Fractal Study On Electromagnetic Wave Through Inhomogeneous Medium

Electromagnetic method is a widespread kind of electrical exploration method of geology exploration. The method makes use of the electromagnetic wave to measure some of the basic physical properties of rock. In 1910, Leimbach and Lowy had presented a patent that use the method to detect underground in Germany. In 1926, Hulsenbeck first had used the pulse technology to detect position of abnormal body, and has pointed out that the technology is superior to seismic methods because that wave source of electromagnetic pulse was direction. After 60 years, the method most used to detect media that energy of electromagnetic wave slowly decrease in exploration, such as ice, salt, etc. With the rapid increasing signal to noise ratio for instruments and data processing technology, the wild range application of electromagnetic method was rapid expansion, such as pollution control, reservoir monitoring, prediction of earthquakes and hydrothermal energy, etc.However, the electromagnetic wave propagation more complex than in the air as that the subsurface media has a characteristics of electromagnetic wave attenuation and its components is diversity. For study of the electromagnetic wave propagation in inhomogeneous media are still some problems on:1) non-uniform medium mainly consist solid particles, liquid and gas that polarization coupling and induction of the electromagnetic wave are not fully reflect the interaction in inhomogeneous media, and the basic characteristics of the theoretical model are based on different assumptions of their mechanism and applicable conditions; 2) the deposition of inhomogeneous media has a layered anisotropic characteristics as to more complex of the electromagnetic wave equation and make the interpretation of the experimental results that have more hydrolysis and lack of quantitative analysis; 3) experimental results are different for the complexity results of the field exploration, so that need to prior calibration to apply the detection place. The fractal theory has been applied to issues of electromagnetic wave propagation in inhomogeneous media. However, the report for application of fractal theory to study the electromagnetic properties is little. Combined with fractal characteristics of inhomogeneous media to study the electromagnetic characteristics (such as conductivity, skin depth, etc.) will get better results and it is important for clearly realizing the electromagnetic waves.In 1967, American mathematician B. B. Mandelbrot first has presented the concept of fractal geometry in his paper "How long is the coastline of Britain? Statistical self-similar fractal dimension". In 1982, Mandelbrot using the Latin "fractus" created this word "fractal" which means irregular, score and broken objects. Mandelbrot proposed a unique judgment "body or dimension geometry is continuous", which means that the fractal dimension of the object can be fractional. Since the 1980s, basic theory of fractal geometry have been rapidly developing in other sciences besides mathematics, computer graphics and applied science. Now, fractal geometry theory has made great success in physics, chemistry, geology and geophysics, computer graphics, biological sciences, medicine, materials science and engineering, communications, energy and environmental science and engineering disciplines, etc.Natural media or systems generally have fractal properties, such as the mountains, islands, rivers, sandstone porosity of reservoirs and fracture systems on Earth and have inhomogeneous nature. It is increasingly to know the world is nonlinear in nature that is fractal geometry where provides a powerful statistical analysis tools to study these complex systems. Therefore the fractal characteristics is an essential nature.In this thesis, the propagation characteristics of electromagnetic wave consist of the conductivity and the skin depth in porous medium (inhomogeneous medium), and on the basis of previous research, analysis microstructure for the effcet of conductivity and the propagation of electromagnetic wave of inhomogeneous medium. I study basic physical characteristics of fractal inhomogeneous medium, and establish the fractal skin depth model with considering effect of inhomogeneous medium. The main work are as follows:The first chapter describes the basic structure and the physical parameters of inhomogeneous media and introduces the basic concepts of fractal theory, including description of fractal mathematical foundations in porous media and an overview of the basic characteristics of fractal inhomogeneous media and apparent two-phase fractal theory; the second chapter provides an overview of fractal conductivity model in inhomogeneous media, we improve the traditional model from the studying for tortuous current flow and present analytical expression between conductivity and porosity based on fractal characteristics in porous media, and a relationship is derived between the fractal dimension and tortuosity fractal dimension from numerical calculation of tortuosity fractal dimension(three-dimensional Sierpinski carpet and Sierpinski sponge) using the finite element method, and fractal dimension of random walk is calculated from the Monte Carlo method. The simulation results clearly show that the pore fractal dimension and tortuosity fractal dimension has a linear relationship which will further promote the study of the transport properties in inhomogeneous medium; the third chapter review analyzed the frequency relativity of the electrical properties, and the relationship between the fractal model of electrical properties and the skin depth are that found the skin depth 8 has a power-law dependence on frequency fby δ∞ f-(?) ((?) indicates how the fractal structures of rocks affect skin depth); the skin depth is a fractal function of fractal dimension of rocks, electromagnetic frequency, porosity and conductivity of solution at a low frequency range, which also indicates the skin depth increases with the decreasing of fractal dimension. The heterogeneous property of rocks reduces the depth resolution of field exploration toward the laboratory measurement. A few comments are also made on the future research directions and possible subjects on the study of skin depth of electromagnetic wave in rocks; the fifth chapter summarizes the main work and innovation of the present thesis, and look forward to the study of the electromagnetic wave propagation of inhomogeneous media in future.