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# Modeling fluid flow with electromagnetic heating

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 ....................(1)

where KT is the thermal conductivity, ρ is the density of the liquid, and CP is the specific heat at constant pressure. In the presence of dissipation of power, an energy balance is described by ....................(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 ....................(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 ....................(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. The fluid velocity, V (we assume that only oil is present) has the following components: ....................(5) ....................(6)

The mass fluid flow per unit area, Q → m, and the temperature-dependent kinematic viscosity, ν, are given by ....................(7)

and ....................(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 ....................(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 ....................(10)

so that we can define a scalar potential Φ ....................(11)

with the current density ....................(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. ....................(13) ....................(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. 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 ....................(15) ....................(16)

and ....................(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 ....................(18) ....................(19) ....................(20)

and ....................(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 ....................(22)

which, assuming spatially independent compressibility and porosity, yields ....................(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 ....................(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 ....................(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.

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.

## Nomenclature

 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