Article outline

This paper presents the development and experimental validation of a method for reconstructing the volumetric distribution of electromagnetic field (EMF) strength in a cylindrical chamber from multi-channel thermometry data of a dielectric medium subjected to microwave irradiation.

The relevance of this work stems from the need for accurate characterization of the spatial EMF distribution in problems of dielectric heating of high-viscosity oils, natural bitumens, and source rocks. As conventional hydrocarbon reserves are progressively depleted and extraction shifts toward unconventional resources — in particular, the Bazhenov formation — electromagnetic heating methods are regarded as a promising alternative to steam-based techniques, which are limited by reservoir depth and low injectivity. The physical mechanism of heating involves dissipation of electromagnetic energy through orientational polarization of polar molecules in the medium — water, asphaltenes, and resins. When applied to source rocks, this leads not only to a reduction in fluid viscosity but also to kerogen pyrolysis with the formation of synthetic oil, as well as to thermal stresses that increase rock permeability. The accuracy of mathematical models of such processes critically depends on reliable knowledge of the EMF distribution inside the reservoir or process equipment.

The objective of this work is to develop a method for determining the volumetric distribution of the squared electric field strength from multi-channel thermometry data of a heated dielectric medium. The proposed method combines two relations: the heat conduction equation with distributed heat sources and the expression for the specific power absorbed by a dielectric medium from an electromagnetic wave, which is directly proportional to the squared electric field strength, the dielectric permittivity, the loss tangent, and the angular frequency of the field. Given spatially distributed temperature measurements in a medium with known properties under electromagnetic irradiation, the squared electric field strength can be calculated at each measurement point.

To implement the method, a dedicated experimental setup was developed. A 2.45 GHz magnetron operating at 800 W irradiated water placed in a 3D-printed, radio-transparent 16-compartment vessel with equal compartment volumes (4 mL each): the compartments provided spatially distributed temperature measurement points required for solving the inverse field reconstruction problem. The vessel was placed inside a 40 cm long metallic pipe with an inner diameter of 10 cm, acting as a waveguide; a metallic mesh at the upper end ensured the formation of a standing wave. The irradiation time was limited to 20 s, which justified a set of simplifying assumptions: heat conduction between adjacent compartments and changes in the dielectric properties of water upon heating were neglected — thereby reducing the inverse problem to a direct calculation of the squared EMF strength from the measured temperature rise in each compartment. Experiments were conducted in two source configurations: lateral (source at the side of the pipe) and radially symmetric (source below the pipe); measurements were taken at different distances from the source, yielding a set of cross-sectional “slices” of the volumetric field distribution. The water temperature in each compartment after irradiation was measured with a thermal imaging camera; quantitative processing of the thermograms employed digital image processing methods — conversion to grayscale, mapping against the camera’s color scale, and interpolation. The full volumetric distribution was reconstructed using the Gauss-Newton method through coordinate-wise optimization of the parameters of approximating functions, selected based on the physically observed periodicity of the field.

In the lateral configuration, the field distribution exhibits a pronounced periodic dependence on the angular coordinate. No clear periodicity along the vertical coordinate was detected, although the authors suggest it might emerge at greater distances from the source. In the radially symmetric configuration, the field is more ordered: angular symmetry is preserved, and a clear periodicity along the vertical coordinate is observed with a period close to the microwave wavelength (12.2 cm). The reconstructed three-dimensional distributions of the squared electric field strength show qualitative agreement with the expected physical behavior of the field in both configurations.

The proposed method enables three-dimensional mapping of the EMF distribution from standard thermal imaging data without placing sensors in the microwave irradiation zone, and is promising for parametric identification of thermophysical models of microwave dielectric heating in porous media.