3, and the component conductance and capacitance values are given next. The resulting distributed circuit is shown in Fig. 2, can be represented by resistances and capacitances connecting to other volume elements. So that we can define a scalar potential Φ 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
COMSOL 5.1 GAS FLOW POROUS SKIN
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. The crossing straight lines indicate that at 60 Hz, the skin depth is of the order of 2 cm, much larger than the diameter of heating wires. frequency in Hz for nickel-chrome alloys. As the frequency increases, we reach the inductive heating regime previously discussed.įig.
1 for typical resistance materials (nickel-chrome alloys with conductivity 10 7 siemens/m). The behavior of the skin depth, as a function of frequency, is shown in Fig. The skin depth δS indicates how far the electromagnetic fields penetrate in a material with conductivity σ, and it is given by The power per unit volume is uniform over the volume of the resistor if the skin depth is much larger than the wire radius. In the case of concentrated resistive heating, where a sinusoidal current of root mean square (RMS) magnitude I (I max = √2) flows through a wire resistance of resistance, R, the total power dissipated is I 2R. The mass fluid flow per unit area, Q → m, and the temperature-dependent kinematic viscosity, ν, are given by The fluid velocity, V (we assume that only oil is present) has the following components: 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 flow equation in the porous media of the reservoir, deemed to be representative of solution-gas-drive production mechanisms, is 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.). 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 In terms of κT the thermal diffusivity, and P PUV the dissipated power per unit volume (electrical in our case). In the presence of dissipation of power, an energy balance is described by Where K T is the thermal conductivity, ρ is the density of the liquid, and C P is the specific heat at constant pressure.
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
5 Model response for vertical wells: concentrated heating vs.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.
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. 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. 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.
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