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In the modeling of any system, one is always faced with the dilemma of choosing the level of complexity that correctly predicts the response of interest. In the case of modeling the electrical heating of wells and reservoirs for heavy or extra-heavy oil at low frequencies (below the microwave range) and considering only one liquid phase and no gas phases, the systems of equations shown in this article are considered sufficient. The problem is still unsolved for the case of microwave heating of reservoirs, in which a complete model, which correctly takes into account the electric losses of a system of solid grains, liquids with dissolved gases and salts (with the corresponding complex geometrical, scaling, and electrochemical properties in the presence of electrical diffusion currents and space charges), is not yet available. For the case of concentrated heating (either resistive or inductive) and distributed heating in the reservoir and surrounding regions (at frequencies below the microwave range) or distributed heating in the metal elements (at any frequency) the equations given next (in a cylindrical coordinate system) are deemed sufficient.

Thermal processes

Heat energy flow per unit area and per unit time (Q → T) in the presence of forced convection because of a velocity, V → , is given by

Vol6 page 0592 eq 001.png....................(1)

where KT is the thermal conductivity, ρ is the density of the liquid, and CP is the specific heat at constant pressure.[1] In the presence of dissipation of power, an energy balance is described by

Vol6 page 0592 eq 002.png....................(2)

in terms of κT the thermal diffusivity, and PPUV the dissipated power per unit volume (electrical in our case). Thus, in a cylindrical coordinate system with axial symmetry with respect to the z axis, the differential equation for a region of spatially constant parameters is

Vol6 page 0593 eq 001.png....................(3)

The third term on the left, the product of temperature multiplied by the divergence of the velocity, has been neglected in many models of heating of reservoirs (it is strictly zero only for incompressible fluids.).

Fluid flow in the porous media

The fluid flow equation in the porous media of the reservoir, deemed to be representative of solution-gas-drive production mechanisms, is

Vol6 page 0593 eq 002.png....................(4)

where P is the pressure, μ(T) is the temperature-dependent viscosity, k is the permeability, c is the compressibility, and Φ is the porosity.[2][3][4] The fluid velocity, V (we assume that only oil is present) has the following components:

Vol6 page 0593 eq 003.png....................(5)

Vol6 page 0593 eq 004.png....................(6)

The mass fluid flow per unit area, Q → m, and the temperature-dependent kinematic viscosity, ν, are given by

Vol6 page 0593 eq 005.png....................(7)


Vol6 page 0593 eq 006.png....................(8)

Electrical processes

In the case of concentrated resistive heating, where a sinusoidal current of root mean square (RMS) magnitude I (Imax = √2) flows through a wire resistance of resistance, R, the total power dissipated is I2R. The power per unit volume is uniform over the volume of the resistor if the skin depth is much larger than the wire radius. The skin depth δS indicates how far the electromagnetic fields penetrate in a material with conductivity σ, and it is given by

Vol6 page 0593 eq 007.png....................(9)

The behavior of the skin depth, as a function of frequency, is shown in Fig. 1 for typical resistance materials (nickel-chrome alloys with conductivity 107 siemens/m). As the frequency increases, we reach the inductive heating regime previously discussed.

In the case of low-frequency heating distributed over the reservoir and the under and overburden regions, the conductivities of the materials involved, are much smaller than metallic conductivities (0.1 to 0.02 siemens/m), the corresponding skin depths are large over the volume elements considered in a given model. Thus, the determination of dissipated energy per unit volume can be obtained from the solution of Maxwell’s equations in the low-frequency limit where

Vol6 page 0594 eq 001.png....................(10)

so that we can define a scalar potential Φ

Vol6 page 0594 eq 002.png....................(11)

with the current density

Vol6 page 0594 eq 003.png....................(12)

Thus, each volume element, shown in Fig. 2, can be represented by resistances and capacitances connecting to other volume elements. The resulting distributed circuit is shown in Fig. 3, and the component conductance and capacitance values are given next.[5]

Vol6 page 0594 eq 004.png....................(13)

Vol6 page 0594 eq 005.png....................(14)

For the frequencies involved in distributed resistive heating at 60 Hz or lower frequencies, the capacitances are neglected. For a given current (or currents) applied at given nodes, the circuit can be solved for all the node voltages by applying Kirchhoff’s current law (KCL), requiring that the sum of all the currents that flow into a node should vanish (charge conservation).

Although not necessary to solve the problem, an equivalent circuit approach can be used for visualizing (and calculating) the solutions of the temperature and fluid flow equations.[5][6][7][8][9] It is particularly helpful in treating regions with space varying parameters.

For the volume elements of Fig. 2, the equivalent circuits for fluid flow in porous media and for heat transfer with convection are shown in Figs. 4 and 5.

In the absence of gravity effects, the element values for the fluid flow are

Vol6 page 0595 eq 001.png....................(15)

Vol6 page 0596 eq 001.png....................(16)


Vol6 page 0596 eq 002.png....................(17)

where the excitation current (IVexc) introduces the initial pressure conditions for the problem.

The element values for the thermal case are equivalent admittances, capacitances, and convection currents

Vol6 page 0597 eq 001.png....................(18)

Vol6 page 0597 eq 002.png....................(19)

Vol6 page 0597 eq 003.png....................(20)


Vol6 page 0597 eq 004.png....................(21)

where the excitation current (ITexc) introduces the initial temperature conditions for the problem and the applied power per unit volume.

The circuital approach is particularly convenient in the case of problems covering regions with different properties. For example, the fluid flow equation for porous media originates from a mass balance of

Vol6 page 0597 eq 005.png....................(22)

which, assuming spatially independent compressibility and porosity, yields

Vol6 page 0597 eq 006.png....................(23)

which gives the equation indicated at the beginning of this section, only if k and μ(T) are space independent. In the case of reservoir heating, if the temperature is space-dependent, the equation for flow in fluid media becomes

Vol6 page 0597 eq 007.png....................(24)

Fluid flow in the well

For steady laminar flow of a single-phase fluid, the velocity of the fluid in the well pipe directed along the z axis, of radius Rwell is given by the Hagen-Poisseuille relation, which is written as

Vol6 page 0597 eq 008.png....................(25)

Model response for vertical wells: concentrated heating vs. distributed heating

We conclude this section with calculations comparing transient concentrated heating vs. distributed low-frequency (60 Hz) heating (according to scheme (b), shown in Fig. 6) for reservoir and well conditions, such as those given in the test case for the Tia Juana field in western Venezuela.[10]

Fig. 7 shows the production response for both cases of heating at different power levels.

It is interesting to see that at low power levels concentrated and distributed heating yield similar responses. As the applied power increases, the distributed heating case shows a better response, although it takes a few more days to reach steady state. In any case, the response is quick, as indicated in most of the field test cases reported.

Fig. 8 shows the production response for both cases of heating at a 30 kW power level.

The concentrated heating cases shown correspond to power applied under different control temperature conditions at the heater. If the temperature exceeds a set point (85°C), the heater is turned off and then restarted if the temperature decreases. The figure determines the importance of reporting real field conditions to properly evaluate the different energy gains.

Fig. 9 shows the behavior of the temperature as a function of time. The temperature decrease, observed for the concentrated heating, is because of the increased convection cooling as the production increases. The energy gains determined from the numerical results in those cases with no temperature limitations are summarized in Table 1.

Additionally, one might note that many control systems for the electrical heating supplies use nonlinear devices (like silicon control rectifiers) that change the nature of the applied power sinusoidal waves so that the harmonic content of the applied voltage (or current) waveforms requires specialized watt meters for the correct evaluation of the applied power levels. The harmonic content implies that excitations at frequencies differing from the fundamental are present.


a = radius of circular waveguide
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


  1. London, E.U. 1967. Heat Transfer. Handbook of Physics, Part 5: Heat and Thermodynamics, Ch. 5, 5-66. ed. E.U. Condon and H. Odishaw, second edition. New York City: McGraw-Hill Book Co. Inc.
  2. Lake, L.W. 1989. Enhanced Oil Recovery, 17-42. Englewood Cliffs, New Jersey: Prentice Hall Inc.
  3. Raghavan, R. 1993. Well Test Analysis, 25-35. Englewood Cliffs, New Jersey: Prentice Hall Inc., Englewood Cliffs.
  4. Bear, J. 1988. Dynamics of Fluids in Porous Media, 65-113. New York City: Dover Publications, Inc.
  5. 5.0 5.1 Callarotti, R.C. and Di Lorenzo, M. 1992. Resistive-Capacitive Tomography. Presented at the SPE Latin America Petroleum Engineering Conference, Caracas, Venezuela, 8-11 March 1992. SPE-23679-MS.
  6. Callarotti, R.C. 1991. Circuit Modeling for the Numerical Calculation of RF Heating of Crude Oil in Pipes and Reservoirs. Proc., Fifth UNITAR/UNDP Conference on Heavy Crudes and Tar Sands, Caracas, 3, 547–559.
  7. Callarotti, R.C. 1995. Circuital Modeling Applied to Transient Flow Problems in Porous Media. Proc., Third Caribbean Congress on Fluid Dynamics and Latin American Symposium on Fluid Mechanics, Caracas, 2, C1–C9.
  8. Callarotti, R.C. 1995. Proper Eigenvalue Solution for the Transient Response of Multidimensional Heat Transfer Systems, Chapter 12—Electromagnetic Heating of Oil VI-607. Communications in Numerical Methods in Eng. 11 (l): 715. .
  9. Callarotti, R.C. 2002. Circuital Modeling from Electronic Devices to Oil Production. Proc., Fourth Intl. Conference on Devices, Circuits and Systems, Aruba, Instrumentation Paper IO28, 1–6.
  10. Callarotti, R.C. 2002. On the Complete Modeling of Transient Electromagnetic Heating of Heavy Crudes in Vertical and Horizontal Wells. Desarrollos Recientes en Métodos Numéricos, ed. Müller-Karger C., Lentini M. and Cerrolaza M., Proc. VI Congreso Internacional de Métodos Numéricos en Ingeniería y Ciencias Aplicadas CIMENICS, Gráficas León SRL, Caracas, CI 1–8.

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See also

Electromagnetic heating of oil

Electromagnetic heating process

Electrical engineering considerations in electromagnetic heating

Field tests of electromagnetic heating

PEH:Electromagnetic Heating of Oil