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    Copyright

    by

    Maylin Alejandra Carrizales

    2010

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    The Dissertation Committee for Maylin Alejandra Carrizales Certifies that this is the

    approved version of the following dissertation:

    Recovery of Stranded Heavy Oil by Electromagnetic Heating

    Committee:

    _________________________________ Larry W. Lake, Supervisor

    _________________________________ Russell T. Johns, Co-Supervisor

    _______________________________________ Kamy. Sepehrnoori

    _______________________________________ Quoc P. Nguyen

    _______________________________________ Isaac C. Sanchez

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    Recovery of Stranded Heavy Oil by Electromagnetic Heating

    By

    Maylin Alejandra Carrizales, B.S.; M.S.

    Dissertation

    Presented to the Faculty of the Graduate School of

    The University of Texas at Austin

    in Partial Fulfillment

    of the Requirements

    for the Degree of

    Doctor of Philosophy

    The University of Texas at Austin

    December 2010

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    Dedication

    To God for all his blessings.

    To my parents, my husband, and my children, whose unconditional love and support

    have made this possible.

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    Acknowledgements

    First, I would like to express my deepest gratitude to my supervising professor,

    Dr. Larry W. Lake, for his continuous guidance, advice, support, and encouragement

    through my research. I have learned from his insightful questions, vast knowledge, and

    all the discussions we held. I considered myself very fortunate to have met him and had

    him as my advisor.

    I would like to thank my co-supervisor, Dr. Russell T. J ohns, for his guidance and

    advice on the important aspects to be considered during the development of this research,

    and for all the questions he asked, which encourage me to a more critical thinking and

    problem solving mind. I want to extend my sincere gratitude to the members of my

    committee, Dr. Kamy Sepehrnoori, Dr. Quoc Nguyen, and Dr. Isaac Sanchez for taking

    the time to review my dissertation and for their invaluable comments at the initial stage

    of my research.I want to express my gratitude to Joanna Castillo for helping me with the format

    of my papers, figures, and this dissertation as a whole, and for installing and keeping the

    software up to date in my office computer. My sincere appreciation goes to Chrissi King

    for sharing her technical knowledge whenever I needed, and for all the conversations we

    had. I want to thank Cheryl Kruzie, Linda Pannell, Esther Barrientes, Cathy Kimbrough,

    Kiki Peckham, and Shelette Paulino, for taking care of registrations, appointments, and

    all other paperwork during my stay at The University of Texas at Austin. Also, I would

    like to thank Roger Terzian for installing COMSOL Multiphysics in my computer.

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    I would like to thank my friends for their company, suggestions and

    encouragement: J esus Salazar, Silvia Solano, Alberto Mendoza, Morteza Sayarpour,

    Maika Gambs, Rouzbeh Ghanbarnezhad, and Hee Jae Lee to name a few.

    I also want to acknowledge PDVSA-CIED for their funding at the beginning of

    this study, and Terratherm for their financial support to part of this work. The Computer

    Modeling Group (CMG) is acknowledged for providing the license for the use of their

    simulator STARS.

    I would like to specially thank my parents Douglas and Cecilia for their love, for

    teaching me to be strong and perseverant, for encouraging me to take on this journey and

    not to give up. To my dear brothers Marlon and Ronald for being understanding and

    supporting during all this years I have been far from them.

    Finally, I want to thank my husband, J oel Payare, for all his support,

    encouragement, love and patience during these years. For believing in me, and being my

    strength during difficult times, and for helping me to take care of our wonderful children,Andres, David and Emma.

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    Recovery of Stranded Heavy Oil by Electromagnetic Heating

    Publication No._____________

    Maylin Alejandra Carrizales, PhD.

    The University of Texas at Austin, 2010

    Supervisor: Larry W. Lake

    Co-Supervisor: Russell T. J ohns

    High oil-viscosity is a major concern for the recovery of oil from heavy-oil

    reservoirs. Introducing energy to the formation has proven to be an effective way of

    lowering the oil viscosity by raising the temperature in the formation. The application of

    low-frequency heating, also known as electrical resistance heating, is limited by water

    vaporization near the wellbore which breaks the conductive path to the reservoir, and

    limits the heating rate as well as the resulting production rates. Electromagnetic (EM)

    heating, also called high-frequency heating, can be used instead.

    Although its potential was recognized during the late 70s, no simulation results

    or detailed modeling studies have yet been published that completely model the complex

    interactions of EM energy and multiphase flow. One of the main drawbacks of proposed

    models is the use of the EM adsorption coefficient as a constant regardless of the

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    properties of the medium, which can obscure the important effect of this parameter on the

    extension of the reservoir area heated.

    This dissertation presents a multiphase, two-dimensional radial model that

    describes the three-phase flow of water, oil, and steam and heat flow in a reservoir within

    confining conductive formations. The model accounts for the appearance and/or

    disappearance of a phase, and uses the variation in temperature and water saturation to

    update the EM absorption coefficient. This model allows determining the temperature

    distribution and the productivity improvement from EM heating when multiple phases

    are present.

    For the numerical simulations of EM heating, I used COMSOL Multiphysics, a

    Lagrange-quadratic finite element simulator, and its partial differential equations (PDE)

    application. Several simulations were made for hypothetical reservoirs with different

    fluid and rock properties. Also, analytical solutions for a single-phase EM heating model

    were developed and used to validate the numerical solutions.Special attention is focused on reservoirs with characteristics for which steam

    injection is not attractive or feasible such as low permeability, thin-zone, and extra-heavy

    oil reservoirs. Results showed that EM heating is feasible based on the power source and

    frequency used to maintain an optimum absorption coefficient and to obtain higher

    production rates. Comparisons showed that cumulative oil production and recovery

    factor obtained by EM heating are better than what is achieved by cyclic steam

    stimulation (CSS) for reservoirs with the above mentioned characteristics.

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    Table of Contents

    List of Tables.......................................................................................................... xii

    List of Figures......................................................................................................... xiv

    Chapter 1: Introduction........................................................................................... 1

    1.1 Description of the Problem............................................................................... 3

    1.2 Research Objectives.......................................................................................... 6

    1.3 Method and Approach Overview...................................................................... 7

    Chapter 2: Literature Review.................................................................................. 12

    2.1 Electromagnetic Waves.................................................................................... 12

    2.1.1 Maxwells Equations......................................................................... 12

    2.1.2 Lamberts Law................................................................................... 15

    2.2 Electromagnetic Heating Methods.................................................................... 15

    2.2.1 Low Frequency Heating or Electrical Resistive Heating (ERH)....... 16

    2.2.2 High Frequency Heating- Radio Frequency (RF) or Microwave

    Frequency (MW) Heating.................................................................. 20

    2.3 Electrical Properties of Oil Sands..................................................................... 28

    2.3.1 Complex Magnetic Permeability....................................................... 30

    2.3.2 Complex Permittivity and Electrical Conductivity............................ 302.4 Physical Properties for Heavy Oil .................................................................... 38

    2.4.1 Viscosity............................................................................................ 38

    2.4.2 Heat Capacity..................................................................................... 38

    2.4.3 Thermal Conductivity........................................................................ 38

    2.5 Thermal Properties for Water and Steam.......................................................... 40

    Chapter 3: Electromagnetic Heating Model ........................................................... 43

    3.1 Thermal Reservoir Model ................................................................................. 43

    3.1.1 Mass Balance..................................................................................... 43

    3.1.2 Energy Balance.................................................................................. 45

    3.2 Electromagnetic Heating Source....................................................................... 49

    3.3 Initial and Boundary Conditions....................................................................... 51

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    3.4 Modeling Using COMSOL Multiphysics......................................................... 52

    Chapter 4: One-Dimensional Single Phase Electromagnetic Heating (EMH)

    Model ...................................................................................................................... 56

    4.1 Overview........................................................................................................... 56

    4.2 One-Dimensional Steady-State Single-Phase EMH Model.............................. 57

    4.2.1 Results................................................................................................ 60

    4.3 One-Dimensional Transient Single-Phase EMH Model................................... 69

    4.3.1 Temperature Distribution Results...................................................... 69

    4.3.2 Productivity Improvement Results.................................................... 73

    4.4 Factors Affecting the Productivity Improvement with Electromagnetic

    Heating.................................................................................................................... 764.4.1 Effect of Heat Transfer Mechanism................................................... 78

    4.4.2 Effect of Oil Viscosity....................................................................... 81

    4.4.3 Effect of EM frequency..................................................................... 83

    Chapter 5: Two-Dimensional Single-Phase Electromagnetic Heating (EMH)

    Model ...................................................................................................................... 87

    5.1 Overview........................................................................................................... 87

    5.2 Two-Dimensional Transient Single-Phase EMH Model .................................. 88

    5.2.1 Temperature Distribution Results...................................................... 90

    Chapter 6: Two-Dimensional Multiphase Electromagnetic Heating (EMH)

    Model ...................................................................................................................... 98

    6.1 Overview........................................................................................................... 98

    6.2 Two-Dimensional Transient Multiphase EMH Model ..................................... 99

    6.2.1 Constitutive Relationships................................................................. 104

    6.2.2 Numerical Implementation................................................................ 107

    6.2.3 Results................................................................................................ 1106.3 Critical Variables for Production Improvement during Electromagnetic

    Heating (EMH) ...................................................................................................... 115

    6.3.1 Electromagnetic (EM) Frequency...................................................... 119

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    6.3.2 Power Source..................................................................................... 121

    6.3.3 Horizontal Permeability..................................................................... 121

    6.4 Comparison of Electromagnetic Heating (EMH) to other Thermal Recovery

    Methods................................................................................................................... 121

    6.4.1 Cyclic Steam Stimulation (CSS)........................................................ 123

    6.4.2 Cases Studied..................................................................................... 125

    a) Thin-Zone Reservoir...................................................................... 125

    b) Low Permeability Reservoirs........................................................ 128

    c) Extra-Heavy Oil Reservoirs........................................................... 130

    6.4.3 Electrical Resistive Heating (ERH) ................................................... 136

    Chapter 7: Summary, Conclusions, and Recommendations................................... 1397.1 Summary........................................................................................................... 139

    7.2 Conclusions....................................................................................................... 139

    7.3 Recommendations............................................................................................. 141

    Appendix A: Analytical Solution for the Steady-State Single-Phase

    Electromagnetic Heating Model ............................................................................. 143

    Appendix B: Productivity Improvement Derivation............................................... 151

    Appendix C: Model Validation............................................................................... 154

    Nomenclature.......................................................................................................... 161

    References............................................................................................................... 164

    Vita.......................................................................................................................... 170

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    List of Tables

    Table 2.1: Coefficients of a linear approximation for the relative dielectric constant

    R with respect to frequency. Values were determined from samples of the Athabasca oil sands in Alberta.................................................. 36

    Table 2.2: Temperature coefficient Ta for electrical conductivity with respect tofrequency for temperatures below 120

    C (248

    F). Values weredetermined from samples of the Athabasca oil sands in Alberta.... 36

    Table 2.3: Temperature coefficients for the electrical conductivity for rich and leanoil sands, and for shales from the Athabasca deposits in Alberta. Valueswere determined within a temperature range from 24 to 240

    C (75 to 482

    F) ................................................................................................... 36 Table 2.4: Penetration depth (Dp) measured at 100

    F for different frequencies, andthe corresponding adsorption coefficient ( ) obtained from equation(2.26) ............................................................................................... 37

    Table 2.5: Empirical constants for calculating oil viscosity as a function of temperature obtained from regression analysis of measured viscosities attwo known temperatures................................................................. 39

    Table 2.6: Enthalpy of water and steam at saturated conditions, and latent heat of

    vaporization. After Prats (1982) .................................................... 41 Table 2.7: Water and steam viscosities as a function of temperature. After Kim

    (1987).............................................................................................. 42

    Table 3.1: Mathematical expression of the boundary conditions in a radial system forthe electromagnetic heating (EMH) model. Conduction heat loss isaccounted for at the bottom (z = 0) and top (z = h) boundaries of thereservoir .......................................................................................... 54

    Table 4.1: Basic data of the hypothetical reservoir used for the application of the

    single-phase electromagnetic heating (EMH) model for heavy-oilrecovery........................................................................................... 62

    Table 4.2: Productivity improvement with time obtained from the cold oil rate (noEM heating), and the oil rate when EM heating was applied......... 77

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    Table 5.1: Basic data used for the solution of the heat conduction equation in theover/underburden formations to include vertical heat loss in the two-dimensional EMH model for heavy-oil recovery........................... 92

    Table 5.2: Energy balance for EMH for a power source of 70 KW (5.732MMBtu/day) considering convective (production) and conductive (verticalloss/outer boundary) heat loss......................................................... 97

    Table 6.1: Relative permeability data for oil-water and gas-liquid system. After Kim(1987).............................................................................................. 105

    Table 6.2: Temperature-dependent irreducible saturations and end-point relativepermeabilities.................................................................................. 105

    Table 6.3: Reservoir and fluid input data used for the multiphase EM heating

    simulations...................................................................................... 113

    Table 6.4: Summary of results from EM heating and CSS for base case reservoir andcases of study.................................................................................. 135

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    List of Figures

    Figure 1.1: Schematic view of EM heating process for counter-current flow. The

    antenna is placed at the center of the producing well in front of thereservoir confined by the adjacent layers. EM energy flows into theformation and it is transformed into heat with the subsequent increase of fluid flowing toward the producing well. Only one well is used for thisEMH process................................................................................... 10

    Figure 1.2: Schematic view of EM heating process for co-current flow. The antenna isplaced at the center of the so-called injector well in front of the target zoneconfined by the adjacent layers. EM energy flows into the formation andit is transformed into heat with the subsequent increase of fluid flowing

    toward the producing well located at the opposite end of the system. Twowells are used for this EMH process............................................... 11

    Figure 2.1: Two typical schemes for resistive heating: (a) less efficient scheme asmost of the current flow is outside the reservoir section, (b) the casing isused as a terminal which improves the scheme by localizing more of theelectrical losses (heating) in the reservoir area. The dotted regionsrepresent the electrically isolated sections of casing...................... 19

    Figure 2.2: Electromagnetic spectrum. Frequency increases from right to left opposite

    to wavelength. Adapted from en.wikipedia en:Electromagnetic-Spectrum.svg ................................................................................... 22

    Figure 2.3: Energy deposition of an electromagnetic (EM) wave in a lossy formation. The skin depth represents the distance traveled by the EM energy into theformation before its energy is reduced to 1/e (0.37) of its original strength.Adapted from Kim (1987) .............................................................. 23

    Figure 2.4: Typical schemes for high frequency electromagnetic power transfer fromthe surface to the reservoir (the hatched area indicates the region whereelectromagnetic power is located). In (a) the energy is transmitted in the

    space between the casing and the production tubing, while in (b) theenergy propagates along the waveguide provided by the production tubing.After Calarotti (2000) ..................................................................... 25

    Figure 2.5: Dielectric Properties of Diatomaceous Rock versus Temperature. AfterKasevich et al . (1994) .................................................................... 29

    xiv

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    Figure 2.6: Dielectric Properties of Diatomaceous Rock at Ambient Temperature (20C) versus Frequency. After Kasevich et al . (1994)...................... 29

    Figure 2.7: Electrical conductivity and relative dielectric constant R for differentfrequencies at 75

    F from Athabasca oil sands. Data measured fromsamples with a water content range from 6.80 to 4.90 in water weightpercent, and densities from 1.99 to 1.87 g/cm 3. Adapted from Chute andVermeulem (1990).......................................................................... 31

    Figure 2.8: Electrical conductivity and relative dielectric constant R for differentfrequencies at 75

    F from Athabasca oil sands. Data measured fromsamples with a water content range from 4.21 to 2.71 in water weightpercent, and densities from 2.01 to 1.89 g/cm 3. Adapted from Chute andVermeulem (1990).......................................................................... 32

    Figure 2.9: Electrical conductivity and relative dielectric constant R for differentfrequencies at 75

    F from Athabasca oil sands. Data measured fromsamples with a water content range from 1.45 to 0.78 in water weightpercent, and densities from 2.02 to 1.88 g/cm 3. Adapted from Chute andVermeulem (1990).......................................................................... 33

    Figure 2.10: Penetration depth Dp from high frequency electromagnetic energy for aconceptual Venezuelan heavy oil reservoir .................................... 37

    Figure 2.11: Semi-log plot of oil viscosity as a function of temperature from typicalreservoirs with different API gravities. Oil viscosity was calculated fromequation (2.26) with the empirical constants presented in Table 2.5. 39

    Figure 2.12: Enthalpy of water and steam at saturated conditions...................... 42

    Figure 3.1: Schematic diagram of the system showing the boundary conditions. Onlyhalf of the reservoir is modeled. This is based on the assumption of symmetry at the wellbore................................................................ 53

    Figure 4.1: Schematic comparison of energy fluxes at steady-state for counter-currentflow (top), and co-current flow (bottom). The sum of the fluxes is aconstant, which value depends on the direction of the flow........... 61

    Figure 4.2: Steady-state temperature profile for a 1-D single phase counter-current(thick curves) and co-current (thin curves) Cartesian flow for differentvalues of the EM absorption coefficient, , with a constant power sourceof 63 kW ......................................................................................... 63

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    Figure 4.3: Steady-state temperature profile for a 1-D single phase counter-currentradial flow for different values of the EM absorption coefficient, with aconstant power source of 63 kW..................................................... 63

    Figure 4.4: Steady-state temperature profile for a 1-D single-phase co-current radialflow for different values of the EM absorption coefficient, , with aconstant power source of 63 kW..................................................... 64

    Figure 4.5: Relative productivity improvement, PI r, for Cartesian counter-current flowas a function of the absorption coefficient, , for different input powervalues used...................................................................................... 67

    Figure 4.6: Relative productivity improvement, PI r, for Cartesian co-current flow as afunction of the absorption coefficient, , for different input power valuesused................................................................................................. 67

    Figure 4.7: Relative productivity improvement, PI r, for radial counter-current flow asa function of the absorption coefficient, , for different input power valuesused................................................................................................. 68

    Figure 4.8: Temperature profile variation with time for radial counter-current flow at aconstant oil production rate (q o)...................................................... 71

    Figure 4.9: Effect of the oil flow rate (q o) on the temperature profile for counter-current radial flow after 30 days..................................................... 71

    Figure 4.10: Transient temperature profile for a radial system with a constant pressurewell for counter-current flow.......................................................... 72

    Figure 4.11: Oil rate with time for EM heating compared to the cold production(without radiation). Production plotted until steady-state is achieved (851days). Note the leveling at a constant rate for the EM heating case afterapproximately 500 days. Peak production is obtained after 128 days of EM heating...................................................................................... 74

    Figure 4.12: a) Pressure profile after 200 days of production for EM Heating comparedto cold production (Top). b) Semi-Log plot of the pressure profile after200 days of EM Heating compared to cold production (Bottom) . 75

    Figure 4.13: Effect of the heat transfer mechanism for steady-state Cartesian counter-current flow..................................................................................... 79

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    Figure 4.14: Effect of the heat transfer mechanism for steady-state radial counter-current flow..................................................................................... 79

    Figure 4.15: Effect of the heat transfer mechanism for radial counter-current flow after

    610 days of EM heating.................................................................. 80

    Figure 4.16: Productivity improvement with time for five different crudes. The topcurve represents the heaviest crude with 7.7 API gravity, and the bottomcurve represents the lightest crude with 21.4 API gravity. Two groups(circled) can be clearly identified. The top group corresponds to the heavyand extra-heavy oils, and the bottom group corresponds to the medium andlight oils simulated.......................................................................... 82

    Figure 4.17: Temperature profile after 315 days of EM heating for radial counter-current flow as a function of the frequency used for a constant power

    source of 70 kW.............................................................................. 84

    Figure 4.18: Oil production in Bbls/day after 315 days of EM heating for radialcounter-current flow as a function of the frequency used for a constantpower source of 70 kW. The vertical line (dashed) at 300 MHz representsthe boundary between RF and MW frequencies............................. 86

    Figure 5.1: Schematic representation on the system modeled for EMH indicating theheat transport mechanisms. EM energy is introduced to the reservoir froma radiating element located in the well at reservoir depth. This energy isconverted to heat within the formation. Fluid production carries heat out of

    the reservoir by convection, while heat conduction accounts for verticalheat loss to the over/underburden formations through the reservoirboundaries....................................................................................... 91

    Figure 5.2: Temperature in

    K without vertical heat loss (top) and with vertical heatloss by conduction included (bottom) after 100 days of EMH....... 93

    Figure 5.3: Temperature in

    K after 100 days of EMH including vertical heat loss byconduction. The arrows represent the direction and magnitude of theconductive heat flux component. Includes mesh elements to appreciatethe heat loss through the reservoir boundaries................................ 94

    Figure 5.4: Comparison of the oil rate in Bbls/day when EMH is applied. The effectof heat loss by conduction becomes more important causing the oil rate todecrease as time increases (bottom curve). For the insulated case (topcurve), more heat remains in the reservoir resulting in higher productionrates................................................................................................. 96

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    Figure 6.1: Electrical conductivity in S/m as a function of temperature for a rich oil

    sand at a frequency of 915 MHz, obtained from the correlation developedby Chute and Vermeulen (1990)..................................................... 103

    Figure 6.2: Electromagnetic (EM) absorption coefficient in 1/m as a function of watersaturation. Initial water saturation is 0.2. As water vaporizes near thewellbore, the EM absorption coefficient increases, which means that moreenergy is absorbed within a few feet from the wellbore. However, at verylow water saturation the EM absorption coefficient decreases dramaticallyto zero.............................................................................................. 103

    Figure 6.3: Water/oil relative permeability curves used for the simulations. It can beinferred from these curves that the reservoir is water-wet.............. 106

    Figure 6.4: Gas/liquid relative permeability curves used for the simulations... 106

    Figure 6.5: Grid used for the multiphase EM heating simulations.................... 112

    Figure 6.6: Temperature in K (left) and water saturation profiles (right) for the basecase ( lateral view/ half reservoir) after applying EM heating for (a) 50days, (b) 100 days, (c) 1 year, and (d) 3 years of heating. The totalhorizontal length shown is 80 ft...................................................... 114

    Figure 6.7: Gas saturation profiles for the base case ( lateral view/ half reservoir) afterapplying EM heating for various simulation times........................ 116

    Figure 6.8: Pressure profiles for various simulation times. Total length of reservoir(164 ft) is shown............................................................................. 117

    Figure 6.9: Oil production in Bbls/day and gas production in Mcf/day for the EMHbase case with water vaporization................................................... 118

    Figure 6.10: Comparison of cumulative oil production in bbls from EMH for differentfrequency values............................................................................. 120

    Figure 6.11: Comparison of cumulative oil production in bbls from EMH for different

    input values of power sources......................................................... 122Figure 6.12: Comparison of cumulative oil production in bbls from EMH for various

    reservoir permeabilities................................................................... 122

    Figure 6.13: Comparison of temperature profiles after 100 days of EMH for variousreservoir permeabilities................................................................... 124

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    Figure 6.14: Cumulative oil recovered in Mbbls for EM heating, cold production (no

    heating) and Cyclic Steam Stimulation for the base case............... 126

    Figure 6.15: Oil rates for EM heating, CSS, and cold production (no heating) for thebase case.......................................................................................... 126

    Figure 6.16: Cumulative oil recovered for EM heating, CSS, and cold production (noheating) for the thin-zone (21 ft) reservoir simulation.................... 129

    Figure 6.17: Cumulative oil recovered for EM heating, CSS, and cold production (noheating) for the low permeability reservoir simulation................... 129

    Figure 6.18: Cumulative oil recovered for EM heating and CCS for an extra-heavy oil(7

    API) reservoir. Without heating no oil production occurs....... 132

    Figure 6.19: Thermal efficiency for EM heating, CSS, and cold production (no heating)......................................................................................................... 134

    Figure 6.20: Productivity improvement (q EMH/qcold) from EM heating for a thin-zoneand a low-permeability reservoir .................................................... 134

    Figure 6.21: Comparison of oil production (Bbls/day) obtained by ERH and EMH fordifferent heat sources. Oil production by EMH is shown for a 10 kWsource (no vaporization), and for a 70 kW source (vaporization) .. 137

    Figure C.1: Comparison of the temperature profiles for the analytical andnumerical solution COMSOL) without heat conduction after 100 daysof EMH .......................................................................................... 155

    Figure C.2: Comparison of the temperature profiles for the analytical andnumerical solution (COMSOL) without heat conduction after 1200 days of EMH................................................................................................ 156

    Figure C.3: Comparison of the temperature profiles for the analytical solution (steady-state) and the numerical (COMSOL) solution without heat conduction.Steady-state was reached after 5,000 days of EMH for the numerical

    simulation........................................................................................ 157Figure C.4: Comparison of the steady-state temperature profiles for the analytical

    and numerical solution (COMSOL) for a 1D single-phase counter-current Cartesian flow. Heat conduction within the reservoir isincluded........................................................................................... 159

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    xx

    Figure C.5: Comparison of the steady-state temperature profiles for the analytical andnumerical solution (COMSOL) for a 1-D single-phase counter-currentradial flow. Heat conduction within the reservoir is included...... 160

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    Chapter 1: Introduct ion

    High viscosity is a major concern for the recovery from heavy-oil reservoirs. A

    very high oil viscosity results in a project being technically challenging, and sometimes

    uneconomical. Introducing heat to the formation has proven to be an effective way of

    lowering the oil viscosity by raising the temperature in the formation.

    Thermal recovery involves well-known processes such as steam injection (cyclic

    steam stimulation or huff and puff, steam drive, and steam assisted gravity drainage), in

    situ combustion, and a more recent technique that consists of heating the reservoir with

    electrical energy (Chakma and Jha, 1992; Soliman, 1997; Sahni et al . 2000; Sierra et al .

    2001). Among these processes, steam injection leads in development and application;

    however, the application of electrical energy is of interest because it offers fewer

    restrictions with respect to candidates for its successful application compared to the

    conventional steam injection methods (steam-drive and cyclic steam stimulation).

    Electromagnetic (EM) heating is a process where high frequency electrical energyis transformed into heat energy by dielectric losses when an electromagnetic wave is

    radiated from antennas into oil-bearing formations. As the electromagnetic energy

    propagates into the formation, fluids and other reservoir materials impede its passage by

    providing resistance to the flow (Chakma, 1992). As a result, the intensity of the

    propagating wave is reduced and the energy is converted to heat increasing the formation

    temperature with the subsequent reduction in oil viscosity.

    In EM heating, energy is propagated by electromagnetic waves that are absorbed

    by the polar molecules (water) in the near wellbore zone, converted to heat, and then

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    transferred by conduction to the oil and reservoir rock. Compared to the case of

    conduction heating, where heat propagates by conduction only, a larger area of the

    reservoir might be covered with EM heating.

    Electrical heating tools and applications can be divided into two categories based

    on the frequency of the electrical current used: (1) Low frequency (less than 60 Hz)

    currents are used in resistive heating (ERH). In this mode it is assumed that resistance

    heating dominates the process and other factors are negligible. Here the depth of

    penetration is high but the intensity low; (2) High frequency currents are used in

    microwave (MW) or radio frequency (RF) heating. A more detailed description of the

    heating methods according to the frequency used and some past applications is discussed

    in Chapter 2 of this dissertation.

    In this study, EM heating refers to RF or MW heating, where heating is produced

    by the absorption of electromagnetic energy by the polar molecules in the formation;

    hence, the amount of heat absorbed will depend on the adsorption coefficient of themedium, a parameter further discussed and analyzed in this work.

    Although several authors have dealt with the possibility of using EM heating to

    enhance recovery from heavy oil reservoirs, there are no comprehensive models or

    commercial tools yet available that includes EM heating to reservoir simulation. This

    research was carried out to develop an EM heating model that couples fluid flow and the

    thermal response of a reservoir when an EM source is applied at a wellbore. Because of

    the complexity in the coupling of the resulting system of equations, numerical solution

    becomes necessary; however, analytical solutions are developed for simplified cases and

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    later used to validate the numerical solutions. Numerical solution of the simultaneous

    equations is achieved by using COMSOL Multiphysics, a commercially licensed

    software. This allowed the calculation of the production improvement resulting from the

    application of EM heating, which can be used as a screening tool for the development of

    future EM heating projects.

    1.1 DESCRIPTION OF THE PROBLEM

    The use of high frequencies for downhole dielectric heating has significant

    potential applications to heavy oil recovery. The heating of formation fluids and porous

    media can lead to improved mobility of the oleic phase, relative to the aqueous and gas

    phases, by reducing its viscosity by several orders of magnitude with a subsequent

    increase in oil production.

    Electromagnetic (EM) energy heats from within and instantaneously; therefore,

    this method is relatively independent of the thermal conductivity of the oil sand andreservoir heterogeneity (Kim, 1987). Applications of EM heating for heavy-oil reservoirs

    can be especially beneficial where conventional methods cannot be used because of large

    depth, formation discontinuity, no water available to make steam, reservoir heterogeneity,

    or excessive heat losses. Chakma and Jha (1992) showed that EM heating is an effective

    way to introduce energy to the reservoir in a controlled manner and that this energy can

    be directed into a specific region. Higher heat efficiency is achieved for thin pay-zones,

    where the use of steam is not economically feasible because of excessive heat loss

    through the overburden.

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    EM heating does not require a heat transporting fluid such as steam or a hot fluid

    injection process, which avoids the complications associated with generating and

    transporting a heated fluid, and allows it to be applied in wells with low incipient

    injectivity. EM heating can apply to situations where generating and injecting steam may

    be environmentally unacceptable (i.e., through permafrost), no wastewater disposal is

    required, and conventional oil field and electrical equipment can be used, which makes

    this technique very attractive for offshore heavy-oil recovery, though it has not been

    applied there. Furthermore, a single well can be used to introduce energy to the

    formation through a power source as well as to recover produced fluids. Production may

    occur during or immediately after EM heating if the formation pressure is large enough

    (Kim, 1987). All of the above are only some of the advantages of EM heating as a

    recovery method for heavy oil reservoirs with respect to the conventional thermal

    processes. However, the intent of this work is not to present EM heating as a preferable

    technology over steam injection for all heavy oil reservoirs, but as an alternative recoverytechnique from heavy oil reservoirs that are not attractive to steam because of steam

    flooding many limitations.

    Although its potential was recognized since the late 70s, there are few field

    applications of EM heating or comprehensive modeling efforts. Some proposed models

    considered the EM adsorption coefficient as a constant regardless of the properties of the

    medium. It is known that as temperature rises near the wellbore, vaporization of connate

    water occurs; then, the adsorption coefficient decreases to zero and no heat energy is

    further absorbed in that zone. Since steam absorbs little EM energy, the latter will travel

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    from the near wellbore zone further into the reservoir where residual water is still present,

    allowing a larger area of the reservoir to be heated than when electrical resistance heating

    is used. Therefore, a good estimation of the absorption coefficient and its variation is

    necessary for an accurate prediction of the response of a reservoir undergoing an EM

    heating process.

    More recent studies have been conducted to couple an EM heating model with

    reservoir simulation (Sahni et al . 2000; Sierra et al . 2001; Ovalles et al . 2002). In these

    studies, the resulting energy distribution from the application of the EM power source is

    calculated separately and then input to a fluid flow simulator to estimate the production

    response. This approach does not integrate the effect of temperature rise and water

    vaporization caused during EM heating to fluid production.

    This dissertation proposes the development of a comprehensive two- dimensional,

    multiphase EM heating model that accounts for the variation of the EM absorption

    coefficient according to the dielectric properties of the medium. These properties areevaluated at each time step as a function of the temperature and the water saturation

    present in the formation. The temperature distribution as well as the water saturation is

    calculated spatially from the simultaneous solution of the overall energy balance and the

    continuity equation for each component. In so doing, I used COMSOL Multiphysics, a

    commercial software that allows building in the model by inputting all the required

    governing differential equations, boundary and initial conditions, and constitutive

    relationships to fully describe the EM heating process. The model includes energy

    transport by the electromagnetic source, thermal convection, thermal conduction and

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    vertical heat loss, coupled to multiphase fluid flow. Water vaporization is accounted for

    by solving for the in situ steam saturation once the boiling temperature of the water at

    reservoir pressure is reached. Analytical solutions were developed for special simpler

    cases to assess the validity of the model. The geometry of the model can be easily

    modified to handle either Cartesian or cylindrical coordinates. Because the EM

    frequency used also plays an important role in the determination of this parameter, the

    existence of an optimum frequency measured from its effect on productivity

    improvement is also studied.

    1.2 RESEARCH OBJECTIVES

    The purpose of this dissertation is the development of an electromagnetic (EM)

    heating model coupled to multiphase fluid flow, for the rapid simulation of heavy oil

    recovery and productivity improvement of a reservoir when an EM heating source is

    applied. This research work will address the following general objectives:

    Develop a two-dimensional EM heating model coupled to multiphase flow

    for the simulation of heavy oil recovery by using EM heating.

    Use the model developed with commercial software to perform numerical

    simulations to calculate the oil production response of a well undergoing

    EM heating.

    Test and validate simulation results by comparing them to simplified

    analytical solutions.

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    Define and evaluate critical variables during EM heating by performing

    sensitivity analysis on variables such as: absorption coefficient ( ), EM

    frequency, input power radiated (P o), and permeability (k).

    Identify improvements in current applications of EM heating to the field.

    Compare the recovery of a heavy oil reservoir when EM heating is applied

    to that obtained by cyclic steam injection and electrical resistance heating.

    1.3 METHOD AND APPROACH OVERVIEW

    To successfully achieve the above objectives, several tasks are completed in the

    different stages of this study. The first stage consists of the development and testing of a

    one-dimensional, single-phase EM heating model integrated to a commercial software

    (COMSOL Multiphysics) to calculate the temperature distribution and the productivity

    improvement obtained when an EM heating source (an antenna) is placed at the well.

    Thermal convection and radial conduction are included in the model; at this stage the

    absorption coefficient to calculate the EM energy is constant. Analytical solutions are

    developed for some limiting cases as well as steady-state solutions to assess the relevance

    of different variables in the EM process. We consider both counter-current flow, in

    which the well with the antenna is also a producer, in other words, fluid flow is opposite

    to EM energy flow, and co-current flow where fluid and EM energy flow in the same

    direction. Figures 1.1 and 1.2 show a schematic view of the EM process for the cases

    considered. In all cases EM energy is introduced during fluid flow.

    The one-dimensional EM heating model is developed in two different coordinate

    systems, Cartesian and cylindrical coordinates, and then solved using COMSOL

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    Multiphysics; a commercial software that allows coupling the different equations that

    describe the EM heating process. Although simple, this model provided useful insights

    about EM heating. Results are validated using the respective simplified steady-state and

    transient analytical solutions for the limiting cases of no conduction allowed, or no fluid

    movement. In the case of no convection, the resulting temperature distribution was

    compared.

    During the second stage of this dissertation, the one-dimensional model is extended to

    two-dimensions. This model is developed in cylindrical coordinates (r,z). Only the

    counter-current case is chosen as the production scheme for the simulations merely for its

    analogy to cyclic steam injection, to which it will be later compared. The center of the

    well, where the EM source (antenna) is located, is a line of symmetry, therefore, only half

    of the reservoir is modeled. Besides the energy transport mechanisms mentioned before,

    transient vertical heat loss through the overburden and underburden is included. The EM

    energy flow from the source is updated at each time step through the dependence ontemperature of some of the electrical properties used to calculate the absorption

    coefficient.

    From this model, temperature profiles and oil production rates are calculated.

    Sensitivity analysis are performed on the model to determine critical variables in the

    design of an EM heating process, such as frequency, input power, permeability, and pay

    thickness, as well as their effect on the productivity improvement from heavy oil

    reservoirs. This process allows the optimization of certain parameters for the field

    application of EM heating. Production response from the EM heating process is also

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    compared to that obtained by steam injection and electrical resistance heating simulated

    using STARS from CMG.

    The last stage of this dissertation consists of the introduction of multiple phases

    flowing to the EM heating model developed. The effect of the relative permeabilities is

    considered, as well as the formation of a vapor phase after the residual water is vaporized

    as a consequence of the rise in temperature near the wellbore zone. The effect of the

    change in water content is assessed through the updating in the dielectric constant of the

    medium used to calculate the absorption coefficient. Again, the productivity

    improvement obtained by the EM heating process is compared to that obtained by cyclic

    steam injection and electrical resistance heating.

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    PowerSupply

    Confining Layer (Overburden)

    Fluid Flow Fluid Flow h = Thickness

    EM flowConfining Layer (Underburden) Antenna

    Figure 1.1: Schematic view of EM heating process for counter-current flow. Theantenna is placed at the center of a producing well in front of the reservoirzone confined by the adjacent layers. EM energy flows into the formationand it is transformed into heat with the subsequent increase of fluidflowing toward the producing well. Only one well is used for this EMheating process.

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    PowerSupply

    h = Thickness

    Confining Layer (Overburden)

    Confining Layer (Underburden)

    Fluid Flow

    Producer

    AntennaEM flow

    Figure 1.2: Schematic view of EM heating process for co-current flow. The antenna isplaced at the center of an injector well in front of the reservoir confined by theadjacent layers. EM energy flows into the formation and is transformed intoheat with the subsequent increase of fluid flowing toward the producing welllocated at the opposite end of the system. Two wells are used for this EMheating process.

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    Chapter 2: Literature Review

    2.1 Electromagnetic Waves

    An electromagnetic (EM) wave is a disturbance of a continuous medium that

    propagates with a fixed shape at a constant velocity. In the presence of absorption the

    EM wave will decrease in size and amplitude as it moves (Griffiths, 1999). The

    propagation and absorption of all EM radiation through a porous medium is described by

    Maxwells equations (Fanchi, 1990; Ayappa et al . 1991; Ayappa et al . 1992). Based on

    the assumption of an infinite medium, a simpler mathematical expression known as

    Lamberts law can also be used to describe the absorption of EM waves (Araque and

    Lake, 2002; Ovalles et al . 2002; Vadivambal and Jayas, 2010).

    2.1.1 Maxwells Equations

    The propagation of EM energy through a porous medium is governed by

    Maxwells equations in the following form:

    f D (2.1)

    0B (2.2)

    BEt

    (2.3)

    f D

    H Jt

    (2.4)

    where the vectors D, B, E, and H, correspond to the electric flux density, magnetic flux

    density, electric field intensity, and magnetic field strength, respectively; f is the free

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    Equations (2.12) and (2.13) admit the plane-wave solutions (Griffiths, 1999), where the

    wave number is complex with the form:

    k i (2.14)

    The imaginary part of results in an attenuation of the wave (decreasing amplitude with

    increasing distance). Then, the electric field intensity (E) can be expressed as

    )0, i kx txE x t E e e (2.15)

    where is the incident electric field intensity, x is the distance coordinate, and t

    corresponds to time. The quantity

    0E

    is the electromagnetic absorption coefficient,

    defined as

    122' 1

    2

    1

    (2.16)

    where is 2 times the frequency, is the real part of the complex permittivity, ' is

    the real part of the complex magnetic permeability, and is the dielectric conductivity

    of the medium.

    The real part of determines the wavelength, the propagation speed, and the

    index of refraction. It is defined as

    122' 1

    2k

    1

    (2.17)

    Using the above equations, the absorption of waves from a power-radiating

    antenna, also known as the power attenuation term, including the reflection effects can be

    described (Fanchi, 1990; Ovalles et al . 2002). Additional discussion on the above

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    derivation can be found in many classical electrodynamics books (J ackson 1975;

    Griffiths, 1999).

    2.1.2 Lambert s Law

    A special solution through a series of simplifications applied to Maxwells

    equations (Barringer et al . 1995) can be used to describe the dependence of the radiation

    power on distance, while ignoring reflection at interfaces; this is known as Lamberts law

    of absorption. This allows modeling the heat generated by EM waves by considering that

    microwave power decreases exponentially as a function of penetration (Vadivambal and

    Jayas, 2010).

    Widely used in spectrophotometry (Bird et al . 2002), Lamberts law states that the

    intensity I of radiant energy passing through a homogeneous absorbing medium

    decreases exponentially according to the equation:

    (2.18) zeII 0

    where I 0 is the incident intensity, the absorption coefficient, and z the distance traveled

    through the medium (Bird et al . 2002). Equation (2.18) is analogous to equation (2.15)

    when the term corresponding to the real part (k) of the wave number is neglected. A

    similar expression for the power attenuation term was developed by Abernethy (1976),and verified by Fanchi (1990).

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    PowerPower

    (a) (b)

    Figure 2.1: Two typical schemes for resistive heating: (a) less efficient scheme asmost of the current flow is outside the reservoir section, (b) the casing isused as a terminal which improves the scheme by localizing more of theelectrical losses (heating) in the reservoir area. The dotted regionsrepresent the electrically isolated sections of casing. After Calarotti(2000).

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    Field testing of electrical resistance heating has been widely reported in the

    literature (Pizarro and Trevisan 1990; Rice et al . 1992; Davison, 1992; Sierra et al . 2001,

    Sahni et al . 2000). Pizarro and Trevisan (1990) presented data of low frequency

    electrical heating from Rio Panom field in Brazil. The field was shallow, targeting a

    single oil-bearing formation of small thickness with a rather high viscosity of 2,500 cp,

    characteristics that made this reservoir an appropriate choice for electrical heating

    opposed to steam injection. Primary production was increased from 1.2 to 13 B/D after

    70 days of applying electrical heating with an averaged power source of 30 kW. Rice et

    al . (1992) reported results of a test conducted by EOR International in the Schoonebeek

    field in Netherlands. Oil production increase from 13 to 31 m 3/D (80 to 194 B/D) was

    observed with a power supply of 60 kW. Low frequency heating has also seen

    applications in environmental remediation (Newmark, 1994).

    Although a rapid response to heating, and encouraging production rates have been

    observed from field applications of low frequency electrical heating, none of the projectswere successful because the heated flow rates could not be maintained as a result of water

    flashing to steam near the wellbore zone.

    2.2.2 High Frequency Heating - Radio Frequency (RF) or Microwave

    Heating (MW)

    In general, radiation with frequencies within a range of 10 to 100 MHz are

    referred to as radio frequencies (RF), and in the range of 300 MHz to 300 GHz as

    microwave (MW), and corresponding wavelengths from 1 to 0.001 m (J ackson, 1975;

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    Ovalles et al . 2002). Figure 2.2 shows the full spectrum of electromagnetic waves. At

    high frequencies dielectric heating dominates (Pizarro and Trevisan, 1990; Chakma and

    Jha, 1992) and heat is generated through the absorption of electromagnetic energy by the

    polar molecules (connate water) in the formation. According to laboratory

    measurements, oil-bearing sands can absorb RF or MW energy and reach very high

    temperatures (300 to 400 C) very rapidly (Ovalles, 2002) as does steam injection. In this

    work will refer to high frequency heating as electromagnetic heating (EM heating).

    Because of the very high absorption of moist soil, most of the introduced energy

    tends to be absorbed in the vicinity of the wellbore (Bridges et al . 1985); however, as

    opposed to ERH that rely on the presence of water to establish a path to maintain

    electrical conductivity, high frequency heating can be used to boil water in situ and

    convert water to steam, creating a dry zone, and increasing pressure in the near wellbore

    area. The amount of energy absorbed will depend on the electrical properties of the

    formation, which are a function of its composition and water saturation (Marsden, 1991)as well as on the operating frequency. The magnitude of the change in temperature in

    the formation will depend on the energy absorbed. McPherson et al . (1985) developed

    the concept of skin depth, which is the distance into the material a wave propagates

    before its field strength is reduced to 1/e (e =2.718) of its initial value, and the wave

    power is reduced to about one third of its original value.

    Figure 2.3 shows a schematic view of the energy deposition of an EM wave, and

    illustrates the concept of skin depth. Since steam does not absorb electromagnetic

    energy, the steam front will move further out into the reservoir as water evaporates for a

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    given heat input (Soliman, 1997). Kim (1987) concluded that at temperatures above the

    boiling point of water, EM heating increases the depth of heat penetration significantly.

    Although, the physical configuration of the well and the components of the power

    system for the delivery of EM waves to the formation will not be discussed in this

    dissertation, the literature reveals different tests that serves as guidance for the well

    configuration assumed in this study (Haagensen, 1965, 1986; J eambey, 1989). The

    Electrothermic Co. of Corpus Christy, TX, developed a process known as the

    electrothermic process. This technology uses a single-well with a system configuration

    very similar to that of a well for EM heating. Their completion design allowed the use of

    high current as well as high power successfully.

    More recently, Calarotti (2000) describes two typical schemes used for high

    frequency heating. Figure 2.4 shows two ways of transferring electromagnetic energyfrom the surface to the reservoir. As can be seen, energy can be transmitted in the

    annular space between the casing and the producing tubing (2.4a) or along the production

    tubing (2.4b). For these a radiating element must be situated at reservoir depth.

    Although its potential was recognized more than two decades ago, there are few

    field applications of EM heating or comprehensive modeling efforts. Abernethy (1976)

    derived an expression for the EM power attenuation term, later studied by Fanchi (1990),

    which allows calculating the temperature profile of a reservoir undergoing EM heating.

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    Ovalles et al . 2002). Okassa et al . (2010) conducted numerical simulations to solve for

    the EM field and the heat transfer through a porous media during EM heating. They

    combined EM heating with the addition of ceramic materials; however, they did not

    include fluid dynamics.

    The IIT Research Institute (IITRI) of Chicago, Illinois, in 1985, developed the

    electromagnetic well stimulation process, for which they hold the basic patents, as a

    thermal process analogous to cyclic steam injection, except that the process is continuous

    and electromagnetic energy is used instead of steam to introduce energy to the formation.

    This process was tested in the Wildmere field of Alberta, producing from the

    Lloydminster sand. Production was reported to increase from 6 to 20 B/D after 20 days

    of applying EM heating.

    Sresty et al . (1986) conducted field experiments in the Asphalt Ridge deposit, UT.

    By inserting tubular electrodes in special arrays and then applying RF energy into the

    deposit, they heated and produced the bitumen by gravity drainage and in situ steamdrive. A recovery of about 35% of the total bitumen in place within the tar sand test

    volume was obtained.

    Kasevich et al . (1994) describe the equipment and method used for well

    completion for their pilot test of a radio frequency heating system conducted in

    Bakersfield, CA. In their pilot testing, Kasevich et al . (1994) used a borehole radio

    frequency (RF) antenna at a depth of 620 ft. crossing the diatomite formation in the North

    Midway field, CA. A diatomaceous earth site was selected because this type of rock does

    not respond well to conventional steam injection methods. The resulting temperature

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    Figure 2.5: Dielectric Properties of Diatomaceous Rock versus Temperature. AfterKasevich et al . (1994).

    Figure 2.6: Dielectric Properties of Diatomaceous Rock at Ambient Temperature (20C) versus Frequency. After Kasevich et al . (1994).

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    Table 2.1: Coefficients of a linear approximation for the relative dielectric constant R with respect to frequency. Values were determined from samples of

    the Athabasca oil sands in Alberta. After Chute and Vermeulem (1990).

    Frequency(MHz) C1 C2

    1 10.0 170

    10 3.7 160

    100 3.3 100

    1000 3.3 82

    Table 2.2: Temperature coefficient Ta for electrical conductivity with respect tofrequency for temperatures below 120

    C (248

    F). Values weredetermined from samples of the Athabasca oil sands in Alberta. AfterChute and Vermeulem (1990).

    T Frequency(MHz) (x 10 -2 1/

    C)

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    (3.1)1,..., 1,...,c p

    Accumulation Netmass Mass inputof iof i in phase j transfer of i in phase j into phase j

    i N j

    N

    Equation (3.1) is the conservation of mass of each component. Neglecting reactions, it

    can be written in differential form as:

    1 01,..., 1,...,

    ij j j j ij s is j ij j j ij

    c p

    S w w u w S K wt

    i N j

    N

    (3.2)

    where is the porosity of the reservoir, is the density of the fluids and rock, S j is the

    saturation of phase j, is the mass fraction of component i in phase j,ijw ju is the Darcy

    velocity vector of each phase, and ijK represents the dispersion tensor. Fluid flow is

    governed by Darcys law:

    (rj j j j j

    kku p

    )g

    (3.3)

    where is the permeability tensor, is the relative permeability of phase j,k rjk is the

    viscosity of phase j, p j is the pressure in phase j, and g is the acceleration because of

    gravity.

    We also use the following additional assumptions: no dispersion

    0ijK ,

    and

    no adsorption 0isw . Assuming there is only one component present in each phase,

    the mass fraction of a component in a phase ijw is either 1 or 0. Then, equation (3.2)

    summed over the phases, yields a form of the pressure equation for each component as

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    Substitution of equation (3.11) into (3.10), gives the total conservation of energy as:

    1 1

    1p p

    eff

    N N

    j j j j s s j j T EM

    j j

    S U U u H k T qt

    (3.12)

    Following a similar treatment to the steam flooding equations, we neglect the pressure-

    volume work by assuming enthalpies equal internal energies and by taking porosity as a

    constant. Then, equation (3.12) becomes:

    1 1

    1p peff

    N N

    j j j j s s j j T EM j j

    S H H u H k T qt

    (3.13)

    Assuming that enthalpies are independent of pressure, we introduce the concept of

    constant pressure heat capacity given by:

    j

    jp

    P

    HC

    T

    (3.14)

    Using the above equation, the derivative of the enthalpies with respect to time in the

    accumulation term can be expressed as:

    j

    j jp

    H H TC

    t T t T

    t

    (3.15)

    Then, equation (3.13) becomes:

    1 1 1

    1

    1p p p

    j s j

    p

    eff

    N N N

    j j j p s p j j j j p j j j

    N

    j j j T EM j

    T S C C H S u C Tt t

    H u k T q

    (3.16)

    Introducing the concept of volumetric heat capacity of each phase jM and solid sM ,

    defined as j j p

    C andss p

    C , respectively; equation (3.16) can be written as:

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    1 1 1

    1

    1p p p

    p

    eff

    N N N

    j j j s j j j j j j j

    N

    j j j T EM j

    T M S M H S M ut t

    H u k T q

    T

    (3.17)

    Equation (3.17) represents a general form of the total energy balanc e for a

    reservoir under EM heating including energy transport by convection, conduction, and

    the EM heating source, for any number of phases. Once the EM heating source term is

    determined and substituted into equation (3.17) , according to the corresponding

    coordinate system used, the temperature distribution of a reservoir undergoing EM

    heating can be estimated. Analytical solutions for both Cartesian and radial coordinates

    of equation (3.17) for some limiting cases are presented in Appendix A.

    Equations (3.4) and (3.17) constitute the system of equations to be solved to

    determ ine the response of a reservoir when an EM source is applied. Equation (3.17) is

    coupled to equation (3.4) through the dependence of the fluid viscosity in the Darcy

    velocity term on temperature to solve for pressure, and the dependence of the temperature

    calculation on the fluid velocity obtained from the solution of the pressure equation.

    Additional relationships depending on the number of phases present in the reservoir are

    used and will be discussed later in this dissertation.

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    adjacent formations is included by setting the continuity of heat as a boundary condition

    between the reservoir and the top and bottom formations.

    A schematic diagram showing the boundary conditions taken for the model, in a

    radial system, is presented in Figure 3.1 and their mathematical expressions are shown in

    Table 3.1 .

    3.4 Modeling using COMSOL Multiphys ics

    Numerical solution of the described system was accomplished by using COMSOL

    Multiphysics, a licensed software that allows coupled physics by solving simultaneously

    any number of differential equations given the boundary conditions, and the relationships

    among variables. When solving the model, COMSOL Multiphysics uses the finite

    element method (FEM), this method has been widely documented and proved in the

    literature.

    Two different ways are available when inputting the governing system of

    equations: 1) the first approach is using the built-in physics modes, which allows the userto build models by choosing among several specific applications with their underlying

    equations already defined, or 2) the second is using the PDE module, where the entire

    system of equations and constitutive relationships describing the model are entered by the

    user following the program structure.

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    Table 3.1: Mathematical expression of the boundary conditions in radial flow for theelectromagnetic (EM) heating model. Conduction heat loss is accountedfor at the bottom (z =0) and top (z =h) boundaries of the reservoir.

    Equation Initial (t=0) r=r w r=r e z=0 z=h

    Pressure(p) i

    p wf p ip 0Pz 0P

    z

    Temperature(T) 0

    T 0 Tr

    0 T , ,eff UB eff res

    UB res T T

    T Tk k

    z z

    , ,eff OB eff res

    OB res T T

    T Tk k

    z z

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    Chapter 4: One-Dimensional Single-Phase Electromagnetic

    Heating (EM heating) Model

    4.1 Overview

    Prior to the development of the two-dimensional, multiphase, radial

    electromagnetic (EM) heating model for heavy oil recovery, we developed a 1-D single-

    phase EM model and obtained both steady-state and transient solutions. This chapter

    discusses the 1-D single-phase EM heating model and the results obtained from its

    application.

    The 1-D single phase EM heating model does not take into account heat loss in the

    vertical direction, and assumes that only the oil phase, with oil as a single component, is

    mobile in the reservoir. The reservoir is horizontal. Rock and fluid properties are

    constant, except for the oil viscosity. The electrical properties of the medium are taken as

    a constant; therefore, the EM absorption coefficient is kept constant and independent of

    the temperature and fluid saturation changes in the reservoir. We considered bothcounter-current flow, in which the well with the antenna is also a producer, similar to

    cyclic steam injection, and co-current flow, where the fluid and EM energy flow in the

    same direction from the source well (injector well in a steamflood process) to a producer

    well.

    The development of the model was carried out in both Cartesian and radial

    coordinates. Although in practice, radial flow is usually used for fluid flow simulation in

    a single-well reservoir, the use of Cartesian coordinates in a linear system allows

    modeling the process for future experimental work.

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    Integration of equation (4.27) gives that EMe q C , this expression indicates the sum of

    the fluxes is a constant. Then, the material energy flux is always opposite to the photon

    flux, where

    0 xEM

    Pe C q C e

    A (4.28)

    where P0 is the incident power radiated at the inlet, A is the cross-sectional area, is the

    EM absorption coefficient, and x is the distance coordinate. The constant C is determined

    by the nature of the flow, as shown in Figure 4.1 . If the flow is counter-current, C=0

    (Figure 4.1 upper); if it is co-current flow, 0PCA

    (Figure 4.1 lower). The depth of

    penetration is entirely governed by , where large values indicate short penetration

    distances.

    Using a variable change 0 T T T , where corresponds to the initial reservoir

    temperature, and substituting equation

    0 T

    (4.26) , in Cartesian coordinates for one-

    dimensional flow, into equation (4.28) yields:

    0 0

    eff eff eff

    x o o

    T T T

    P C M u dTe TAk k k dx

    (4.29)

    for counter-current flow, and

    0 0

    eff eff eff

    x o o

    T T T

    P C M u dTe

    Ak k k dx T (4.30)

    for co-current flow. Equations (4.29) and (4.30) were solved analytically to obtain the

    temperature profile for one-dimensional single-phase flow when an EM source is applied.

    The following expressions were obtained:

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    Figure 4.1: Schematic comparison of energy fluxes at steady-state for counter-currentflow (top), and co-current flow (bottom). The sum of the fluxes is aconstant, which value depends on the direction of the flow.

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    Figure 4.4: Steady-state temperature profile for a 1-D single-phase co-current radialflow for different values of the EM absorption coefficient, , with a

    constant power source of 63 kW.

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    improvement (PI) in this work. Appendix B gives the formulas used to calculate the

    steady-state productivity improvement.

    Using the rock and fluid properties given in Table 4.1 , we calculated the

    maximum production achievable for different input power values. Figure 4.5 shows a

    PI for counter-current linear flow of about 2.5 times for a power input of 20 kW. The PI

    increases as the power in the source increases. For an input power of 150 kW, the PI

    improvement was about 14. For all the values used for the input power, the peak in the

    PI occurred within the range of =10 -3 and 10 -1 m -1, which indicates the existence of an

    optimum absorption coefficient. When the increases beyond this point, a considerable

    reduction in the PI occurs. For linear co-current flow the PI was even greater for the

    same values of input power than in counter-current flow. Figure 4.6 shows a maximum

    PI of about 40 times for a power of 150 kW. This is more than three times the value

    obtained for the counter-current flow case. However, when increases beyond 10 -1 m-1,

    production declines back to the initial production rate.

    For radial counter-current flow the maximum PI increase was about 11 times the

    cold production for an input power of 150 kW. Figure 4.7 shows the PI for radial

    counter-current flow for different values of input power. Unlike linear flow, the

    maximum production for most of the power sources used occurred for within the range

    of 10 -2 to 10 m -1.

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    well were done. For constant flow rate, the temperature increases with time until it

    reaches the steady-state solution.

    Figure 4.8 shows the temperature profile with time for counter-current flow in a

    radial system for a heavy oil reservoir (11 API), with 28% porosity, 30 ft thickness, and

    a drainage radius of 50 ft. A constant production rate of 30 BOPD was imposed at the

    well. Steady-state flow is reached within a year of production. Figure 4.9 shows the

    temperature profile after 30 days of heating for different flow rates using a source of 63

    kW for radial counter-current flow. The initial reservoir temperature was 100 F.

    Temperatures in the near wellbore zone as well as the extension of the heated zone into

    the formation are affected by the production rate for counter-current flow. As the

    amount of oil produced increases, more of the heat generated by EM is extracted from the

    formation; therefore, lower temperatures at the wellbore are obtained. This decreases the

    temperature rise in the formation, and moreover the effectiveness of EM heating because

    a lower temperature means less mobility of the fluids. Then, the EM energy is reheatingthe near wellbore area instead of propagating the heated front into the reservoir.

    Figure 4.10 shows the temperature distribution with time for counter-current

    radial flow with a constant pressure at the well as a boundary condition. Steady-state

    takes longer to be reached compared to the constant rate well case. For a constant

    pressure well, the steady-state solution is reached after approximately 850 days, while it

    takes less than a year for the constant rate well case (See Figure 4.8 ).

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    Figure 4.10: Transient temperature profile for a radial system with a constant pressurewell for counter-current flow.

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    These results suggest that the overall efficiency of EM heating can be diminished

    by a high production rate based on the pressure drop imposed, without even considering

    other operational problems associated with high-rate production.

    4.3.2 Productiv ity Improvement Results

    To estimate the productivity improvement with time obtained from the application

    of EM heating, the production rate was plotted until the steady-state condition was

    achieved. Figure 4.11 shows the oil rate produced when EM heating was applied, and

    the cold production with time. The oil rate curve for EM heating shows a continuous

    increase of the production rate with time at early times, this correlates with the fact that

    as the initial temperature in the reservoir is raised about 100

    F, the reduction in the

    original viscosity is in the order of 10,000, so fluid flow becomes much easier than it was

    originally in the heated area, which increases production rapidly. This increment in the

    production rate is also a consequence of the effect of EM heating on the pressure at the

    near wellbore region. The pressure drop at the near wellbore is almost eliminated withEM heating (See Figure 4.12 a-b ).

    However, as the oil production increases more of the heat is continuously

    withdrawn from the reservoir, until it reaches a point where the rate no longer increases

    but instead begins to decrease as time passes, approaching a constant value at late times.

    Although the final oil rate is higher than we expected, this is attributed to the assumption

    of constant pressure and constant temperature at the external boundary for the model

    solution.

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    Figure 4.11: Oil rate with time for EM heating compared to the cold production(without radiation). Production plotted until steady-state is achieved at851 days. Note the leveling at a constant rate for the EM heating caseafter approximately 500 days. Peak production is obtained after 128 daysof EM heating.

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    0

    200

    400

    600

    800

    1000

    0 50 100 150 200

    Distance from wellbore, ft

    P r e s s u r e ,

    p s

    i

    EM Heating

    Cold (No heating)

    0

    200

    400

    600

    800

    1000

    0.1 1 10 100 1000

    Distance from wellbore, ft

    P r e s s u r e , p s

    i

    EM Heating

    Cold (No heating)

    r w

    Figure 4.12: a) (Top) Pressure profile after 200 days of production for EM Heating

    compared to cold production. b) (Bottom) Semi-log plot of the pressureprofile after 200 days of EM Heating compared to cold production.

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    The productivity improvement, which is a measurement of the EM heating effect

    on the production, was calculated based on the cold rate, which is the rate obtained when

    no EM energy is applied. Therefore, as time progresses the difference between the cold

    rate and the heated rate becomes larger, because of the fast decline of the oil rate when no

    heating is provided to the reservoir. Some of the results are shown in Table 4.2 .

    Results showed an increase in oil production from 35 BOPD to 104 BOPD after a

    period of 850 days of heating continuously. The highest production rate was found to be

    123 BOPD after 128 days of EM heating. Results indicate the feasibility of raising oil

    production to approximately four times the initial cold oil production after 2 years of

    heating.

    4.4 Factors affecting the Productiv ity Improvement with EM Heating

    To evaluate the response of EM heating to different parameters involved in the

    model we conducted several sensitivities to identify some of the critical variables in the

    process. This analysis was later used to select the optimum characteristics for ahypothetical reservoir for the application of this technique. The performance of the

    application of EM heating was analyzed based on the temperature profiles as well as the

    production rates obtained for the different scenarios considered. The productivity

    improvement with EM heating is directly related to the temperature distribution obtained

    when EM heating is applied; therefore, a qualitative response of the productivity

    improvement can be estimated by simply analyzing the temperature response for each

    specific set of parameters in some cases.

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    Table 4.2: Productivity improvement with time obtained from the cold oil rate (noEM heating), and the oil rate when EM heating was applied.

    Time,days

    q o,coldBOPD

    q o,heated

    BOPD

    PI r

    0 35.9 35.9 1.0010 25.1 58.8 2.3530 23.0 107.1 4.6650 22.1 122.8 5.57

    101 20.9 123.7 5.92126 20.6 123.8 6.01250 19.6 116.1 5.93306 19.3 112.9 5.84700 18.3 107.2 5.87851 18.0 106.3 5.89

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    4.4.1 Effect of Heat Transfer Mechanism

    To evaluate the effect of the heat transfer mechanism when EM heating is applied,

    two different scenarios were considered: 1) heat transfer by convection and radiation, and

    2) heat transfer by conduction and radiation. Simulations were run in Cartesian and

    cylindrical coordinates, respectively, with a constant rate condition at the well. The

    constant rate condition allowed validating the results with the analytical solutions

    developed (See Appendix A). The EM heating response is presented in the form of

    temperature distribution plots for each of the cases considered.

    Figure 4.13 shows the effect of the heat transfer mechanisms on the temperature

    distribution for steady-state Cartesian counter-current flow. Figure 4.14 shows the

    effect of the heat transfer mechanisms on the temperature distribution for steady-state

    radial counter-current flow. A constant rate of 30 BOPD was assumed for the

    convection-radiation cases, and a constant absorption coefficient ( ) for a 915 MHz

    frequency (0.13 m-1

    ) was used. Results in both coordinate systems indicated that even

    though a higher temperature near the EM sourc