneous differential equations is also proposed. First, the non-homogeneous partial
differential equation, Equation 4.35, is transformed to homogeneous format. The
following transformation (Equation 4.103) is considered to reduce the non-homogeneous
part that is the microwave energy dissipation component of Equation 4.35.
-x
u + c e Dp = T
(4.103)
The differentiation of Equation 4.111 with regard to time gives Equation 4.112.
du
dT
~di = ~ d i
(4-104)
The first and second derivation with respect to length, x, results:
du
c
dx
Dp
d2u
c
dx
(4.105)
TT_
dx + D ] S
2
dT
vr~
~ dx
(4.106)
2
Substitution of derivatives 4.104 to 4.106 into the partial differential equation
(Equation 4.35) will result in:
du
(0 ki + ( W
d2u
) * ,)
c
^7
sT = ( 1 - 0 ){p CP), + 0 ( p C p), Cdx2 + D\ 6 P]
{
P e D' 1 - e
&x
~ D„
0
(pCp^u,
du
c
( 1 - 0 ) ( p c p)s +<j> (pCp), - hdxr - - —
D e
]+■( 1 - 0 ){pCP)s +0 ( p C p),
60
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(4.107)
Converting Equation 4.107 into homogeneous format results in:
fyk,+(\-<j> )kt)
(W
(P Cp),u,
'] +
) {p C P)s +</> ( p C p),
D
p ji
^ P
f
^
Ax
1 - exp
1Poee
{ \ - 4 ) ( j >Cp ) s +<I><j , C p) S D
(4.108)
(1 - ^ ) ( p C P)s + ^ ( p C p)l
From Equation 4.108 C may be determined:
f
-DP
2Po 1 - exp
Ax
V\
c=
(4.109)
kl +(l-<^)ks) + Dp( p C p)lul
The partial differential equation is transformed into:
du
dt
(M +(W )^)
&u
(1 -<j>)(pCP)s +</){pCp)l dx2
( p C p),ul
du
(\-</>)(pCp)s + $ { p C P)l dx
(4.110)
This is a homogeneous equation. The new boundary conditions are:
-x
u(x, 0) = T0 —ceDp
(4. I l l )
u(0,t) = T0 - c
(4.112)
du(x,0)
dx
(4.113)
du(0,t)
dt
(4.114)
________(P Cp )/ u{_______
2
(4.115)
(1 -<t> )(P CP) S +<f> (p Cp ),
61
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k, + (\-<f> ) k s)
(4 . 116)
K l = {\-</> )(P Cp )s + <j>( p C p ),
From these boundary conditions , the following can be determined:
du
(fu
du
— = K , — t - K 2—
dt
dx
dx
(4 . 117)
Further assuming that:
K tx
.2K,
u —v e '
(4. 118)
it is possible to show:
K 2 du
K 2 dv
K \
= -—
e 2Kl - v
2T - e 2Kx
K l dx
K x dx
2K l
(4 . 119)
and
d 2u
d 2v
dx2
dx1
I
^
■+
du
1 dv
K x dt
K x dt
dv K 2
— — e
dx 2 K x
1+ v
K 22
\~e
4K \
dv K 2
1 + --------- — e
dx 2 K
(4 . 120)
(4 . 121)
Substitution gives:
K2
2
dv
K] 8x2 ~ 4Kt V~ K] dt
(4 . 122)
The boundary conditions for the new assumption are:
u (0,0 = v(0,0
(4 . 123)
K 2x
'
2
K,
u ( x , 0 ) = v ( x , 0)
(4 . 124)
62
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dv(x, 0)
(4.125)
dx
dv(0 ,t)
0
=
dt
(4.126)
To reduce the equation to a more soluble form, the following are considered:
ML
v = we
4Kt
dv
dw
dt
dt
dv
dw
dx
dx
(4.127)
-
e
Kit
4 K,
K:
Ml
4 K,
(4.128)
4 K,
Kit
e
d 2v
d 2w
dx2
dx2
4 K,
(4.129)
Kit
4 K-,
(4.130)
This allows equation to be transformed as:
cdw
dw
1 dx2
dt
(4.131)
The initial and boundary conditions are:
Kit
v ( 0 , t ) = w 0e AKl
(4.132)
v(x, 0) = w(x, 0)
(4.133)
dw{x, 0) _ Q
(4.134)
dx
dw{0,t) _ q
(4.135)
dt
w = 0.5Vne u '
erfc {■
2y[kJ
4 k.
- t ) + e ’ 1 erfc{
-o
63
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(4.136)
v = 0.5Vn
1 erfc(-
4 k,
2^JkJ
-t) + e ' ' erfc(-
2 Jk~t
■o
V4 ^
(4. 137)
x
u = 0 .5 (T0 - c) erfc{-
-t ) + e /c‘ erfc{-
2V v
V
2T V
+
y 4^j
-o
(4 . 138)
v = 0.5(T0 - c )
erfc(;
k2
2yJkJ ]j 4kt
r"
-Z>2P
-L/)-r0 1- exp
Ax
xj-rpt) + e * 1 er/c(
x
■o
2Vv v4^
(4. 139)
A
v Dp j j
-eDp +
+ ( l - ^ ) * J) + Z)1,(p C ;,),M/
r
r
V\
Ax
D2P
^ p 10 1- exp
v Dp j j
x-kJ
x + fc,f
v
0.5(7; +
)x e//c(— f=^) + e 1erfc{— jf=-)
{krf> +{\-<j>)ks) + Dp{pCp)lul
2^Kt
T(x,t) =
(4. 140)
2Vv
- k2 ^ U - h(rf ~ T0)
(4. 141)
Applying the boundary condition shown in Equation 4.141 gives:
(
T(x,t)-T0
Tf - T 0
-D
P 0 1- exp
^p1
r
Ax
s\
v Dp jj
-eDp +
(k{j) +{\-<f>)ks) + Dp{pCp)lul
x
r
A
Ax
1- exp
/>
V Dp j j
x-—h/x
x
hJk/
0.5(7; +
erfc{—
p=-)
+
e
2 erfc{—
+ --------)
■)x
(krf +{\-<f>)ks) + D (pC \u,
'2 tJIcJ
(4. 142)
The outlet temperature is defined as the following in this method:
64
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T(!,t) = 1/2
1+
Tn
(Tf - T 0)
I
lT~+,^ir
I
hJkl
erfc(— j=-) + e 2 kl erfc{— j = + — ---- )
(4.143)
4.8.1 Analytical Solution to Model with Microwave at Outlet
Similar to the solution for the microwave at the inlet a transformation from nonhomogeneous to homogeneous is required when the source of microwave energy is
located at the outlet o f the sandpack. Since the second method of analytical solution is
more straightforward, it was used for the microwave at the outlet as follows:
u + ce p = T
(4.144)
The derivation with respect to time, and the first and second derivatives with
respect to distance yield Equations 4.145 to 4.147.
du
dt
dT
dt
du
dx
c
Dp
d2u
(4.145)
dT
dx
(4.146)
dT
dx2
c
~d7+lol
(4.147)
Applying the overall heat transfer equation, Equation 4.35, to the derivatives of
Equations 4.145 to 4.147, the homogeneous form of the heat transfer equation is derived:
dT
dt
k, + (1 -<j> )ks)
d2T
{ \ - ^ ) { p C p)5 +<f> {pCp)l dx2
To exp
v
D.p
1 - exp
J
dT
( l ^ ) f r C , ) 5 + ^ ( p C f )( 0.
i P C p ) l Ul
+
■
dx
v
(4. 35)
Dp j j
+ -
(1-
0)0oCrX+tipC,,),
65
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du
dt ~
( M + (i ~<t> )*,)
du
[ T T
){pCP)s +<f> ( p C P), Ldxl
c
D
7t
+ T r e l
Ax
1- e
-x
du
(P Cp) iui
+
h
r( l - 0 ) ( p C P)s + 0 ( p C P), dx
(4.148)
jr
D
e ' ]+
( W ) { p c P)s +0 ( p CP),
which can be further simplified to:
r r * ']■
( 1 - ^ ) ( p C P\ + </>{p CP)t bLDp
f
f
S'
Ax
P
1 - exp
1 oe
e
v
+ -
Pp jj
(p c P\ ui
i-TT* ‘’ ]
(1 -<j>)(pCp)s +<f>{pCP) , LD
(4.149)
=0
(1—
^){pcP)s +<t>( p C p),
The constant C must reduce the non-homogeneous form into a homogeneous format:
f
-D2P
p o 1 -e x p
V
Ax
W
j
j
c(4.150)
\^kl + ( X - < ^ ) k ) - D p{pCp)lul
This reduces the heat transfer equation to:
du
dt
(^*,+(1-0)*,)
{\-(/>){pCP)s +</> (pCP)l dx
( p C p)lul
du
(\-</>)(pCp)s +<j)(pCp)l dx
(4.151)
The conductive heat transfer is small compared to the convective heat transfer
exponent of Equation 4.151 meaning that the conductive heat transfer may or may not be
taken into account. For instance if the fluid flow passes metallic balls, the conductive heat
transfer plays a more dominant rule:
66
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(4.152)
(1 - ^ )(P Cp )s + <f> {p Cp ),
__________ (P c P)iui________
2 (1 -</> ) { p C P) s +</> { p C P)l
(4. 153)
Using these constants allows simplification of Equation 4.151
du
<fu
du
— = K ,— t + K 2—
dt
dx
dx
(4.154)
K 2x
Multiplying both sides of equation by e 1 gives:
1 du
d2u f
=
~ e
K x dt
K'
dx
+
K2 ^d u _
(4.155)
dx
which reduces to:
3m 7 ^
■e 1 =
K x dt
dx
1
K, x
du
dx
(4.156)
^ du
1— e 1
dx
ax v
(4.157)
K 7x
du
■= K e
1
dt
Equation 4.157 is solved by integration methods. Carslaw [58] provided the
following solution.
u = e rf
\K £
v
(4.158)
, K 2 J
This yields:
67
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/
- D 2P
1- exp
Ax
Y\
V D p ))
T(x,t)-T0
(krf + {\-<j>)ks) ~ D ( p C ),u,
Tf ~T 0
- e Dp +
(4.159)
erfc(P J W }
Equation 4.167 is the result of microwave heating at the outlet of the sandpack,
and can simulate microwave heating of the reservoir with the power source at the
production well.
4.9 Fluid and Rock Property Data and Supporting Data for Dielectric Properties
To solve the model, the physical and dielectric properties for sand, water, brine
and oil are required.
4.9.1 Correlations Used for Physical Properties of Water
p w= [0.01602 + 0.000023(-6.6 + 0.0325r+0.000657r)r (lbnVft3) T<400°F
9.97 - 0 . 0 4 6
- 3o2o \
f T
rr
1.8
Pw=
0. 00306
ct
-
(4 160)
3 2 ' x2
1.8
16
T > 400°F
(4.161)
Fhw, = 2.02 8 92 - 0. 01423 9 4 T + 4. 30191 x 1 0 ~5T 2- 5. 99485 x 10~ST 3
+ 3.9781 1 x 10~n T 4- l . 0 2089 x 10“14T 5 (B t u / h r - f t - ° F )
4 9 2 < T < 11 6 l°i?
(4.162)
Cw=4.482-1.5x10^7+3.44xl0_7r 2 +4.26xl0'873 (KJ/KgC)
(4.163)
(4.164)
.
E W ~ S wo +
g,o ~g
/-\2
1+{2nsJ)
(4.165)
1 , /o
2 ^ o /( ^ o “ O
l + ( 2 ^ 0/ )
,
- +
cr,^ o /
-
(4.166)
68
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where s w0 = 8.854x10“ F / m , is the absolute perm ittivity,/is the frequency in Hertz,
and s w = 4.9 is the high frequency limit for pure water and brine with any concentration.
The dielectric constant o f water ew0, the relaxation time r w, and the ionic conductivity cr.
can be expressed in terms of salinity of water and temperature.
s w0= s [ [ \ T ) x A { T , S w)
(4.167)
t w = t I p\ T ) x B ( T , S J
(4.168)
e~9’<
'T,SJ
(4.169)
cr. = crw (s) x
£(p) (T) = 87.134- 0.1949T -1.276 x 10“2T 2 + 2.491 x 10“4T 3
(4.170)
wQ
,4(7, S /) = 1.0 +1.613 x 10“57SW- 3.656 x l0 “35^+ 3.210 x l0 “5S 2 - 4.232 xlO“7S3
(4.171)
x l . l l l x l 0 “10- 3. 82 4x l 0“12r + 6.938xl0“,4T2-5 .0 9 6 x l0 “16r 3
T(j» (T) = -------------------------------------------- — ----------------------------------------------
(4 . 1 72 )
B(T, S J = 1.0+ 2.282 x 1QTsTSw- 7.638 xlO“X ~ 7.76 xlO“6S 2 + 1.105 x l 0 “X
(4.173)
o f ; (S) = 0.1825Sw -1.4619 x 10“X
+ 2.093 x 10“X
“ 1.282 x 10“' S*
(4.174)
<b(T, S ) = 2.033 x 10“2(25 - T) +1.266 x 10“4(25 - T )2 + 2.464 x 1 0“6(25 - T )3
(4.175)
-1.849 x 10”s(25 - T ) S W- 2.551 x 10“7(25 - T ' f ^ + 2 .5 5 1 x 1 0“8(25 - T ) 3Sw
4.9.2 Oil Physical Properties
Po = P oR- C / r - 6 0 ) + C 2( T - 6 0 ) 2
( l b m / f t 3)
p oR = 62.4278[141.5/131.5 + API]
(4.176)
(4.177)
69
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Cj = 0.0133+ 152.4/? ~^'45
(4. 178)
C2 =0.0000081-0.0622 xlO~00764/^
(4.179)
CPo =(0.3881 + 0 .0 0 0 4 5 ? ) / ^
(4.180)
= [1 .6 2 (1 -0 .0 0 0 3 (7 -3 2 ))]/^
(Btu/lbm-T)
(Btu/hr-ft-’F )
(4.181)
[35]
= O'0005
[35]
4.9.3 Correlations Used for Sand
khsand = 0-735-1.3^ + 7 ^ "
(Btu/hr-ft-°7)
CpsaM= 0.715 + 0.0017077-1.98 x lO^T2 (KJIKgQ
P sand
“
£ 'r-Sand
1 1 0
Ibmlf?
=3.87(F//?l) [35]
=0.000387
[35]
70
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(4.182)
(4> 183)
CHAPTER 5 RESULTS OF MATHEMATICAL MODEL STUDIES
5.1 Introduction
The ultimate intention of mathematical and computational modeling is to facilitate
a way for scientists and engineers to explore unknown areas of their investigations,
applying mathematical methods and today’s powerful computational abilities.
The goal of this study is to demonstrate the effects of different parameters, such
as frequency and power level, and/or characteristics of material under microwave
irradiation, such as initial temperature and salinity. A module-based program was written
utilizing VISUAL C++ programming language to simulate the entire microwave heating
process. The full code is provided in Appendix C.
5.2 Penetration Depth
The penetration depth of microwave radiations in different materials is an
important part of this investigation. Its importance stems from the ability of microwave
radiation to reach certain distances within the reservoir. In fact, this boundary is the
stimulation depth of the reservoir for which remedial operations will be effective for the
removal of wax depositions. Figure 5.1 demonstrates the microwave penetration depth
comparison for various media such as, oil, water, and sand, and different fractional flow
for o il a nd w ater flows at 2 450 M Hz frequency. T he s imulation r esults s how t hat t he
penetration depth for dry sand is greater than the length of the sandpack (i.e. 30 cm). The
laboratory results of experiments with sand confirmed these results.
71
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20
Power (watts)
15
10
•sand
•Water+sand
sand+oil+water
sand+oil
sand+10%watwe+oil
5
0
0
sand+5%water+oil
0.05
0.1
0.15
Length,(m)
0.2
0.25
0.3
lPenetration depth versus length for sand, oil, water and different concentration
inlet. Phi=30% initial Temp=20 °C, Incident Power=22 watts.
Figure 5.
72
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The results show that the penetration depth in porous media with lower water
saturation is greater than that for higher water saturations. The effect of microwave
frequencies for dry sand and sand saturated with water are shown in Figures 5.2 and 5.3,
respectively.
73
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Power (Watts)
Penetration Depth Vs. Frequency
950 MHz
1350 MHz
2450 MHz
16 -
0
0.05
0.1
0.15
0.2
0.25
0.3
Depth (m)
Figure 5.2 Penetration depth for sand versus length for different frequencies. Phi=30%
initial Temp=20 °C, Incident Power=22 watts.
74
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20
-
-
2450 MHz
- 1350 MHz
-------------950 MHz
Power (watts)
15
10
5
0
0
0.02
0.04
0.06
0.08
0.1
0.12
Length (m)
Figure 5.3 Penetration depth for water saturated sand versus length for different
frequencies. Phi=30% initial Temp=20 °C, Incident Power=22 watts.
75
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0.14
This shows that lower frequencies have better penetration depth compared to the
higher frequencies. On the other hand, as Figures 5.4 and 5.5 illustrate, despite narrower
penetration depth, heating effects of higher frequency microwaves are higher in
comparison to lower microwave frequencies for identical microwave inlet power.
76
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Power (Watts)
1°
Absorbed
14
6
8
-
— “ - — 950 MHz
4 -
—
-
—
- 1350 MHz
■— 2450 MHz
2
00
0.05
0.1
0.15
Length
0.2
0.25
Figure 5.4 Absorbed power after 1000 sec. irradiation for sand versus length for different
frequencies. Phi=30% initial Temp=20°C, Incident Power=22 watts.
77
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0.3
3000
2500 -
— 2450 MHz
\
-
1350 MHz
950 MHz
g
a>
o 1500
Q_
O
| 1000
500
0.005
0.01
0.015
0.02
0.025
Length (m)
0.03
0.035
0.04
0.045
Power absorption for water saturated sand versus length after 1000 sec. for
different frequencies. Phi=30% initial Temp=20 °C, Incident Power=22 watts.
Figure 5 .5
78
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0 .0 5
Additionally, a closer examination of the results shown in Figures 5.1 through 5.5
indicates that water absorbs microwave irradiations better than either oil or sand. The
microwave energy will be absorbed by the liquids, especially water, that are present in
the experimental environment and consequently, the penetration depth decreases as the
water saturation of porous media increases.
5.2 Effect of Microwave Power
Heating effects and penetration depth of microwaves are power dependent. This
means that for similar frequencies, higher microwave power will produce more heat and
will penetrate deeper into porous media. As indicated, the penetration depth inside the
reservoir is important because it is the depth which microwaves can be induced to reduce
wax precipitation for conformance control and/or well stimulation purposes after build up
of wax. Figure 5.6 shows the heating effect, as the outlet temperature profile, for a
sandpack with water flow rate of 10 ml/min and different inlet powers. No oil was
present for this simulation and the injection and microwave were co-current.
79
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250
power=10
200
power=30
power=40
power=75
150
CL
100
0
1000
2000
3000
4000
5000
6000
Tim e(s)
Effect of microwave power on outlet temperature. Phi=30% initial Temp=20
°C, injection flow rate=10 ml/min.
Figure 5 .6
80
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Figure 5.7 illustrates the temperature profile in the case wherein microwave and flow
directions were counter-current. These results confirm the fact that higher power inlet
will increase the outlet temperature profile.
81
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210
190
170
Temperature (C)
150
130
110
70
*■■20 watts
■ " 3 0 Watts
— 60 Watts
10
0
1000
2000
3000
4000
5000
6000
Tim e (Sec.)
Figure 5.7 Water temperature profile at the outlet of sandpack. Injection rate:10 ml/min
and microwave and flow directions are co-current. Phi=30% initial Temp=20°C.
82
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The temperature distributions inside the sand pack for different power levels are
displayed in Figure 5.8. These results suggest that inlet microwave powers in the range of
10 to 30 watts are desirable for the conditions represented in the experiments.
83
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400
power=KD
350
•power=30
•power=40
■power=75
300
Temp, (c)
250
150
100
0
0.05
0.2
0.1
0.25
Figure 5 .8 Temperature distribution in sandpack saturated with water and water flow
m l/m in after 6000 seconds. Phi=30% initial Temp=20°C, Injection rate=10 ml/min.
84
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0.
at
5.4 Effect of Flow Rate
Flow rate plays an important role in microwave heating of reservoirs because of
its effects on the convection and forced convection elements of heat transfer in porous
media. In order to simulate the effects of flow rate on microwave heating and temperature
out put, water and oil at different flow rates were simulated and these are displayed in
Figures 5.9 and 5.10.
85
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80
-
w a te r u = 2 0
w a te r u = 1 0 + oil u= 10
P.30
20
1000
2000
3000
4000
5000
Time (sec)
6000
The effect o f fractional flow rate on outlet temperature profile. Phi=30% initial
Temp=20 °C, Incident Power=22 watts.
Figure 5 .9
86
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120
100
a .
40
MW a t outlet 10 ml/min
MW a t outlet 5 ml/min
“ ■ ^ M W outlet 20 ml/min
0
1000
2000
3000
4000
5000
6000
Time (se c )
Effect of flow rate on outlet temperature profile for the case microwave and
fluid flow are counter-current. Phi=30% initial Temp=20 °C, Incident Power=22 watts.
Figure 5.10
87
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Figure 5.11 is the outlet temperature profile for water and water-oil mixture in
different flow rates. Whenever the water-oil mixture is injected the outlet temperature
profile is close to the flow rate of water portion of water-oil mixture. This confirms that
the microwave adsorption of oil and its products is lower compare to water.
88
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= 50
Q.
■water u=20
water u=10 + oil u=10
■water u=20 + oil u=20
•water u=10
H 30
1000
2000
3000
4000
5000
6000
Time (sec)
Outlet temperature profile versus time for various injection rates and fractional
flow rates, phi=30%, Incident power= 22 watts, Initial Temp=20 °C.
Figure 5.11
89
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Figure 5.12 shows the comparison between the co-current and counter-current
flow and effect of microwave power for the co-current and counter-current conditions.
Here again the higher flow rates reduce the temperature
90
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160
............... MW a t inlet
140
“ ■“ ^ ^ M W a t outlet 10 ml/min
MW outlet 5 ml/min
------------ MW a t outlet 20 ml/min
100
Q.
E
-------- —
<D
b-
—
60
40 -
20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Length (m)
Temperature distribution profile for different water flow rates and microwave
at the outlet and comparison with microwave at the inlet. Phi=30% initial Temp=20 °C,
Incident Power=22 watts.
Figure 5.12
91
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5.5 Effect of Inlet Temperature
Another important condition occurs when the inlet temperature varies. In this case,
although the microwave power is constant, the outlet temperature must increase as shown
in Figure 5.13.
92
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180
160 -
cp=30 %
time =6000
initial temp=lo
initial temp=20
initial temp=30
tem p (c)
initial temp=40
3000
Tim e(s)
6000
Effect of inlet temperature on outlet temperature profile when microwave and
fluid flow are co-current. Phi=30%, Incident Power=22 watts.
Figure 5.13
93
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Figure 5.14 shows temperature distribution when the microwave is located at the
inlet of the sandpack. The frequency for the above simulations is 2450MHz. The
simulation shows that higher inlet or initial temperature results in a higher outlet
temperature profile. An ambient temperature of 20 °C was selected for the experiments.
94
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250
initial temp=1Q
‘initial temp=30
■initial temp=40
■initial temp=50
Temperature (c)
200
150
100
0
0.05
0.1
0.15
Length (m)
0.2
0.25
0.3
Temperature distribution profile for various inlet temperatures for co-current
microwave and fluid flow. Phi=30%, Incident Power=22 watts.
Figure 5.14
95
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5.6 Effect of Salinity
Salinity of water has a big impact on the heat production in sandpack and
reservoir. Figure 5.15 shows the effect of salinity on the outlet temperature profile. The
simulation was conducted using fresh water, 1 wt% brine, and 2 wt% brine. The results
show that the presence of ionic compounds such as NaCl cause an increase in outlet
temperature. This increase is due to the current induced by ionic transport in fluids. The
results suggest that pure water and
1
wt% brine are suitable choices for the experiments
in order to maintain the temperature within reasonable bracket.
96
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120
100
■- - Pure water
— - 1% NaCI solution
— 2% NaCI Solution
60
a.
40
0
1000
2000
3000
Time (S)
4000
5000
Outlet temperature profile versus time for different salinities, Phi=30%,
Incident Power=22 watts, initial temperature=20 °C.
Figure 5.15
97
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6000
CHAPTER 6
Experimental Results and Discussions
6.1 Introduction
Porous media consists of pore space and a solid matrix. The pore spaces are
typically connected, allowing transport processes such as fluid flow to take place inside
porous media.
The results and procedures for the laboratory experiments are presented in this
chapter. Primary experiments include the tests for material selection, microwave power
level selection, and optimal flow rate settings. The sandpack material selections are made
in order to test the compatibility of material with microwave radiation.
6.2 Primary Experiments
The primary experiments included two sets of tests for sandpack material selection
and a series of experiments for choosing microwave power level, flow rate, and salinity.
6.2.1 Material Selection Experiments and Procedures
The orientation of investigations on microwave heating and the microwave
characteristics requires utilizing materials for the sandpack that are resistant to the high
temperatures and are transparent to microwave radiation. This transparency allows the
sandpack to convey microwave radiation, with minimum absorption, to the water and oil
in porous media. The importance of determining the transparency of sand stems from the
98
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fact that higher penetration depth increases the microwave sweep for near wellbore wax
removal.
6.2.1.1 Sandpack Material Selection
A total of nine experiments with different materials were performed to select the
material possessing the appropriate resistance and transparency to microwave radiation
and the mechanical properties from which the sandpack can be constructed. Table 6.1
shows the materials, results of the experiment, and the length of time for each ran of the
experiments.
99
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No.
1
2
3
4
5
6
7
8
9
Name of Material
Under
Investigation
Plexiglas
Polycarbonate
Nylon
Polyurethane,
Heavy
Polyurethane,
Medium
Polyurethane, Light
Power, Watts
Melamine
Medium Teflon®
Heavy Teflon®
Time, Min
Result
50
50
30
30
20
Burned
Burned
Burned
Degraded
30
15
Degraded
30
8
50
50
50
40
60
60
10
12
10
Deformed and
Degraded
Burned
Good
Good
Table 6 . lThe material under investigation for the sandpack and the results of
experiments.
100
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In these experiments, a cylindrical sample with a height of 2 cm and a diameter of
1 cm was placed in a 30 cm long waveguide and then radiated with the desired power
level (Table 6.1).
6.2.1.2 Results of Experiments with Sand
The first series of experiments with sand was designed to investigate the
applicability of available sand samples as filling material for the sandpack, and to
measure the penetration depth of the microwave in dry sand. Sand primarily consists of
silica so it is expected that sand is not a good absorber for microwave radiation.
After assembling the sand-filled wave guide and microwave device, microwave
with 764 watts incident power was irradiated, and the same time, the inlet, outlet, and
reflected amounts of microwave were recorded. Figure 6.1 shows the diagram for
experiments with a sand-filled waveguide.
101
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Power
Figure 6.
lThe setup for experiment with sand-filled waveguide
102
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The energy balance for calculating the amount of absorbed power for experiments
with sand is as follows.
power absorbed = incident power - reflected power - power passed
Since the incident power = 764 Watts and reflected power = 615 and power passed = 91
The amount o f microwave absorption in sand is equal to 7.6% of total incident
power. These results confirm that sand is not a good absorbent material for microwave
radiations.
A second experiment was conducted resulting in the following measurements:
incident power = 764 Watts
reflected power = 615
power passed = 96
Thus, using the same energy balance, the power absorbed is 53 Watts.
Again, the amount of power absorption is approximately 7.2% of the total.
6.2.2 Primary Flow Experiments with Fresh Water
The first experiment was conducted with fresh water at 20 °C and 1 ml/min flow
rate. For t his e xperiment, t he m icrowave p ower w a s s e t a t 1 5 0 w atts. The m icrowave
incident power was absorbed by the water in the sandpack. The effluent temperature
increased dramatically after 5 minutes and large volumes of steam formed at the outlet.
This indicated that the microwave power should be reduced to control the temperature
within the porous medium. After a series of experiments, an incident power of 22 watts
was determined to be optimal.
103
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In the second set of experiments the outlet temperature of the effluent was
measured using a mercury thermometer and a temperature profile for the outlet
temperature was developed. The temperature was recorded every 60 seconds and the
recordings were transferred to a Microsoft ™ Excel© spreadsheet. Figures 6.2 and 6.3
show the temperature output profiles for injection rates of 10 and 5 ml/min, respectively.
104
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45
40
Temperature (c)
35
30
25
20
15
0
1000
2000
3000
4000
5000
Time (S)
Figure
6
.2 The outlet temperature profile for water injection of 10 ml/min.
105
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6000
50
45
Temperature (c)
40
35
30
25
20
15
0
500
1000
1500
2000
2500
3000
3500
Time (S)
Figure 6 .3
The outlet temperature profile for water injection of 5 ml/min.
106
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4000
To increase the accuracy of the measurements, a thermocouple was installed at
the outlet o f the sandpack. Figure 6.4 gives an example of the results of one of the
experiments in which the temperature profile was recorded using the thermocouple.
107
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5000
10000
15000
Time (S)
Figure 6 .4 The first experiment using thermocouple. 10 ml/min injection rate, fresh water,
initial temperature=20 °C, incident power=22
108
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6.3 Experiments with Brine
This series o f experiments was designed and conducted to investigate the effect of
salinity on the temperature profile in porous media under microwave radiation. Injection
fluids with brine concentration of 0.2 and 1 wt% brine were used. The microwave power
was set at 22 and 15 watts for these tests. The results show that the outlet temperature for
brine is higher compared to the outlet temperature for fresh water (Figures 6.5 and 6 .6 ).
109
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Temperature (C)
60 50 40 -
20
-
0
2000
4000
6000
8000
10000
12000
14000
Time (S)
Figure 6 .5 Outlet temperature profile for experiment with 1 wt% brine, injection flow
rate= 1 0 ml/min, incident power = 2 2 watts
110
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80
70
Temperature (C)
60
50
40
30
20
10
0
0
2000
4000
6000
Time (S)
8000
10000
12000
Figure 6 . 6 Outlet temperature profile for experiment with 0.2 wt% brine, injection flow
rate= 1 0 ml/min, incident power = 2 2 watts
111
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14000
6.4 Experiments with Paraffin
To simulate the paraffin removal from the near wellbore area, a series of
experiments was conducted with a blend of 5% paraffin and 95% sand. The sandpack
was filled with the selected paraffin-sand mixture and was then saturated with water.
Water injection was initiated in this porous medium while the pressure drop was
monitored across the sandpack and the temperature at the outlet was measured.
11
was
hypothesized that if applying the electromagnetic heating is effective, the pressure drop
will decrease as the temperature increases.
To simulate the microwave stimulation in production wells, the microwave
incident power was applied at the outlet of the sandpack (Figure 6.7). In this experiment
the o utlet t emperature i ncreased, w hich i s t he i ndicative o f h eating d ue t o t he applied
electromagnetic energy. Also, the pressure drop across the sandpack diminished due to
the melting and removal o f the paraffin from the sandpack. The presence of paraffin was
observed in the effluent collected from sandpack. Solid paraffin particles were formed
upon the cooling o f the effluent. The inspection of the sandpack after the paraffin
removal experiments showed that the sand near the outlet, analogues to the near wellbore
region, was clean from paraffin while further away from the outlet the paraffin was still
present.
112
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P re s s u re
F lo w ra te = 1 0 m l/m in
Incident P o w er* 4 0 w atts
5% paraffin in sa n d
Pressure (psi)
Temperature (c)
‘T e m p e ra tu re
1000
Tim e (S)
Figure 6 .7 The experiment for wax removal. The experiment shows lower pressure
gradients for after wax removal procedure.
113
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CHAPTER 7 CONCLUSION AND RECOMMANDATIONS FOR
FUTURE STUDIES
7.1 CONCLUSIONS
The modeling and experiments procedure proved the effectivity of microwave
applications for wax removal for better flow and production in near wellbore
conformance control. A list of achievements is as followed.
1. A mathematical model was developed for the heating of porous media under
electromagnetic heating conditions.
2. Analytical solutions were provided for the proposed mathematical model.
3. The heating effect of microwave in porous media is a function of flow rate, as
experiments at 5, 10 and 15 ml/min flow rates indicated.
4. Sand absorbs about 7% of the incident power of microwave energy.
5. Microwave heating is applicable for porous media under fluid flow.
6
. Experiments indicated that microwave irradiation could be used effectively for
melting and removing paraffin.
7.2 RECOMMENDATIONS
1. Applying a new method of measuring the temperature inside the waveguide such
as liquid crystal or infrared detectors in order to study the temperature gradient
inside the porous media.
2. Apply implicit and explicit numerical methods for investigating the temperature
gradient and outlet temperature.
114
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3. Apply larger scale lab methods and non-cylindrical sandpacks in order to examine
the 2 -D a nd 3 -D e ffects, a nd s olve t he g oveming e quations u sing a nalytical o r
numerical methods.
4. Investigate using a variable frequency microwave device for both wax and
asphaltene scales.
115
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Properties of Athabasca Oil Sands", Canadian Journal of Earth Science, 16, 20092021
49- Bisisio R. G., Cambon J.L, Chavarie C., Klvana D. (1977), “Experimental Results on
Heating of Athabasca Tar sand Samples with Microwave power”, Journal O f
Microwave Power, 12, 4
50-E.R. Abemethy (1976), “Production Increase of Heavy Oils by Electromagnetic
Heating”, Journal of Canadian Petroleum Tech., Montreal
51-Davison R.J. (1995), “Electromagnetic Stimulation of Lloyd Minster Heavy Oil
Reservoirs: Field Test Results”, Journal of Canadian Petroleum Technology
52-McPherson R.G. (1985), Chute F.S., Vermeulen F.E., “Recovery of Athabasca
Bitumen with Electromagnetic Flood (EMF) Process”, Journal of Canadian Petroleum
120
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Technology
53 - Wadadar, S.S., Islam, M.R., (1994), "Numerical Simulation of Electromagnetic
Heating of Alaskan Tar Sands Using Horizontal Wells", Journal of Canadian
Petroleum Technology, 33, 7, 37-43
54- Weingarten, J.S., Euchner, J.A. (1986), "Methods for Predicting Wax Precipitation
and Deposition", Society of Petroleum Engineers, Society of Petroleum
Engineering, 15654
55- Hsu, J.J; Santamaria, M.M; Brubaker (1984), "W ax Deposition of Waxy Live
Cmde’s Under Turbulent Flow Conditions", Journal of Petroleum Engineering,
Society o f Petroleum Engineering, 28480
56- Hamouda, A. A., Davidsen, S. (1995), "A n Approach for Simulation of Paraffin
Deposition in Pipelines as a Function of Flow Characteristics With a Reference to
Teesside Oil Pipeline", Society of Petroleum Engineers, 28966
121
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APPENDIX A Pressure Transducer Calibration
The calibration of the pressure transducer requires two steps. The first step is to
measure the voltage outlet o f the transducer for two or more known pressures. In order to
find these values, a 4 ’, 1/8” tubing is attached to one of the inlets of the pressure
transducer and the other inlet is set at atmospheric pressure as the reference pressure. The
outlet voltage is measured for two different water column heights.
The second step is to calculate the calibration factor. The calculation for
determination of the voltage difference for one psi pressure gradient is as follows:
Height
mVolt
Ap
0
-2.4
0
29"
3.2
0.8265
Ap = 4.3926 m V ! psi
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APPENDIX B
THE SIM ULATION PROGRAM
// Analytical SOl.h : main header file for the ANALYTICAL SOL application
//
#if
!defmed(AFX_ANALYTICALSOL_H_335B3F8A_B6Dl_437C_866B_0CA923D2D2
21 _ I N CLUDED_)
#defme
AFX_ANALYTICALSOL_H_335B3F8A_B6Dl_437C_866B_0CA923D2D221_INC
LUDED_
#if_MSC_VER > 1000
#pragma once
#endif// _MSC_VER > 1000
#ifndef_A F X W IN _H _
#error include 'stdafx.h' before including this file for PCH
#endif
#include "resource.h"
// main symbols
/////////////////////////////////////////////////////////////////////////////
// CAnalyticalSOlApp:
// See Analytical SOl.cpp for the implementation of this class
//
class CAnalyticalSOlApp : public CWinApp
{
public:
CAnalyticalS01App();
// Overrides
// ClassWizard generated virtual function overrides
// {{AFX_VIRTUAL(C AnalyticalSOlApp)
public:
virtual BOOL Initlnstance();
//}} AFX_VIRTUAL
// Implementation
//{{AFX_MSG(CAnalyticalS01App)
// NOTE - the ClassWizard will add and remove member functions here.
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// DO NOT EDIT what you see in these blocks of generated code !
//}}AFX_MSG
DECLARE_MESSAGE_MAP()
};
/////////////////////////////////////////////////////////////////////////////
//{{AFX_INSERT_LOCATION} }
// Microsoft Visual C++ will insert additional declarations immediately before the
previous line.
#endif //
!defined(AFX_ANALYTICALSOL_H_335B3F8A_B6Dl_437C_866B_0CA923D2D2
21 INCLUDED_)
/ Analytical SOl.cpp : Defines the class behaviors for the application.
//
#include "stdafx.h"
#include "Analytical SOl.h"
#include "Analytical SOlDlg.h"
#ifdef _DEBUG
#defme new DEBUG_NEW
#undefTHIS_FILE
static char THIS_FILE[] = _ F I L E _ ;
#endif
/////////////////////////////////////////////////////////////////////////////
// CAnalyticalSOlApp
BEGINJV1ES SAGEJVlAP(CAnalyticalS OlApp, CWinApp)
//{{AFX_MSG_MAP(CAnalyticalS01App)
// NOTE - the ClassWizard will add and remove mapping macros here.
// DO NOT EDIT what you see in these blocks of generated code!
//}}AFX_MSG
ON_COMMAND(ID_HELP, CWinApp ::OnHelp)
END_MES SAGE_MAP()
lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll/lllllllllllllll
II CAnalyticalSOlApp construction
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CAnalyticalSOlApp:: CAnalyticalS 01App()
{
// TODO: add construction code here,
// Place all significant initialization in Initlnstance
}
/////////////////////////////////////////////////////////////////////////////
// The one and only CAnalyticalSOlApp object
CAnalyticalSOlApp theApp;
/////////////////////////////////////////////////////////////////////////////
// CAnalyticalSOlApp initialization
BOOL CAnalyticalSOlApp::lnitlnstance()
{
if (! AfxSocketInit())
{
AfxMessageBox(IDP_SOCKETS_INIT_F AILED);
return FALSE;
}
// Initialize OLE libraries
if (!Afx01eInit())
{
AfxMessageBox(IDP_OLE_INITJF AILED);
return FALSE;
}
AfxEnableControlContainer();
// Standard initialization
// If you are not using these features and wish to reduce the size
// of your final executable, you should remove from the following
// the specific initialization routines you do not need.
#ifdef_AFXDLL
Enable3dControls();
DLL
#else
Enable3dControlsStatic();
#endif
// Call this when using MFC in a shared
// Call this when linking to MFC statically
// Parse the command line to see if launched as OLE server
if (RunEmbeddedO || RunAutomated())
{
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// Register all OLE server (factories) as running. This enables the
// OLE libraries to create objects from other applications.
COleTemplateServer: :RegisterAll();
}
else
<
// When a server application is launched stand-alone, it is a good idea
// to update the system registry in case it has been damaged.
COleObj ectF actory: :UpdateRegistryAll();
}
CAnalyticalSOlDlg dig;
m_pMainWnd = &dlg;
int nResponse - dlg.DoModal();
if (nResponse = IDOK)
{
// TODO: Place code here to handle when the dialog is
// dismissed with OK
}
else if (nResponse = IDCANCEL)
{
// TODO: Place code here to handle when the dialog is
// dismissed with Cancel
}
// Since the dialog has been closed, return FALSE so that we exit the
// application, rather than start the application's message pump,
return FALSE;
}
// Analytical SOlDlg.cpp : implementation file
//
#include
#include
#include
#include
#include
"stdafx.h"
"Analytical SOl.h"
"Analytical SOlDlg.h"
"DlgProxy.h"
"Math.h"
#ifdef_DEBUG
#defme new DEBUG_NEW
#undef THIS_FILE
static char THIS_FILE[] = FILE
#endif
;
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FILE *filel, *file2, *file3, *file4;
/////////////////////////////////////////////////////////////////////////////
// CAboutDlg dialog used for App About
class CAboutDlg : public CDialog
{
public:
CAboutDlg();
// Dialog Data
// {{AFX_DAT A(C AboutDlg)
enum { IDD = IDD_ABOUTBOX };
//}} AFX_DATA
// ClassWizard generated virtual function overrides
//{{AFX_VIRTUAL(C AboutDlg)
protected:
virtual void DoDataExchange(CDataExchange* pDX);
//}} AFX_VIRTUAL
// DDX/DDV support
// Implementation
protected:
// {{AFX_MSG(C AboutDlg)
//}}AFX_MSG
DECLARE_MESSAGE_MAPO
};
CAboutDlg::CAboutDlg(): CDialog(CAboutDlg::IDD)
{
//{ {AFX_DATA_INIT(CAboutDlg)
//} }AFX_DATA_INIT
}
void CAboutDlg: :DoDataExchange(CDataExchange* pDX)
{
CDialog: :DoDataExchange(pDX);
//{{AFX_DATA_M AP(C AboutDlg)
//} }AFX_D ATA_M AP
}
BEGIN_MESSAGE_MAP(CAboutDlg, CDialog)
//{{AFX_MSG_MAP(C AboutDlg)
// No message handlers
//}} AFX_MSG_MAP
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END_MESSAGE_MAP()
/////////////////////////////////////////////////////////////////////////////
// CAnalyticalSOlDlg dialog
IMPLEMENTJ)YNAMIC(CAnalyticalS01Dlg, CDialog);
CAnalyticalS01Dlg::CAnalyticalS01Dlg(CWnd* pParent /*=NULL*/)
: CDialog(CAnalyticalS01Dlg::IDD, pParent)
{
// {{AFX_D ATA_INIT (CAnalyticalS OlDlg)
m jphi = 0 .0 ;
m_initemp = 0 .0 ;
m_t = 0 .0 ;
m_deltax = 0 .0 ;
m_velo = 0 .0 ;
m_p 0 = 0 .0 ;
m_sat = 0 .0 ;
m_length = 0 .0 ;
m_api = 0 .0 ;
m_salt = 0 .0 ;
m_deltat = 0 .0 ;
m_uoil = 0 .0 ;
m f r e c = 0 .0 ;
m_lll = _T("");
//}} AFX_DATA I N I T
// Note that Loadlcon does not require a subsequent Destroylcon in Win32
m_hIcon = AfxGetApp()->LoadIcon(IDR_MAINFRAME);
mjpAutoProxy = NULL;
}
CAnalyticalSOlDlg: :~CAnalyticalS01Dlg()
{
// If there is an automation proxy for this dialog, set
// its back pointer to this dialog to NULL, so it knows
// the dialog has been deleted,
if (mjpAutoProxy != NULL)
m_pAutoProxy->m_pDialog = NULL;
}
void CAnalyticalS01Dlg::DoDataExchange(CDataExchange* pDX)
{
CDialog::DoDataExchange(pDX);
//{{AFX_DATA_MAP(CAnalyticalS01Dlg)
DDX_Text(pDX, IDC EDIT1, m_phi);
DDX_Text(pDX, IDC_EDIT2, m_initemp);
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DDX_Text(p.DX, IDC_EDIT3, m_t);
DDX_Text(pDX, BDC_EDIT4, m_deltax);
DDX_Text(pDX, IDCJEDIT5, m_velo);
DDX_Text(pDX, E)C_EDIT 6 , m_pO);
DDX_Text(pDX, IDC_EDIT7, m_sat);
DDX_Text(pDX, IDCJBDIT 8 , m jength);
DDX_T ext(pDX, IDC_EDIT9, m_api);
DDX_Text(pDX, IDC_EDIT10, m_salt);
DDX Text(pDX, E)C_EDIT11, m_deltat);
DDX_T ext(pDX, IDC_EDIT12, m_uoil);
DDX_Text(pDX, IDC_EDIT13, m_frec);
DDX_Text(pDX, IDC_EDIT14, m_lll);
//}} AFX_D AT A_MAP
}
BEGIN_MESSAGE_MAP(CAnalyticalS01Dlg, CDialog)
//{{AFX_MSG_MAP(CAnalyticalS01Dlg)
ON_WM_SYSCOMMAND()
ON_WM_CLOSE()
ON_WM_P AES1T0
ON_WM_QUERYDRAGICON()
//}} AFX_MSG_MAP
END_MESSAGE_MAP()
/////////////////////////////////////////////////////////////////////////////
// CAnalyticalSOlDlg message handlers
BOOL CAnalyticalSOlDlg: :OnInitDialog()
{
CDialog::OnInitDialog();
// Add "About..." menu item to system menu.
// IDM_ABOUTBOX must be in the system command range.
ASSERT((IDM_ABOUTBOX & OxFFFO) = IDM_ABOUTBOX);
ASSERT(IDM_ABOUTBOX < OxFOOO);
CMenu* pSysMenu = GetSystemMenu(FALSE);
if (pSysMenu != NULL)
{
CString strAboutMenu;
strAboutMenu.LoadString(IDS_ABOUTBOX);
if (!strAboutMenu.IsEmpty())
{
pSysMenu->AppendMenu(MF_SEPARATOR);
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pSysMenu->AppendMenu(MF_STRING, IDM_ABOUTBOX,
strAboutMenu);
}
}
// Set the icon for this dialog. The framework does this automatically
// when the application's main window is not a dialog
SetIcon(m_hIcon, TRUE);
// Set big icon
SetIcon(m_hIcon, FALSE);
// Set small icon
// TODO: Add extra initialization here
return TRUE; // return TRUE unless you set the focus to a control
}
void CAnalyticalSOlDlg: :OnSysCommand(UINT nID, LPARAM IParam)
{
if ((nID & OxFFFO) = IDM_ABOUTBOX)
{
CAboutDlg dlgAbout;
dlgAbout.DoModal();
}
else
{
CDialog: :OnSysCommand(nID, IParam);
}
}
// If you add a minimize button to your dialog, you will need the code below
// to draw the icon. For MFC applications using the document/view model,
// this is automatically done for you by the framework.
s dens CAnalyticalSOlDlg::dens(double temp, double api)
{
s_dens ro;
double roor,cl,c 2 ,tk;
tk=temp* 1.8+32;
roor=62.4278*141.5/(131.5+api);
cl=0.0133+152.4*pow(roor,-2.45);
c2=0.000008 l-0.0622*pow(10,-0.0764*roor);
ro.o=(roor-cl*(tk-60)+c2*(tk-60)*(tk-60))*16;
if (tk < 400) ro.w=16/(.01602+.000023*(-6.6+.0325*tk+.000657*tk*tk));
if (tk >= 400) ro.w=(9.97-.046*temp-.00306*pow(temp,2));
ro.s=1760;
ro.l=(ro.w*m_sat+ro.o *( 1 -m_sat)) *16;
retum(ro);
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}
s_term CAnalyticalSOlDlg: :termal(double temp, double api)
{
s_term kk;
double tk;
tk=temp* 1.8+492;
kk.s=(l .735-1.3*m_phi)* 1.730734666;
kk.o=1.62*(l-.0003*(temp-32))*(131.5+api)/141.5;
kk. w=2.02892-0.0142394*tk+4.30191E-5*pow(tk,2)-5.99485E8*pow(tk,3)+3.9781 IE -11 *pow(tk,4)-1.02089E-14*pow(tk,5);
kk.l=(kk.w*m_sat+kk.o*(l-m_sat))* 1.730734666;
retum(kk);
s_htc CAnalyticalSOlDlg: :heatcapa(double temp, double api)
{
s_htc cp;
double tk;
tk=temp* 1.8+32;
cp.o=4.19*(.3881+.00045*tk)/sqrt(141.5/(131.5+api));
if (temp < 240) cp.w=4.482-.00015*temp+3.44E-7*pow(temp,2)+4.26E8*pow(temp,3);
if (temp >= 240) cp.w=l 1.55-.064518*temp+1.5087E-4*pow(temp,2);
cp.s=.715+0.001707*temp-l .908E-6*pow(temp,2);
cp .l=cp.w*m_sat+cp.o *( 1 -m_sat);
retum(cp);
}
s_visc CAnalyticalS01Dlg::viscosity(double temp, double api)
{
s_visc mu;
double tkk,tk;
tkk=temp* 1.8+492;
tk=temp* 1.8+32;
mu. w=-121.3274+48768.8/tkk-7.62292E7/pow(tkk,2)+5.91509E10/pow(tkk,3)2.28157E13/pow(tkk,4)+3.53226E15/pow(tkk, 5);
mu.o= 1 0 0 ;
retum(mu);
s_dpp CAnalyticalSOlDlg: :loss(double temp, double s)
{
s_dpp dp;
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double
fft,ff,bt,at,epswp,epswz,ewo,sigi,sigmais,tw,pp,salt,ewop,twp,t,delt,deltax,epswpp,epswz
z,tpf;
t=temp;
salt=m_salt;
deltax=m_deltax;
ewop=87.134-.1949*t-0.01276*t*t+.0002491*pow(t,3);
at=1.0+.00001613*t*salt-.003656*salt+.0000321*pow(salt,2).0000004232*pow(salt,3);
ewo=ewop*at;
twp=0.0000000000176821141-0.6086085E-12*t+l. 1042E-14*pow(t,2)~
8.1105359E-17*pow(t,3);
bt=1.0+.00002282*t*salt-.0007638*salt.00000776*pow(salt,2)+.00000001105*pow(salt,3);
tw=twp*bt;
sigmais=.18225*salt-.0014619*pow(salt,2)+.00002093*pow(salt,3).0000001282*pow(salt,4);
delt=t-25;
pp=0.002033*delt+.0001266*pow(delt,2)+2.464E-6*pow(delt,3).00001849*delt*salt+2.551E-7*pow(delt,2)*salt+2.551E-8*pow(delt,3)*salt;
sigi=sigmais*exp(-l *pp);
// if (m_frec = 0) m_frec=2450;
tpf=m_frec*6.2831* 1 0 0 0 0 0 0 ;
epswp=4.9+(ewo-4.9)/(l+pow(tpPtw,2));//15386000000=2*p*f
epswz=tpf:tw*(ewo-4.9)/(l+pow(tpf|stw,2))+sigi/tpf;
epswpp=mjphi* (m_sat* epswp+( 1 -m_sat) * 1 .9)+( 1 -m_phi)* 3.87;
epswzz=m_phi*(m_sat*epswz+(l-m_sat)*0.0005)+(l-m_phi)*0.0001;
if (m _ sa t= 0 ) {
ep swpp=( 1-m_phi) *3 . 8 7 *m_frec/2450;
epswzz=(l-m_phi)*0.0001*m_frec/2450;
}
ff=epswpp*(sqrt(l+pow(epswzz/epswpp,2 )-l));
fft=0.0097441802*sqrt(2/ff);
dp.p=fft;
retum(dp);
s_coe CAnalyticalS01Dlg::factor(double t, double x)
{
double api,bet;
s_coe z;
api=m_api;
s_dens ro;
sjhtc cp;
s_term kk;
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s_dpp dp;
ro=dens(t,api);
cp=heatcapa(t,api);
kk=termal(t,api);
dp=loss(t,api);
z.beta=(l-m_phi)*ro.s*cp.s+m_phi*ro.l*cp.l;
z.del=kk.l*m_phi+kk.s*( 1 -m_phi);
z.gama=ro.l*cp.l*m_velo;
if ( m_uoil > 0 ) {
bet=m_velo/(m_velo+m_uoil);
if (m_velo = 0 ) bet= 0 ;
z.beta=(l-mjhi)*ro.s*cp.s+m_phi*(bet*ro.w*cp.w+(l-bet)*ro.o*cp.o);
z.del=(kk.w*bet+(l-bet)*kk.o)*m_phi+kk.s*(l-m_phi);
z.gama=ro.w*cp.w*m_velo+ro.o*cp.o*m_uoil;
}
/* if ( m_velo = 0 ) {
z.beta=(l-m_phi)*ro.s*cp.s;
z.del=kk.s*(l-m_phi);
z.gama=0 ;
} */
z .k2 =(z .gama)/z .beta;
z.aa=pow(z .k2 ,2 );
z.kl=z.del/z.beta;
z.epp= 1 -exp (-m_deltax/dp .p);
z.c2 =(z.del+dp.p*z.gama);
z.cl=(m_p 0 *pow(dp.p,2 )*z.epp)/z.c2 ;
z.ee=exp(-x/dp.p);
z.c3=z.cl*z.ee;
z.c4=m_p0/22;
z.c5=dp.p;
retum(z);
}
double CAnalyticalSOlDlg::erfc(double w)
{
double u,v,erf;
u=l/(l+.3275911*w);
v=.254829592*u-.284496736*pow(u,2)+1.421413741*pow(u,3)1.453152027*pow(u,4)+l-061465429*pow(u,5);
erf=v*exp(-1*pow(w,2))-1. 5e-7;
retum(erf);
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void CAnalyticalSOlDlg::OnPaint()
{
if (IsIconicO)
{
CPaintDC dc(this); // device context for painting
SendMessage(WM_ICONERASEBKGND, (WPARAM)
dc.GetSafeHdc(), 0);
// Center icon in client rectangle
int cxlcon = GetSystemMetrics(SM_CXICON);
int cylcon = GetSystemMetrics(SM_CYICON);
CRect rect;
GetClientRect(&rect);
int x = (rect.Width() - cxlcon + 1) / 2;
int y = (rect.Height() - cylcon + 1) / 2;
// Draw the icon
dc.DrawIcon(x, y, m_hIcon);
}
else
{
CDialog: :OnPaint();
}
}
// The system calls this to obtain the cursor to display while the user drags
// the minimized window.
HCURSOR CAnalyticalSOlDlg: :OnQueryDragIcon()
{
return (HCURSOR) m_hIcon;
}
// Automation servers should not exit when a user closes the UI
// if a controller still holds on to one of its objects. These
// message handlers make sure that if the proxy is still in use,
// then the UI is hidden but the dialog remains around if it
// is dismissed.
void CAnalyticalSOlDlg::OnClose()
{
if (CanExit())
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CDialog: :OnClose();
}
void CAnalyticalSOlDlg::OnOK()
{
CAnalyticalSOlDlg: :UpdateData(TRUE);
long double txot,x,t,s,txt,bb,bbb,tt,ebb,ebbb,tss,tsss,ppp,tf,x 1 ,ttl ,x2 ,tt2 ;
t=m_initemp;
s_coe z;
s=m_salt;
txot= 0 ;
x=m_length;
tt= 0 ;
// z=factor(t,x);
filel = fopen("dfell.dat", "wt");
do{
z=factor(t,x);
bb=x/ ( 2 *sqrt(z .k 1 *tt));
bbb=bb+2.470*sqrt(tt*z.kl)/z.k2;
txt=(2.470*x/z.k2)+tt*z.kl*pow(2.470/z.k2,2);
ebb=erfc(bb);
ebbb=erfc(bbb);
if (z.k2 == 0 ) {
txt= 0 ;
// ebbb=l;
}
txot=t*(l+z.c4*tP(ebb-ebbb*exp(txt)));
if (m_sat < 1 && m_sat > 0) txot=t*(l+tfs(ebb-ebbb*exp(txt))/4);
if (m_velo < 10 ) txot=t*(l+tP(ebb-ebbb*exp(txt))/2.5);
if ( m_uoil > 0) txot=t*(l+(tf-t)*(ebb-ebbb*exp(txt))/4);
txotl=20+55*(ebb-ebbb*exp(txt))/2; //for sandand liquids
ts=88-44*erfc(2.74*sqrt(z.kl*tt)/z.k2)*exp(tt*2.470*pow(2.74/z.k2,2));
tsl=tf+t*(tf-t)*(ebb-ebbb*exp(txt));
tss=m_initemp+z.c4*85*erfc(x/(2*sqrt(z.kl*tt)));
tss= 1 0 *(l+erfc(x/(2 *sqrt(z.kl *tt))+sqrt(z.kl*tt)/z.k 2 ));
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tss 1 =t+(tf-t) *(-z .c3+z.c 1 *( 2 0 +(ebb-ebbb *exp(txt)))/2 );
tss2=l-exp(tt*z.kl*pow(2.470/z.k2,2))*erfc(2.470*sqrt(tt*z.kl)/z.k2);
tss3=l 0*(l+erfc(x/(2*sqrt(z.kl *tt))));
tss4=10*(l+exp(tt*pow(z.k2/z.kl,2))*erfc(2*sqrt(z.k2)*tt/z.kl));
ppp=m_pO*z.ee;
fprintf(filel,"%f,%f,%f'n",tt,txot,tss);
tt+=m_deltat;
}while (tt<=m_t);
fclose(filel);
x l= 0 ;
ttl=m_t;
file2 = fopen("dfel 2 .dat", "wt");
do{
z=factor(t,xl);
bb=x 1 / ( 2 *sqrt(z .k 1 *tt 1 ));
bbb=bb+2.470*sqrt(ttl *z.kl)/z.k2;
txt=(2.470*xl/z.k2)+ttl*z.kl*pow(2.470/z.k2,2);
ebb=erfc(bb);
ebbb=erfc(bbb);
if (z.k2 = 0 ) {
txt= 0 ;
// ebbb=l;
}
txot=t*(l+z.c4*tfl:(ebb-ebbb*exp(txt))/2);
txotl=20+55*(ebb-ebbb*exp(txt))/2; //for sandand liquids
ts=88-44*erfc(2.74*sqrt(z.kl*tt)/z.k2)*exp(tt*2.470*pow(2.74/z.k2,2));
ts 1 =tf+t*(tf-t)*(ebb-ebbb*exp(txt));
tss=65*erfc(x/(2*sqrt(z.kl*tt)));
tssl= 1 0 *(l+erfc(x/( 2 *sqrt(z.kl*tt))+sqrt(z.kl*tt)/z.k 2 ));
tss2=t+(tf-t)*(-z.c3+z.cl*(20+(ebb-ebbb*exp(txt)))/2);
tss3=l-exp(tt*z.kl*pow(2.470/z.k2,2))*erfc(2.470*sqrt(tt*z.kl)/z.k2);
tss4=10*(l+erfc(x/(2 *sqrt(z .k 1*tt))));
tss5=10*(l+exp(tt*pow(z.k2/z.kl,2))*erfc(2*sqrt(z.k2)*tt/z.kl));
ppp=m_pO*z.ee;
fprintf(file2 ,"%f,%f,%f\n" ,x 1 ,txot,ppp);
xl+=m_deltax;
}while (xl<= m_length+m_deltax);
fclose(file2 );
136
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
file3 = fopen("dfel3.dat", "wt");
do{
z=factor(t,x 2 );
bb=x 2 /(2 *sqrt(z.kl *ttl));
bbb=bb-2.470*sqrt(ttl *z.kl)/z.k 2 ;
txt=(2.470*x2/z ,k2)+tt 1*z.k 1*pow(2.470/z.k2,2);
ebb=erfc(bb);
ebbb=erfc(bbb);
if (z.k 2 = 0 ) {
txt= 0 ;
// ebbb=l;
}
tss=m_initemp;
//do {
tss=m_initemp+ 8 5 *erfc(x2/(2 *sqrt(z.kl *tt2/z .k2)));
if(m_sat < 1 && m_sat > 0) tss=m_initemp+85*erfc(x2/(2*sqrt(z.kl*tt2)));
if (m_velo < 1 0 ) tss-m__initemp+85*erfc(x2/(2*sqrt(z.k 1 *tt2)))/.7;
if (m_velo < 2 ) tss=m_initemp+85*erfc(x2/(2*sqrt(z.kl*tt2)))/(m_velo/3.5);
if (m_velo > 10 ) tss=m_initemp+850*erfc(x2/(2*sqrt(z.kl*tt2)))/(m_velo-5);
x 2 +=m_deltax;
Iprintf(file3,"%f,%f\n",x2,tss);
} while (x2 < z.c5+50*m_deltax);
// tsss= t *(1 +z.c4 *tP (ebb+ebbb *exp(txt)));
//
}while (x2 <= m_length+m_deltax);
fclose(file3);
m_lll="completed";
CAnalyticalS01Dlg::UpdateData(FALSE);
void CAnalyticalS01Dlg::0nCancel()
{
if (CanExit())
137
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CDialog: :OnCancel();
}
BOOL CAnalyticalSOlDlg: :CanExit()
{
// If the proxy object is still around, then the automation
// controller is still holding on to this application. Leave
// the dialog around, but hide its UI.
if (m_p AutoProxy != NULL)
{
ShowW indow(S WJHIDE);
return FALSE;
}
return TRUE;
}
// Analytical SOlDlg.h : header file
//
#if
!defmed(AFX_ANALYTICALSOLDLG_H__CB06F7ED_07C9_42EFJB2E8_EBB4F6
76A 523_IN CLU D ED J
#define
AFX_ANALYTICALSOLDLG_H__CB06F7ED_07C9_42EFJB2E8JEBB4F676A523_
_INCLUDED_
#if_M SC_VER> 1000
#pragma once
#endif // _MSC_VER > 1000
class CAnalyticalSOlDlgAutoProxy;
/////////////////////////////////////////////////////////////////////////////
// CAnalyticalSOlDlg dialog
typedef struct density {
double l,s,o,w;
} s_dens;
typedef struct thermal {
double l,s,o,w;
} s_term;
138
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
typedef struct heatcapa {
double l,s,o,w;
} s_htc;
typedef struct viscosity {
double o,w;
} s_visc;
typedef struct loss {
double p;
} s_dpp;
typedef struct factor {
double beta,gama,del,aa,kl,k2,cl,c2,c3,c4,c5,ee,epp;
} s_coe;
typedef struct tor{
double xt;
}s_fa;
/*
typedef struct erorfc{
double c;
}s_err; */
class CAnalyticalSOlDlg : public CDialog
{
DECLAREJDYNAMIC(CAnalyticalSOlDlg);
firiend class CAnalyticalSOlDlgAutoProxy;
// Construction
public:
CAnalyticalS01Dlg(CWnd* pParent = NULL);
virtual ~CAnalyticalS01Dlg();
s_dens dens(double, double);
s_term termal(double, double);
s_htc heatcapa(double, double);
s_visc viscosity(double, double);
s_dpp loss(double, double);
s_coe factor(double, double);
double erfc(double w);
// s_fa tor(double, double, int);
// standard constructor
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
// Dialog Data
11{{AFX_D AT A(C A nalytical OlDlg)
enum { IDD = IDD ANALYTICALS OL_DIALOG };
double mjphi;
double m_initemp;
double m_t;
double m_deltax;
double m_velo;
double m_pO;
double m_sat;
double m_length;
double m_api;
double m_salt;
double m_deltat;
double m_uoil;
double m_frec;
CString
m_lll;
//}}AFX_DATA
// ClassWizard generated virtual function overrides
//{ (AFX_VIRTUAL(CAnalyticalS01Dlg)
protected:
virtual void DoDataExchange(CDataExchange* pDX);
//} }AFX_VIRTUAL
// DDX/DDV support
// Implementation
protected:
CAnalyticalSOlDlgAutoProxy* m_pAutoProxy;
HICON m_hIcon;
BOOL CanExit();
// Generated message map functions
// {{AEX_MSG(CAnalyticalS01Dlg)
virtual BOOL OnInitDialog();
afx_msg void OnSysCommand(UINT nID, LPARAM IParam);
afx_msg void OnPaint();
afx_msg HCURSOR OnQueryDragIcon();
afxjmsg void OnClose();
virtual void OnOK();
virtual void OnCancel();
//}}AFX_MSG
DECLARE_MESSAGE_MAP()
140
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
// {{AFX_INSERT_LOC ATION}}
// Microsoft Visual C++ will insert additional declarations immediately before the
previous line.
#endif //
!defined(AFX_ANALYTICALSOLDLG_H_CB06F7ED_07C9_42EF_B2E8_EBB4F6
76A 523_IN CLU D ED J
// stdafx.h : include file for standard system include files,
// or project specific include files that are used frequently, but
//
are changed infrequently
//
#if
!defmed(AFX_STDAFX_H_691CE22D_B57B_465C_AB41_BFBAD645DA58_INC
LU D ED J
#define
AFX_STDAFX_H_691CE22D_B57B_465C_AB41_BFBAD645DA58_INCLUDED_
#if_M SC_VER> 1000
#pragma onee
#endif // _MSC_VER > 1000
#define VC_EXTRALEAN
// Exclude rarely-used stuff from Windows headers
#include <afxwin.h>
// MFC core and standard components
#include <afxext.h>
// MFC extensions
#include <afxdisp.h>
// MFC Automation classes
#include <afxdtctl.h>
// MFC support for Internet Explorer 4 Common Controls
#ifndef _AFX_N 0_AFXCMN_SUPPORT
#include <afxcmn.h>
// MFC support for Windows Common Controls
#endif// _AFX_NO_AFXCMN_SUPPORT
#include <afxsock.h>
// MFC socket extensions
// This macro is the same as IMPLEMENT_OLECREATE, except it passes TRUE
// for the bMultilnstance parameter to the COleObjectFactory constructor.
// We want a separate instance of this application to be launched for
// each automation proxy object requested by automation controllers.
#ifhdef IMPLEMENT OLECREATE2
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
#define IMPLEMENT_0LECREATE2(class_name, extemal_name, 1, w l, w2, b l, b2,
b3, b4, b5, b6, b7, b8) \
AFX_DATADEF COleObjectFactory class_name::factory(class_name::guid, \
RUNTIME_CLASS(class_name), TRUE, _T(extemal_name)); \
const AFX_DATADEF GUID class_name::guid = \
{ 1, w l, w2, { b l, b2, b3, b4, b5, b6, b7, b8 } };
#endif// IMPLEMENT_0LECREATE2
// {{AFX INSERT LOCATION} }
// Microsoft Visual C++ will insert additional declarations immediately before the
previous line.
#endif //
!defined(AFX_STDAFX H__691CE22D_B57B_465C_AB41_BFBAD645DA58_INC
LU D ED J
// DlgProxy.h : header file
//
#if
!defmed(AFX_DLGPROXY_H_3F65AA8A_ECAE_4D08_AD63_288E807DC02E_I
NCLUDEDJ
#define
AFX_DLGPROXY_H_3F65AA8A_ECAE_4D08_AD63_288E807DC02E_INCLUD
ED_
#if _MSC_VER > 1000
#pragma once
#endif // _MSC_VER > 1000
class CAnalyticalSOlDlg;
/////////////////////////////////////////////////////////////////////////////
// CAnalyticalSOlDlgAutoProxy command target
class CAnalyticalSOlDlgAutoProxy : public CCmdTarget
{
DECLARE_DYNCREATE(CAnalyticalS01DlgAutoProxy)
CAnalyticalS01DlgAutoProxy();
creation
// protected constructor used by dynamic
// Attributes
142
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
public:
CAnalyticalSOlDlg* mjpDialog;
// Operations
public:
// Overrides
// ClassWizard generated virtual function overrides
// {{AFX_VIRTUAL(C AnalyticalSOlDlgAutoProxy)
public:
virtual void OnFinalRelease();
//}} AFX_VIRTUAL
// Implementation
protected:
virtual ~C AnalyticalS 01DlgAutoProxy();
// Generated message map functions
// {{AFX_MSG(CAnalyticalS01DlgAutoProxy)
// NOTE - the ClassWizard will add and remove member functions here.
//}}AFX_MSG
DECLARE_MESSAGE_MAP()
DECLAREOLECREATE(CAnalyticalSOlDlgAutoProxy)
// Generated OLE dispatch map functions
// {{AFX_DISPATCH(C AnalyticalSOlDlgAutoProxy)
// NOTE - the ClassWizard will add and remove member functions here.
//}} AFXJDISPATCH
DECLARE_DISP AT CH_MAP()
DECLARE_INTERFACE_MAP()
};
/////////////////////////////////////////////////////////////////////////////
// {{AEX_IN SERT_LOC ATION }}
// Microsoft Visual C++ will insert additional declarations immediately before the
previous line.
#endif //
!defined(AFX_DLGPROXY_H_3F65AA8A_ECAE_4D08_AD63_288E807DC02E_I
NCLUDEDJ
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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