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# Electrical engineering considerations for electromagnetic heating of oil

Transmitting electrical current to the subsurface can create special considerations. Successful application of electromagnetic heating often requires a multi-disciplinary approach combining electric engineering and petroleum engineering. To assist petroleum engineers considering this approach, this article identifies some of the issues that an electrical engineer might normally anticipate and address.

## Basics

In most practical situations, we are concerned with fields that vary periodically in time (the sinusoidal steady state generally). In these cases the electrical phenomena are properly described by Maxwell equations in terms of complex vector field intensities of electric and magnetic fields (E → and H →); complex vector field electric, magnetic, and current densities (D → ,B → ,J →); complex charge concentrations (ρc); and complex material parameters: conductivity, permittivity, and permeability (σ, ε, μM). For the case of sinusoidal excitations [exp (jωt)] and in the absence of diffusion currents, these equations are listed next. ....................(1) ....................(2) ....................(3) ....................(4) ....................(5) ....................(6) ....................(7)

As indicated, the material parameters are frequency-dependent complex numbers. Of course, ω = 2 π f, where the frequency, f is given in cycles per second (Hz).

The physical fields vectors correspond to the real part of the complex field vector times [exp ( jωt)]. For example, the physical electric field vector is given by Real[E → exp ( jωt)]. The power radiated per unit area is given by the complex Poynting vector, S → , defined as ....................(8)

where * indicates complex conjugate.

For the volume, V, enclosed by a surface, A, indicated in Fig. 1, the volume integral of the divergence of the complex Poynting vector, S → , yields ....................(9)

This equation represents a balance of power for the volume shown. The left-hand side represents the total power entering the volume through its surface. The first two right hand terms represent the power stored in the volume, while the last two terms represent the power dissipated in the volume. This dissipated power depends on the real part of the conductivity and the magnitude of the local electric field (third term), the imaginary parts of the magnetic permeability and electrical permittivity of the material enclosed in the volume, and the magnitudes of the magnetic and electric fields (fourth term). If the permeability and permittivity are real, the only power loss is owing to the real part of the conductivity and the magnitude of the local electric field.

The power radiated into the volume is equal to the rate of increase of the stored energy in the volume, plus the power dissipated (because of the real part of the conductivity and the imaginary parts of permittivity and permeability, and the magnitudes of the corresponding fields). Thus, the real power dissipated per unit volume, PPUV, is given by ....................(10)

This is the term that enters the differential equation for the heat distribution. Among other factors,the power depends on the magnitudes of the local electric and magnetic fields. In those cases where the electrical power is generated at the surface of the earth, these magnitudes will depend on the transmission power losses from the surface to the reservoir.

### The vector wave equation for exp(jwt) excitation, space independent(σ, ε, μM), no diffusion currents, and ρC=0.

Proper knowledge of the issues involved in the transmission and dissipation of electrical power for the heating of wells and reservoirs can only be obtained by examining the solution of the vector wave equations. The wave equations are derived from Maxwell’s equations. ....................(11)

The propagation constant, γ, is defined as ....................(12)

For their relevance to the heating oil problem, we will consider solutions in rectangular Cartesian coordinates x, y, z and in cylindrical coordinates, r,Φ ,z. We will first discuss the rectangular Cartesian coordinate case.

### Rectangular coordinates: plane wave propagation in the z direction(an assumed direction of power or energy flow).

Under these conditions, we have: ....................(13) ....................(14) ....................(15)

Then, the wave equations have these solutions: ....................(16) ....................(17)

A+ and A– are integration constants defined by the boundary conditions of the problem. Thus, the solutions for propagation in free space along the positive z-axis (σ = 0, ε = ε0 = 8.854 × 10–12 farads/m, the permittivity of vacuum, μM = μ0 = 4π × 10–7 henrys/m, the permeability of vacuum) are ....................(18)

These equations represent waves propagating in the +z direction with velocity, V0 (the speed of light in vacuum), a wavelength, λ0, and a wave impedance, Z0. ....................(19) ....................(20) ....................(21) ....................(22) ....................(23)

Frequency-wavelength relations for free space shows the variety of wavelengths for propagation in free space as a function of frequency, f.

The effect of losses (σ = σ′ – jσ", ε = ε′ – jε", μM = μ′ + jμ") on the solutions makes the propagation constant, γ, become ....................(24)

where j is the complex unit, and the fields are ....................(25)

and ....................(26)

If α < 0, the amplitude of the wave will decrease as it travels in the +z direction. The real power radiated per unit area (PPUA) is ....................(27)

### Cylindrical coordinates: wave propagation in the r direction (an assumed direction of poweror energy flow).

In this case and for uniform wave propagation in the r direction, we have: ....................(28) ....................(29) ....................(30)

The wave equations become ....................(31)

and ....................(32)

Bessel’s equations provide solutions for wave propagation in the positive r direction in terms of Hankel functions H0(2) and H1(2). ....................(33) ....................(34)

The nature of the waves propagating in the +r direction is clearly shown observing the limit of the Hankel functions for large arguments (γ r >> 1). ....................(35)

In essence, this is the far-field approximation used by Abernethy. The real power per unit area radiated in the positive r direction is ....................(36)

## Electrode structures that allow transverse electromagnetic (TEM) wave propagation

The metal electrode configurations, shown in Fig. 2, allow TEM wave propagation for all values of frequency. Both the single-phase (two wires) and three-phase transmission systems shown (normally used for 60 Hz power transmission) have an electromagnetic energy distribution over the cross section of the wires and in the surrounding space. The metal cables simply guide the electromagnetic energy along. In the case of the other structures shown, the fields and the energy are enclosed within the metallic electrodes. The coaxial structure exists naturally in vertical and horizontal wells. It can be used to transfer electromagnetic energy at high frequencies from the surface to the reservoir.

## TEM transmission in coaxial lines

The details of the solution for a z-directed coaxial cable (inner radius = ra, outer radius = rb) are found in Ref. 8 . For an inner electrode material with μM = μ0 and σ = 0, the fields and the propagation constant are ....................(37) ....................(38) ....................(39)

If we include losses due to the conductivity of the metallic conductors (see details in the section on waveguides below), and if the losses are small, the coaxial propagation will be given by ....................(40) ....................(41)

and ....................(42)

where the attenuations factors for the coaxial structure due to the finite metal walls conductivity (αM ) and the imaginary part of the permittivity (αD), are ....................(43)

and ....................(44)

The coaxial cable metal losses will be a minimum for (rb /ra) ≅ 4.

## Waveguides

The waveguide metal structures, shown in Fig. 3, were developed for electromagnetic energy transmission in the microwave frequency range (3 to 300 GHz), in view of their small losses at these frequencies. The circular waveguides are potentially important for the excitation of reservoirs and wells.

The waves that can be transmitted along these systems have field components along the direction of propagation, here assumed as the z-axis. Transverse magnetic modes (TM) have zero Hz field components, and transverse electric field modes (TE) have zero Ez components. The propagation constant, γ, depends both on material properties (σ, ε, and μM of the enclosed material and σmw of the surrounding metal walls) and the dimensions of the waveguide. According to the value of the applied frequency, a given waveguide will either support waves that travel along the guide or attenuated waves (evanescent modes). We will examine the case of cylindrical waveguides in view of their importance in energy propagation in threaded oil pipes or in coil tubing.

### Cylindrical waveguide: TM modes (Hz = 0, σ = 0, ε" = 0, μM =μ0, radius = a, σmw = ∞)

The propagation constant for these modes is ....................(45)

The constants pn,mTM arise from the imposed boundary conditions at the metal surfaces. The smaller values are p0,1TM= 2.405, p1,1TM= 3.382, and p0,2TM = 5.135. For each root, we have a given mode of solution. According to the value of the frequency, the propagation constant will be real (propagation) or imaginary (evanescence). The cutoff frequency, ωC, corresponds to the first value of ω, where propagation occurs (β real). ....................(46)

### Cylindrical waveguides: TE modes (Ez = 0, σ = 0, ε" = 0, μM = μ0, radius = a, σmw = ∞).

The propagation constant for these modes is ....................(47)

Again, the constants pn,mTE arise from the imposed boundary conditions at the metal surfaces. The smaller values are p1,1TE = 1.841, p1,2TE = 3.054, and p0,1TE = 3.832. For each root, we have a given mode of solution. According to the value of the frequency, the propagation constant is real (propagation) or imaginary (evanescence). The cutoff frequency for TE modes is ....................(48)

Thus, for cylindrical waveguides, the first mode to propagate is the TE11 mode because it has the lowest cutoff frequency. If the waveguide is empty or filled with materials with electrical properties similar to a vacuum (μ0 and ε0), this mode will have a cutoff wavelength λC0,TE11 = 3.41a. The situation for this and other modes is illustrated in Fig. 4.

## Attenuation caused by metallic walls

The above relations were obtained assuming infinite conductivity for the cavity walls σmw = ∞. In fact, real metallic walls have a finite conductivity, σmw, so that the waves will attenuate as they travel along the waveguides. The attenuations for TE and TM modes are ....................(49)

and ....................(50)

where ....................(51)

and ....................(52) ....................(53) ....................(54)

and ....................(55)

These values of attenuation determine the wall losses for microwave energy transmitted along a pipe (this energy is transformed into heat) and the amount of power available at a distance, L, from the source. Fig. 5 shows the frequency dependence of the attenuation for modes TE11, TE01, and TM01 in an empty (μM = μ0,ε = ε0,σ = 0) circular waveguide of radius a. The figure also shows the attenuation for an equally empty coaxial cable with an external conductor with radius, a, and an internal conductor of radius, a/4.

The TE01 mode has an attenuation that decreases monotonically as the frequency increases, thus, indicating that this mode is convenient for transmission at high frequency. Transmitted power depends on (E,H) products so that the attenuation for power transmission doubles the field attenuation values.

Attenuation can be measured in nepers/m or in decibels, and the relationship between these units is ....................(56)

Fig. 6 shows the fraction of the power applied to a waveguide or a coaxial cable that will arrive at the end of a 1 km-long line. The waveguides are much better transmitters at higher frequencies. The power “lost” is converted into heat at the line metallic walls. As shown by the figure, waveguides are much better transmitters of power than coaxial cables at higher frequencies.

In both type of structures, the attenuation is caused by the induction of electrical currents at the surface of the metallic conductors. As these conductors have finite electrical conductivities, the flow of the induced currents results in power losses as in the case of any current-carrying wire. The power dissipated at the walls is, thus, conveniently converted into heat. By selecting a given frequency value, the ratio of power transmitted to power dissipated in the walls can be chosen at will.

## Inductive concentrated heaters

Inductive heating has been widely applied outside of the oil industry in steel foundries for light alloy melting (aluminum and magnesium), and in copper and zinc foundries.  It occurs when a metal core is excited by a coil of wire carrying an alternating current, as shown in Fig. 7. The metal is characterized by a high conductivity (σ = 107 siemens/m) and real permittivities and permeabilities (μ ≅ μ0, ε ≅ ε0).

For an applied magnetic field, H0, in the z direction along the core axis, the electric and magnetic fields inside the solid core with radius a, are ....................(57) ....................(58)

J0 and J1 are Bessel functions of complex arguments. Since the applied frequency, ω, is chosen so that σ > > ωε0, we have ....................(59)

The real time-averaged power entering the core at r = a (in the –r direction) is ....................(60)

and ....................(61)

In terms of an effective relative permeability, μR, refer to Fig. 8.  The real power per unit area, input to the core, is ....................(62)

The real power per unit area grows as (ω)1/2 at high frequency, and as the frequency grows, the power is more concentrated near the core surface, as shown in Fig. 9. Thus at high frequencies, the heating occurs closer to the surface.

## Energy gain in electrical heating processes used in enhanced oil production

It is important to consider the energy gain (EG) of the particular electrical heating system used in enhanced oil recovery. If the production is increased by (ΔQ) barrels per day, under the effect of PE kW of electrical power (generated from a process with energy efficiency, η), we define the energy (or power) gain, EG, as ....................(63)

where η is of the order of 30% for thermal electrical energy generation, 85% for hydroelectric generation, and 15% for microwave generation. The efficiency factor, η, represents the ratio of the 60 Hz electrical energy produced in a given process (thermal/hydroelectric) to the energy input in each process. Electrical energy in the microwave range is produced from 60 Hz electrical energy with an approximate efficiency of 50%.

The factor 68.366 comes from the equivalence of 1 standard bbl of oil with 5.6 million Btu as heating content and the equivalence of 1 kW-h = 3,413 Btu. Thus, 1 bbl of oil per day is equivalent to 68.366 kW. For the different types of electrical power generation processes, the different energy gains will be approximately ....................(64)

The energy (or power) gain concept is useful in the relative evaluation of different electrical heating systems and the comparison of electrical heating with steam injection characterized with EG < 10. [Personal communication with H. Mendoza, PDVSA Production and Exploration Division, Lagunillas, Venezuela (1998).] For more details, refer to the chapter on steam injection in the Reservoir Engineering and Petrophysics section of this Handbook.(this needs to be re-worded and linked)

Because electrical heating is considered as a viscosity reduction process for the production of heavy and extra-heavy oil, it is very pertinent to compare the energy efficiency of this process with steam injection—perhaps the more accepted process used in the industry for the production of hydrocarbon during the last four decades.

## Sources of dielectric losses

In the electrical heating of oil wells or reservoirs, we can disregard magnetic losses (including those in the metallic structures), because μ ≅ μ0. The power dissipated in a given volume, V, is then given by ....................(65)

Losses, because of the presence of ε"(ω), are defined as dielectric losses. They are caused by several processes: electronic, ionic or atomic, orientation, and space charge or interfacial. All the processes occur because of the electrical field influence on the charge distributions existing in matter. The nature of these mechanisms is sketched in Fig. 10.

Mechanisms (a) and (b) imply field-induced dipoles, while (c) occurs in polar systems with permanent dipolar structures. Process (d) occurs because of complex charge distributions at interfaces (as between a solid and a liquid). Each of these mechanisms implies energy losses caused by “frictional” forces as the dipoles follow the changes in the electric field. The losses occur in different frequency ranges as shown schematically in Fig. 11.  Losses in the microwave frequency range occur either because of the presence of polar molecules (like water) or possible complex space charge effects.

Because petroleum (without water) is a collection of mostly nonpolar molecules, it is practically transparent to electromagnetic energy in all the frequency ranges (ε′ /ε0 = 2 to 4, ε″ ≅ 0). The situation is different in a reservoir where water is associated with a solid matrix, either absorbed in material capillaries, adsorbed on grain surfaces, or is chemically associated to other molecules. The resonance of bound water occurs at lower frequencies than the free-water resonance that occurs near 20 GHz at room temperature.

The basic curve for each resonance is given by the derivation because of Debye, in terms of relative permeabilities at zero frequency and at very large frequencies, and a time constant, τ, ....................(66)

The previous expressions yield a semicircular Cole-Cole plot, as shown in Fig. 12, similar to the plot for a parallel combination of a capacitance and a resistance. The dotted ellipse shows the behavior generally found experimentally. This behavior is interpreted as being caused by the contribution of several time constants.

## Nomenclature

 a = radius of circular waveguide A = surface A+ , A– = integration constants A1, A2 = constants for liquid hydrocarbon B → = magnetic density vector c = compressibility Cp = specific heat at constant pressure Cr = electrical capacitance along the r axis Cz = electrical capacitance along the z axis CT = thermal capacitance CV = fluid capacitance D → = electric density vector E → = electric field vector EG = power (energy) gain f = frequency, Hz H → = magnetic field vector H0(2),H0(2) = Hankel functions i → = unit vector I = electrical current, amperes Icr = thermal convection current in the r direction Icz = thermal convection current in the z direction IVexc = excitation current for the fluid model ITexc = excitation current for the thermal model j = imaginary numbers unit J0, J1 = Bessel functions of complex arguments J → = vector current density k = permeability, darcy KT = thermal conductivity L = distance P = space and time-dependent pressure, Pa PPUV = electrical power per unit volume, watts/m3 PPUA = electrical power per unit area, watts/m2 p n,mTM = constants due to waveguide TM boundary conditions p n,mTE = constants due to waveguide TE boundary conditions PE = applied electrical power, kW Q(t) = time-dependent oil production Q → T = heat energy flow per unit area and per unit time Q → m = mass fluid flow per unit area and per unit time r, φ, z = cylindrical coordinates ra, rb = inner and outer coaxial radii Rwell = well pipe radius S → = complex Poynting vector t = time T = absolute temperature, Kelvin TE = waveguide transverse electric mode TM = waveguide transverse magnetic mode V → = fluid velocity V = volume Vr = component of fluid velocity along the r axis Vz = component of fluid velocity along the z axis V0 = wave velocity in free space, 3.0 × 10+ 8 m/s x, y, z = Cartesian coordinates Yr = electrical conductance along the r axis :Yz = electrical conductance along the z axis YTr = thermal conductance along the r axis YTz = thermal conductance along the z axis YVr = fluid conductance along the r axis YVz = fluid conductance along the z axis Z = wave impedance Z0 = free space wave impedance, 377 ohm :α = attenuation constant αD = coaxial cable attenuation due to dielectric losses αm = coaxial cable attenuation due to wall metallic losses α n,mTE = waveguide attenuation due to wall metallic losses for TE modes α n,mTM = waveguide attenuation due to wall metallic losses for TM modes β = real propagation constant β n,mTM = waveguide TM propagation constant β n,mTE = waveguide TE propagation constant γ = propagation constant γ0 = propagation constant for free space γcoax = propagation constant for coaxial cable δS = skin depth ΔQ = increase in oil production, SBLD ε = permittivity ε0 = free space permittivity, 8,854 × 10–10 farad/m ε ′ = real part of the permittivity ε″ = imaginary part of the permittivity η = process efficiency κT = thermal diffusivity λCO,TE11 = cutoff wavelength for waveguide TE11 mode μ = viscosity μ ′ = real part of the magnetic permeability μ″ = imaginary part of the magnetic permeability μM = magnetic permeability μ0 = free space magnetic permeability, 4π × 10–7 henry/m μ(T) = temperature-dependent viscosity ρ = density of hydrocarbon ρc = electrical charge per unit volume σ = conductivity σ′ = real part of the conductivity σ″ = imaginary part of the conductivity σM = metal conductivity σmw = metal wall conductivity ν = kinematic viscosity Φ = porosity Φ = potential, volts ω = angular frequency, radians/s ω CTM = cutoff frequency for TM modes ω CTE = cutoff frequency for TE modes