摘要
混合介质电性谱的研究对于矿产勘探、材料设计、医学检查和环境监测都有很重要的意义。介电常数和电导率的频散现象广泛地存在于各种天然介质和人工介质中。近来,越来越多的实验发现一些普通混合介质的介电常数在低频域会出现异常大的增强现象,对此人们争议还比较大,还没有统一的解释。
本文采用三维有限差分方法求解混合介质内部的电场分布,进而提取混合介质的等效介电常数和电导率,在差分网格的划分中采用了一种近似方法来解决含薄膜层的介质的数值模拟问题,通过经典解析公式的验证,表明本文对膜结构的近似处理是合理的,没有引起大的误差。本文主要从以下几个方面对混合介质的等效电性参数及其频散特性进行了深入研究:
(1)对无损耗介质的等效介电常数进行了简单的分析,阐述了其等效介电常数的主要控制因素和变化规律。
对无损耗介质的模拟计算表明:在包裹体体积较小时,而且包裹体和宿主的介电常数比值不大的情况下,Maxwell-Garnett公式和FDM的计算结果非常接近。当包裹体体积较大时,Maxwell-Garnett公式就不太适用了,所以其与FDM的计算结果差异也较大。在包裹体介电常数大于宿主介电常数的情况下,嵌入比越大,等效介电常数随包裹体介电常数增长而增长的速度越快;球形包裹体嵌入比在0.4到0.7之间时,等效介电常数随嵌入比增长速度最快。
(2)对一般混合介质的电性谱进行了分析,研究了包裹体的几何参数和电性参数对混合介质电性谱的影响规律,并分析了弱介电增强和弱电导率增强现象及其产生机理。
对一般混合介质的模拟表明:在某些情况下,等效介电常数会在低频端出现微弱的增强;某些情况下,等效电导率会在高频端出现微弱的增强。等效介电常数的增强和介质各组分电导率的差异有关;等效电导率的增强和介质各组分介电常数的差异有关。等效介电常数的增强是由于等势线在传导电流为主导时(低频端)在局部汇集引起的;等效电导率的增强是由于等势线在位移电流为主导时(高频端)在局部汇集引起的。其中,等效电导率的增强现象尚未见有报道。
对于各组分介电常数相同的混合介质,包裹体和宿主的电导率比值控制着等效介电常数在低频端的增强,等效电导率介于包裹体和宿主电导率之间。对于各组分电导率相同的介质,包裹体和宿主的介电常数比值控制着等效电导率在高频端的增强,等效介电常数介于包裹体和宿主介电常数之间。
当各组分介电常数同时增大但比值不变时,等效电导率在高频端的增强幅度不变,但是电性谱的驰豫频率会发生平移。当各组分电导率同时增大但比值不变时,等效介电常数在低频端的增强幅度不变,但是电性谱的驰豫频率会发生平移。
等厚的两层介质模型的计算表明:对于介电常数相同,电导率不同的两层介质,随着频率的降低,等势线会逐步集中到电导率较小的介质中,同时,在低频端等效介电常数会增大一倍;对于电导率相同,介电常数不同的两层介质,当频率升高时,等势线会逐步集中到介电常数较小的介质中,同时,在高频端等效电率会增大一倍。
对于介电常数相等,中间一层电导率较低的三层介质模型,发现中间一层厚度越小,低频端的介电增强作用越强,同时电导率曲线整体向左上方平移。
对于含球形包裹体的模型,当包裹体嵌入比R小于0.5时,低频端的等效介电常数随着R的增加而增加;当包裹体嵌入比R大于0.5时,低频端的等效介电常数随着R的增加而降低。所以说R=0.5是一个临界状态,因为当包裹体的嵌入比R大于0.5时,这些包裹体就相互连接起来,如果包裹体是导电的,那么混合体系的导电性就会极大地增强。所以当R大于0.5后,混合介质的等效电参数会有很大的变化。
椭球包裹体的倾角、椭球包裹体的纵横比和圆柱包裹体的纵横比都对电性谱有一定影响,其中相似的一个特点是:包裹体在所施加电场方向的延伸越长,低频端的介电增强作用越大。
混合介质中的组分越多,其电性谱越复杂,对于含三种不同组分的混合介质,其电性谱会出现两次驰豫。
在准静态电场的假设下,等比例缩放模型时,电性谱不发生改变。
含八个单元和一个单元的模型的电性谱完全一致,说明基于混合定律对周期结构混合介质的简化处理是正确的。
(3)对含薄膜层的介质进行了分析,发现低电导率薄膜会引起等效介电常数在低频端出现异常大的增强。薄膜电导率、薄膜厚度、薄膜面积对低频端的介电增强作用有很大影响。包裹体的几何形状、倾角等因素对介质的电性谱也有一定影响。类似地,低介电常数薄膜会引起等效电导率在高频端出现很大的增强。
当膜的厚度大于10~(-8)m时,介电增强作用随着膜厚度的减小而增强;当膜的厚度小于10~(-8)m时,介电增强作用随着膜厚度的减小而减弱。因为当薄膜厚度太薄时,薄膜就起不到阻断传导电流的作用了。
随着膜电导率的降低,低频端的介电增强作用不断增强,但是这种增强作用随着膜电导率的降低逐步变弱,最后当膜的电导率小到一定程度后,低频端的介电常数趋向一个饱和值(如:膜电导率σ_m=10~(-8)S/m和σ_m=0对应的曲线基本重合)。因为膜电导率越低,等势线越向膜层中汇集,当膜电导率低到一定程度后,等势线就全部汇集到膜层中了,这样就到达一种饱和状态了。
随着膜面积(水平层状薄膜)的增大,低频端的介电常数越来越大(介电增强作用越来越强);当膜层完全将模型上部和下部隔开时,(即:将导电通路完全切断),这时传导电流很难穿过整个模型,而低频时,以传导电流为主,所以低频端的等效电导率就特别低;高频时以位移电流为主,所以高频端的等效电导率未受影响。
含膜的两层介质的计算表明:低频端的介电增强幅度和薄膜参数之间存在着定量关系;同样地,高频端的电导率增强幅度和薄膜参数之间也存在着定量关系。等效介电常数在低频端的增强是电容模型的有效厚度在低频端减小引起的;等效电导率在高频端的增强是电阻模型的有效长度在高频端减小引起的。所以可以认为混合介质等效电参数的频散现象是因为样本的有效尺寸随频率发生改变而引起的。
(4)对比了经典频散模型和本文数值计算的电性谱的变化规律,分析了经典公式和数值方法中模型参数对电性谱的控制作用。
介电常数和电导率的经典频散模型(Cole-Cole模型)分别用了4个参数来控制介电常数和电导率频散曲线的形态和位置,但是其中3个参数的物理意义不明确。本文的数值模方法通过改变混合介质中各个组分的电性参数和几何参数也可以实现对散曲线形态和位置的控制。数值方法计算频散曲线的速度远远慢于经典解析模型,而且引入的模型参数更多,但是数值方法中采用的参数都有明确的物理意义,更方便分析介质频散特性和介质本身的物理参数之间的关系。
(5)对几种岩石模型的电性谱进行了分析,证实在含有机质页岩、含金属颗粒的岩石、亲油岩石等岩石中会出现巨大的介电增强现象。
对于饱水纯砂岩模型,在砂岩孔隙相互连通的情况下,等效介电常数和电导率都不会随频率发生变化。对于完全饱油的纯砂岩模型,电性谱会出现频散现象,而且低频端会出现较弱的介电增强现象。饱含低电导率流体的裂缝模型的计算表明:这种介质会出现很大的介电增强现象。含有机质页岩的模型的计算表明:低电导率的有机质层片会引起很大的介电增强现象。对于含金属颗粒的岩石模型,如果金属颗粒表面存在低电导率膜层(氧化壳),该介质就会在低频端出现很大的介电增强。亲油岩石由于岩石颗粒表面覆盖着油性膜层,所以在低频端也会出现很大的介电增强,但其等效电导率基本不随频率发生变化。
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.
引文
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