# Microwave heating for wax precipitation prevention and near wellbore conformance control

код для вставкиСкачатьNOTE TO USERS This reproduction is the best copy available. ® UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Library and Archives Canada Bibliotheque et Archives Canada Published Heritage Branch Direction du Patrimoine de I'edition 395 Wellington Street Ottawa ON K1A 0N4 Canada 395, rue Wellington Ottawa ON K1A 0N4 Canada Your file Votre reference ISBN: 0-494-06014-X Our file Notre reference ISBN: 0-494-06014-X NOTICE: The author has granted a non exclusive license allowing Library and Archives Canada to reproduce, publish, archive, preserve, conserve, communicate to the public by telecommunication or on the Internet, loan, distribute and sell theses worldwide, for commercial or non commercial purposes, in microform, paper, electronic and/or any other formats. AVIS: L'auteur a accorde une licence non exclusive permettant a la Bibliotheque et Archives Canada de reproduire, publier, archiver, sauvegarder, conserver, transmettre au public par telecommunication ou par I'lnternet, preter, distribuer et vendre des theses partout dans le monde, a des fins commerciales ou autres, sur support microforme, papier, electronique et/ou autres formats. The author retains copyright ownership and moral rights in this thesis. Neither the thesis nor substantial extracts from it may be printed or otherwise reproduced without the author's permission. L'auteur conserve la propriete du droit d'auteur et des droits moraux qui protege cette these. Ni la these ni des extraits substantiels de celle-ci ne doivent etre imprimes ou autrement reproduits sans son autorisation. In compliance with the Canadian Privacy Act some supporting forms may have been removed from this thesis. Conformement a la loi canadienne sur la protection de la vie privee, quelques formulaires secondaires ont ete enleves de cette these. While these forms may be included in the document page count, their removal does not represent any loss of content from the thesis. i *i Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant. Canada Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. SUBSCRIPTS r = Relative b= Brine w = Water / = Liquid s = Sand, Solid Portion 0 = Incident distance, The Inlet / = Final x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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- 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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, 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 9 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. '^ '^ - - ^ 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 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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. 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z Az > Ax Figure 4.1 Element of volume in Cartesian coordinates used for conservation of mass 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ^ (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z >• Ax Figure 4 . 2 Element of volume in Cartesian coordinates used for conservation of energy 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Az z u Ay Ax Figure 4 .3 Element of volume in Cartesian coordinates used for conservation of mass 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. z Ay Ax Figure 4 .4 Element of volume in Cartesian coordinates used for conservation of energy 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / - 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Power Figure 6. lThe setup for experiment with sand-filled waveguide 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CH A PTER 8 REFERENCES 1- Green D. W., Willhite G. P. (1998), “Enhanced Oil recovery”, Society of Petroleum Engineering 2- Craig F.F. Jr. (1980), “The Reservoir Engineering Aspects of Waterflooding”, Society of Petroleum Engineering 3- Prats M. (1986), “Thermal Recovery”, Society of Petroleum Engineering 4- Rose Stephen C., Buckwalter John F. and Woodhall Robert J. (1989), “Design Engineering Aspects of Waterflooding”, Society of Petroleum Engineering 5- Butler R.M. (1991), “Thermal Recovery of Oil and Bitumen”, Prentice Hall 6 - El-feky S.A. (1977), “Theoretical and Experimental Investigation of Oil Recovery by Electrothermic Technique”, PhD dissertation, Penn State 7-Taabbodi L. (2003), “Near Wellbore Jell Placement” Master’s thesis, University of Regina 8 - Todi S. (2003), “Wax Deposition in Crude Oil Carrying Pipelines”, Master’s Thesis, University of Utah 9- Mansoori G. A., Lindsey Bames H., Webster G. M. (2003), “Petroleum Waxes” Chapter 19 in "Fuels and Lubricants Handbook", American Society for Testing Material International, West Conshohocken, PA, 525-558 10- Mansoori G.A. (2001), "Deposition and Fouling o f Heavy Organic Oils and Other Compounds", Invited paper, Proceedings of the 9th International Conference on Properties and Phase Equilibria for Product and Process Design, Kurashiki, Okayama, Japan 11- Jessen F.W., Howell J.N. (1958), “Effect of Flow Rate on Paraffin Accumulation in 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Plastic, Steel, and Coated Pipe”, Petroleum Transactions, The American Institute of Mining, Metallurgical, and Petroleum Engineers, 213, 80-84 12- Hunt E.B. (1982), “Laboratory Study of Paraffin Deposition “, Journal of Petroleum Technology, Paper 279, Nov., 1259-1269 13- Bott T.R. (1977), Gudmundsson J.S., “Effect of Velocity on Deposition of Wax”, The Canadian Journal of Chemical Engineering, 55, 381-385 14- Burger E.D., Perkins T.K., Striegler J.H. (1981), “Studies of Wax Deposition in the Trans Alaska Pipeline”, Journal of Petroleum Technology, Paper 8788, June, 10751086 15- Coutinho J.A.P., Edmonds B., Moorwood T., Szczepanski R., Zhang X. (1998), "Reliable Wax Predictions for Flow Assurance", Society of Petroleum Engineering, 78324 16- Freemann S.A., Booske J. H., Cooper R. F. (1998), “Modeling and Numerical Simulations of Microwave-induced Ionic Transport”, Journal of Applied Physics, 8311 17-Rajagapol and Tao (1998), Report, US Department of energy 18- Metaxas A.C., Meridith R J. (1983), “Industrial Microwave Heating”, Peter Peregrinus, London, 19- Saltiel C., Datta A. (1997), “Heat and mass transfer in microwave processing”, Advancement in Heat Transfer, 30, 1-94 20- Zhao H., Turner I.W., Torgovnikov G. (1997), “An experimental and numerical Investigation of the microwave heating of wood”, Journal of Microwave Power Electromagnetic Energy, 33, 121-133 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21- Jackson C. (2002), “Upgrading Heavy Oil Using Variable Frequency Microwave Energy”, Society of Petroleum Engineering 78982 22- Kaviany M. (1995), “Principles of Heat Transfer in Porous Media (Mechanical Engineering Series)”, Springer Verlag; 2nd edition 23- Carman P.C. (1956), ‘Flow of Gases through Porous Media” Academic Press, New York 24- Leverett M.C., Lewis W.B. (1972), “Steady flow of gas-oil-water mixtures through unconsolidated sands”, Transactions SPE of The American Institute of Mining, Metallurgical, and Petroleum Engineers Institute of Mechanical Engineering, 142, 107-116 25- Lauwerier H.A. (1955), “The Transport of Heat in an Oil Layer Caused By injection of Hot Fluid”, Applied Scientific Research, A-5, 145 26-J.W. Marx; R.H. Langenheim (1959), “Reservoir Heating by Hot Fluid Injection”, Transactions, The American Institute of Mining, Metallurgical, and Petroleum Engineers ,216, 312-315 27- Ramey H.J. (1959), “Reservoir Heating by Hot fluid Injection”, Transactions The American Institute of Mining, Metallurgical, and Petroleum Engineers , 216, 312 28- Spillette A.G. (1956), “Heat transfer during Hot Fluid Injection into an Oil Reservoir”, Journal o f Canadian Petroleum Technology, 59, 213-218 29- Klinkenberg A. (1948), “Numerical Evaluation of Equations Describing Transient Heat and Mass Transfer in Packed Solids”, Industrial Engineering Chemistry, 40, 10, 1992-1994 30- Preston F.W., Hazen R.D. (1959), “Further Studies on Heat Transfer in 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Unconsolidated Sands during Hot Water Injection”, Producers Monthly, 24 31- Schild A. (1957), “A Theory of Heating Oil-producing Wells”, Transactions AIME, 210 , 1 32- Poisson S.D. (1835), “Theorie Mathmatique de la Chaleur”, Bachelier, Paris 33- Copson D. A. (1975), “Microwave Heating”, Avi Publishing Company 34- Puchner H. (1989), “The Microwave Heating of Agricultural Material”, PhD Dissertation, Denmark 35- Okress E.C. (1968), “Microwave Power Engineering”, Academic Press, New York & London 36- Hill J.M., Marchant T.R. (1996), “Modeling Microwave Heating”,Journal of Applied Mathematical Modeling, 20, 3-15 38- Housova J., Topinka P., Hoke K. (1996), “Mathematical Model of Temperature Distribution in Food Materials Heated by Microwaves”, Potrav Vedy, 14, 329-346 39- Stogryn A. (1971), "Equations for calculating the dielectric constant of saline water", IEEE Transactions Microwave Theory Techniques, MTT-19, 733-736 40- Ratanadecho P., Aoki K., Akahori M. (2001), “A Numerical and Experimental Investigation o f the Modeling of Microwave Melting of Frozen Packed Beds Using a Rectangular Wave Guide”, International Community of Heat Mass Transfer 28 ,751762 41- Klein L.A., Swift C.T. (1977), “An improved model for the dielectric constant of sea water at microwave frequencies”, IEEE Trans Antennas Propagation, AP-25, 104-111 42- Nguyen C. (1994), " the Penetration Depth o f Radars in Wet and Dry Sands", Masters Thesis, Korea 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43- Keenan J.H., Keyes F.G. (1955), “Thermodynamic Properties of Steam Including Data for the Liquid and Solid Phases”, NY. John Wiley & Sons 44- Farouq Ali S.M. (1970), “Oil Recovery By Steam Injection”, Producers Publishing Co. Inc., Bradford, PA 45- Gros R.P. (1984), “Steam Soak Predictive Model”, MS thesis, University of Texas at Austin 46- Boberg T.C. (1988), “Thermal Methods of Oil Recovery”, John Wiley Publishing Company New York City 47- Kislitsin A. A., Fedeev A.M. (1996), “Dielectric Properties of Oil-Saturated Sand and Water-in-Oil Emulsion from High-Viscosity and High-Paraffinaceous Oils”, Russian Academic Society, International Conference Oil & Bitumen’s, Kazan, Tatarstan 48- Chute F.S., Vermeulen F.E., Cervenan M.R., McVea F J. (1979), "Electrical 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 122 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. // 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 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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()) { 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. // 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 ; 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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); 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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); 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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); 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. } 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; 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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; 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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); 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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()) 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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 )); 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 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. 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