Computation Of Electrical Spectra Of A Periodic Composite Medium Using A 3D-FDM Method

The study of electrical spectra of composite materials is of great significance for mineral exploration,material design,medical examination and environmental monitoring.The dielectric constant and conductivity of both natural and artificial materials can show dispersion phenomena in frequency domain.In recent years,more and more experiments prove that some common composite materials can display very large dielectric constant(effective dielectric constantε_eff is greater than 1000) at low frequencies(f＜100 KHz).This thesis computes the inner electrical field of composite materials and extracts the effective dielectric constant and conductivity using a three dimensional finite difference method(3D-FDM).A special grid is used to handle membrane structures in the 3D-FDM method.A comparison with classical analytic method indicates that our numerical method is correct.In this thesis,the study of effective electrical parameters of composite materials and their dispersion characters are as follows:(1) Analyze briefly the effective dielectric constant of lossless media and study the parameters that can affect the effective dielectric constant of the lossless media.The simulations of lossless media show that the results from Maxwell-Garnett formula and the 3D-FDM are pretty close when the volume fraction of the inclusion is small.The Maxwell-Garnett formula is not suitable for mixture with high inclusion volume fraction,so the results from Maxwell-Garnett formula and the 3D-FDM are not very close when the insertion ratio of the inclusion is high.When the dielectric constant of the inclusion is greater than that of the host,the growth speed of effective dielectric constant with the increase of inclusion dielectric constant becomes higher when the insertion ratio of the inclusion increases.The growth speed of effective dielectric constant with the increase of inclusion insertion ratio is faster when the insertion ratio ranges from 0.4 to 0.7.(2) Analyze the electrical spectra of general mixtures,and study how the geometrical and electrical parameters of the mixtures affect the electrical spectra of the mixtures and the physical mechanics of weak dielectric and conductivity enhancement.The simulations of general mixtures show that effective dielectric constant can display weak enhancement at low frequencies in some situations and effective conductivity can display weak enhancement at high frequencies in some situations.The enhancement of dielectric constant is related to conductivity differences of the components of the mixtures.The enhancement of conductivity is related to the differences of dielectric constant of the components of the mixtures. The enhancement of dielectric constant is due to the local concentration of equipotential lines when displacement current is dominant at low frequencies.The enhancement of conductivity is due to the local concentration of equipotential lines when conductive current is dominant at high frequencies.There is no report on conductivity enhancement until now.When the dielectric constant of the components of the mixture is the same,the conductivity ratio of the components of the mixture controls the dielectric enhancement at low frequencies; the effective conductivity is between the conductivities of the inclusion and the host.When the conductivity of the components of the mixture is the same,the dielectric constant ratio of the components of the mixture controls the conductivity enhancement at high frequencies.When the dielectric constant of the components of the mixture increases and their ratio is a constant,the enhancement level of effective conductivity is the same and the relaxation frequency shifts.When the conductivity of the components of the mixture increases and their ratio is a constant,the enhancement level of effective dielectric constant is the same and the relaxation frequency shifts.The computation of a two-layer model shows that:when the dielectric constant of the two layers is the same,the equipotential lines will converge in the low-conductivity layer at low frequencies and the effective dielectric constant is two times of that of the media;when the conductivity of the two layers is the same,the equipotential lines will converge in the low-permittivity layer at high frequencies and the effective conductivity is 2 times of that of the media.For a three-layer model,when the dielectric constant of the three layers is the same and the conductivity of the middle layer is low,the dielectric enhancement will increase at low frequencies and the shift of conductivity spectrum towards left and up.For the model which consists of spherical inclusions,the dielectric enhancement level increases as the insertion ratio(R) increases when R is less than 0.5;the dielectric enhancement level will decreases as the insertion ratio(R) increases when R is greater than 0.5.0.5 is a critical condition,because the inclusions will connect to each other when R is greater than 0.5.This will make the conductivity of the whole system increase if the inclusions are conductive.The dip angel of the ellipsoidal inclusion and the aspect ratio of the ellipsoidal and cylindrical inclusion can affect the electrical spectra of the mixtures.These cases show that the larger the span of the inclusion in electrical field direction,the greater the enhancement is.The more components the mixture contain,the more complicated the electrical spectra are. The mixture which consists of three components usually shows two relaxations.Under the conditions of a quasistatic assumption,the electrical spectra will not change when we scale up or down the model.The models consist of eight units and one unit show the same electrical spectra,so the reduction of the model of a periodic medium is reasonable.(3) The study of the composite medium contains membrane structures shows that the low-conductivity membrane can cause large dielectric enhancement at low frequencies. Membrane conductivity,membrane thickness and membrane area can affect the dielectric enhancement level significantly.Similarly,the low-permittivity membrane can cause large conductivity enhancement at high frequencies.When the membrane thickness is greater than 10^-8 m,the dielectric enhancement level increases with the decrease of membrane thickness.When the membrane thickness is less than 10^-8 m,the dielectric enhancement level decreases with the decrease of membrane thickness. Because the membrane seems cannot block the conductive current if the membrane is too thin.The dielectric enhancement level increases with the decrease of membrane conductivity. However,the enhancement effect will become weaker and weaker as the membrane conductivity decrease.When the membrane conductivity is less than 10^-7 S/m,the dielectric enhancement level approaches to a constant.For example,the spectra from the model with membrane conductivityσ_m =10^-8 S/m andσ_m =0 overlap.Because the lower the membrane conductivity is, the denser the equipotential lines in the membrane are.When the membrane conductivity is low enough,all the equipotential lines are convergent to the membrane and the dielectric enhancement level approaches to a constant.The dielectric enhancement level increases as the membrane area increases.The conductive current can not flow through the model at low frequencies when the membrane separates the model to two parts completely,so the effective conductivity becomes very low.The high frequency conductivity is uninfluenced,because displacement currents are dominant at high frequencies.The computation of a two-layer model with a membrane shows that:the dielectric enhancement level has a quantitive relation with the membrane parameters at low frequencies; the conductivity enhancement level has a quantitive relation with the membrane parameters at high frequencies.The dielectric enhancement is caused by the decrease of the effective thickness of the capacitor model at low frequencies.The conductivity enhancement is caused by the decrease of the effective length of the resistor model at high frequencies.Thus,the frequency dispersion of electrical parameters is caused by the frequency dependence of the effective sample size.(4) Compare the variation rule of electrical spectra from the classical model and our numerical computation and analyze the effect of model parameters of the two methods on the electrical spectra.The classical model(Cole-Cole model) uses four parameters to control the shape and position of the electrical spectra.The physical meaning of the four parameters is ambiguous.The numerical method we use can control the shape and position of the electrical spectra by adjusting the geometric and electrical parameters of the components of the model.The numerical method is much slower than the classical analytic method,but the physical meaning of the parameters used in the numerical method is specific.So it is more convenient to use the numerical method to study the relationship between the physical parameters of the model and its dispersion characters.(5) Study the electrical spectra of several kinds of rocks by the 3D-FDM method.The results prove that the organic-rich shale,metal-bearing sand and oil-wet rock can display large dielectric enhancement.The effective dielectric constant and conductivity will not change with frequency for a pure sand model when the pore space of the sand is connective.The electrical spectra are frequency dependent and a weak dielectric enhancement will present at low frequencies when the sand is fully saturated by oil.The oil-bearing fracture can cause large dielectric enhancement.The metal-bearing rock can also show large dielectric enhancement when the metal particles are coated by low conductive membrane(oxide shell).Because the oil membrane is low conductive, the oil-wet rock can display large dielectric enhancement,and the effective conductivity of the oil-wet rock is not frequency dependent.