VI Model Calculations

[68.1.2.1] This section returns to eq. (3.2) and presents numerical solutions. [68.1.2.2] This is intended as a case study exploring the relationship between the statistics of local geometries and bulk dielectric behavior. [68.1.2.3] The main focus will be on dielectric enhancement. [68.1.2.4] To solve eq. (3.2) for $\varepsilon(\omega)$ , one must know the geometric input functions $\mu(\phi)$ , $\lambda(\phi)$ and local dielectric responses $\varepsilon _{C}(\omega;\phi)$ , $\varepsilon _{B}(\omega;\phi)$ . [68.1.2.5] Unfortunately, no experimental data are available at present,[27] and geometric modeling has to be used instead.

A Local dielectric response

[68.1.3.1] The hypothesis of local simplicity states that the local geometries are simple and that the effective local dielectric constants are insensitive to geometrical details other than local porosity. [68.1.3.2] The simplest isotropic local geometry is spherical. [68.2.0.1] For conducting local geometries, a water-coated spherical rock grain will serve as the local model. [68.2.0.2] For blocking geometries a rock-coated spherical water pore is employed. [68.2.0.3] In the notation of Sec. IV, this means

$\displaystyle\varepsilon _{C}(u;\phi)$	$\displaystyle=\varepsilon _{W}\left(1-\frac{1-\phi}{(1-\varepsilon _{R}/\varepsilon _{W})^{{-1}}-\tfrac{1}{3}\phi}\right),$		(6.1)
$\displaystyle\varepsilon _{B}(u;\phi)$	$\displaystyle=\varepsilon _{R}\left(1-\frac{\phi}{\left(1-\varepsilon _{W}/\varepsilon _{R}\right)^{{-1}}-\frac{1}{3}(1-\phi)}\right).$		(6.2)

[68.2.0.4] In the low-frequency limit, one obtains for the conducting geometry

$\sigma^{\prime}_{C}(0;\phi)=\sigma^{\prime}_{W}\frac{2\phi}{3-\phi}=\tfrac{2}{3}\sigma^{\prime}_{W}\phi(1+\tfrac{1}{3}\phi+\dots),$

(6.3)

thereby identifying $C_{1}$ in eq. (4.8) as $C_{1}=\frac{2}{3}\sigma^{\prime}_{W}$ . [68.2.0.5] The real dielectric constant is found as

$\varepsilon^{\prime}_{C}(0;\phi)=\varepsilon^{\prime}_{W}-\frac{1-\phi}{(1-\tfrac{1}{3}\phi)^{2}}\big[\varepsilon^{\prime}_{W}(1-\tfrac{1}{3}\phi)-\varepsilon^{\prime}_{R}\big].$

(6.4)

[68.2.0.6] For the blocking geometry, the dc limit gives $\sigma^{\prime}_{B}(0;\phi)=0$ , in agreement with eq. (4.6), and

$\varepsilon^{\prime}_{B}(0;\phi)=\varepsilon^{\prime}_{R}\frac{1+2\phi}{1-\phi},$

(6.5)

for $\phi<1$ , identifying $B_{1}=1/\varepsilon^{\prime}_{R}$ in eq. (4.9), and $\varepsilon^{\prime}_{B}(0;\phi)=\varepsilon^{\prime}_{W}$ , for $\phi=1$ . [68.2.0.7] Note the presence of the thin-plate divergence in the $\phi\to 1$ limit.

B Local porosity distribution

[68.2.1.1] It was mentioned repeatedly that no experimental data for $\mu(\phi)$ are available to the author at present. [68.2.1.2] A qualitative guideline for porous media resulting from spinodal decomposition might be the shape of the order-parameter distribution calculated in Ref. [28] which suggests in particular that $\mu(\phi)$ can be bimodal.

[68.2.2.1] For the subsequent calculations, a simple mixture of two $\beta$ distributions has been used. [68.2.2.2] The analytic expression reads

$\mu(\phi)=\omega\frac{\Gamma(\mu _{1}+\nu _{1})}{\Gamma(\mu _{1})\Gamma(\nu _{1})}(1-\phi)^{{\mu _{1}-1}}\phi^{{\nu _{1}-1}}+(1-\omega)\frac{\Gamma(\mu _{2}+\nu _{2})}{\Gamma(\mu _{2})\Gamma(\nu _{2})}(1-\phi)^{{\mu _{2}-1}}\phi^{{\nu _{2}-1}},$

(6.6)

where $0\leq\omega\leq 1$ , $\mu>0$ , $\nu>0$ and $\Gamma(x)$ denotes Euler’s $\Gamma$ function. [68.2.2.3] The bulk porosity is then given as

$\overline{\phi}=\omega\frac{\nu _{1}}{\mu _{1}+\nu _{1}}+(1-\omega)\frac{\nu _{2}}{\mu _{2}+\nu _{2}}.$

(6.7)

[68.2.2.4] For $\mu,\nu>1$ , the $\beta$ densities are bell shaped, and for $\mu,\nu>1$ they diverge at the limits.

[68.2.3.1] Eight different local porosity distributions are compared in the calculations. [68.2.3.2] All of them are chosen such that they give the same bulk porosity $\overline{\phi}=0.1$ . [68.2.3.3] The values of the parameters are listed in Table 1, and the densities [page 69, §0] themselves are displayed graphically in fig. 1. [69.1.0.1] Each distribution is identified by a number and a line style as indicated in the inset of Fig. 1. $\overline{\phi}_{i}$ ( $i=1,2$ ) are the partial porosities $\nu _{i}/(\nu _{i}+\mu _{i})$ in eq. (6.7). [69.1.0.2] The uniform distribution carries number $1$ and is identified by a thin dot-dashed line. [69.1.0.3] Number $2$ represents the strongly peaked case and is identified by a wide dashed line, and so on.

Figure 1: Eight porosity densities with parameters given in Table 1.

Table 1: Paramenter values for the eight different forms of $\mu(\phi)$ displayed in Fig. 1 and used in the model calculations. The parameters $\omega$ , $\mu _{1}$ , $\nu _{1}$ , $\mu _{2}$ and $\nu _{2}$ are those of Eq. (6.6). $\overline{\phi}_{1}$ , $\overline{\phi}_{2}$ , $\mathrm{var}_{1}$ and $\mathrm{var}_{2}$ are the average and variance for $\mu _{1}$ , $\mu _{2}$ , if $\mu$ in Eq. (6.6) is written as $\mu=\omega\mu _{1}+(1-\omega)\mu _{2}$ . The rows labeled $\overline{\phi}$ , variance, and skewness contain the mean, variance and skewness for the eight forms of $\mu(\phi)$ .


Curve No.	1	2	3	4	5	6	7	8

$\omega$	1	1	2/3	1	1	1	2/3	2/3
$\mu _{1}$	1.0	360.0	191.1	7.2	4.5	1.8	28.8	58.6
$\nu _{1}$	1.000	40.000	3.900	0.800	0.500	0.200	0.087	0.176
$\mu _{2}$			1423.0				13.9	2.24
$\nu _{2}$			500.00				6.00	0.96
$\overline{\phi}_{1}$	0.1	0.1	0.02	0.1	0.1	0.1	0.003	0.003
$\overline{\phi}_{2}$			0.26				0.294	0.294
$\mathrm{var}_{1}$	0.00333	0.00024	0.00010	0.00010	0.01500	0.03000	0.00010	0.00005
$\mathrm{var}_{2}$			0.00010				0.01000	0.05000
$\overline{\phi}$	0.1	0.1	0.1	0.1	0.1	0.1	0.1	0.1
Variance	0.00333	0.00024	0.01290	0.01000	0.01500	0.03000	0.02213	0.03541
Skewness	0	0.2657	0.6993	1.6004	1.8679	2.3124	1.1588	2.1176

[69.1.1.1] The choices presented in Fig. $1$ are arbitrary. [69.1.1.2] The reader should bear in mind, however, that $\mu(\phi)$ is easily measureable and cannot be adjusted to fit experimentally observed dielectric data when comparing the theory with experiment. [69.1.1.3] The choices for $\mu(\phi)$ presented here are intended as a case study illustrating different possibilities that might occur in real or artificial experimental systems. [69.2.0.1] The highly peaked $\mu(\phi)$ (curve $2$ ) represents the limit of weak disorder. [69.2.0.2] Remember that $\mu(\phi)=\delta(\phi-\overline{\phi})$ for ordered systems. [69.2.0.3] Curve $2$ gives a reference to which other distributions can be compared. [69.2.0.4] The distributions 4, 5 and 6 have been chosen divergent at $\phi=0$ with exponents 0.8, 0.5 and 0.2 as examples for distributions whose inverse first moment does not exist. [69.2.0.5] Curves 3, 7 and 8 demonstrate the fact that $\mu(\phi)$ itself might be “of percolation type.” [69.2.0.6] This occurs if the porous medium contains two types of porosity or regions of very different porosities. [69.2.0.7] For curve 3 the ratio between the two porosities is roughly 100, and the inverse first moment of $\mu(\phi)$ exists. [69.2.0.8] For curves 7 and 8, the ratio roughly 1000, and the densities diverge at $\phi=0$ . [69.2.0.9] In all cases the distributions were chosen critical in the sense that the weight for the higher-porosity component is $\tfrac{1}{3}$ .

C Local percolation probability

[69.2.1.1] The local percolation probabilities $\lambda(\phi)$ can be measured simultaneously with the local porosity distribution $\mu(\phi)$ . [69.2.1.2] However, such a measurement is more difficult because it requires the approximate reconstruction of the three-dimensional pore space from parallel two-dimensional sections. [69.2.1.3] For this reason geometric modeling of porous media is most important for this quantity.

[69.2.2.1] Three simple models will be compared in the calculations: the uniformly connected model (UCM), the central pore model (CPM), and the grain consolidation model (GCM).

1 Uniformly connected models

[69.2.3.1] In these models $\lambda(\phi)$ equals a constant, i.e.,

$\alpha(\phi)=p.$

(6.8)

[69.2.3.2] In the simplest case, the fully connected model, $p=1$ . [69.2.3.3] This means that all local geometries are assumed to be [page 70, §0] conducting. [70.1.0.1] In addition, the case $p=\tfrac{1}{2}$ will also be investigated.

2 Central pore model

[70.1.1.1] Consider a cubic cell of volume 1 filled with rock. [70.1.1.2] Inside the cubic cell a centered cubic pore of side length $a$ ( $0\leq a\leq 1$ ) is cut out. [70.1.1.3] Now a random process is used to drill cylindrical pores with square cross section from the faces of the cube toward the central pore. [70.1.1.4] Sometimes these pores will connect to the central pore, and sometimes not. [70.1.1.5] The random process starts with the choice of an arbitrary face of the cube. [70.1.1.6] Now choose a random number $r$ between $0$ and $\tfrac{1}{2}$ . [70.1.1.7] If $r>\tfrac{1}{2}(1-a)$ , a pore with square cross section of side length $b$ ( $0\leq b\leq a$ ) is drilled from the center of the face all the way to the central pore. [70.1.1.8] The central pore had volume $a^{3}$ , and the connection pore has the volume $\tfrac{1}{2}b^{2}(1-a)$ . [70.1.1.9] If the random number fulfills $f<\tfrac{1}{2}(1-a)$ , then the face is not pierced, but instead the same volume $\tfrac{1}{2}b^{2}(1-a)$ is removed from the wall in such a way that the resulting pore space remains disconnected from the pore space connected to the central pore. [70.1.1.10] This process is repeated for all six faces of the cube. [70.1.1.11] The cubic symmetry is not essential, and a model with different symmetry can be defined similarly.

[70.1.2.1] The result of the process described above is a cubic cell whose porosity can be expressed in terms of the side length $a$ of the central pore and the ratio $R=b/a$ as

$\phi=a^{3}+3b^{2}(1-a)=a^{3}(1-3R^{2})+3R^{2}a^{2}.$

(6.9)

[70.1.2.2] According to the definitions in Section II, the cell is called percolating if there exist at least one path within the pore space connecting a face to a face different than itself. [70.1.2.3] To obtain $\lambda(\phi)$ the probability that either no or exactly one face is pierced has to be calculated. [70.1.2.4] This probability euqals $1-\lambda(\phi)$ . [70.1.2.5] Clearly,

$1-\lambda=(1-a)^{6}+a(1-a)^{5}=(1-a)^{5}.$

(6.10)

[70.1.2.6] Thus, in the central pore model,

$\lambda(\phi,R)=1-[1-a(\phi,R)]^{5},$

(6.11)

where $a(\phi,R)$ is that root of eq. (6.9), which fulfills $0\leq a\leq 1$ for all $0\leq\phi\leq 1$ and $0\leq R\leq 1$ . [70.1.2.7] For $R=1$ , i.e., $b=a$ , it follows that $\phi\propto a^{2}$ and thus $a\propto\phi^{{1/2}}$ , resulting in $\lambda(\phi)\propto\phi^{{1/2}}$ for small $\phi$ . [70.1.2.8] On the other hand, for $R\to 0$ one finds $a\propto\phi^{{1/3}}$ for $\phi\to 0$ and thus $\lambda(\phi)\propto\phi^{{1/3}}$ for $\phi\to 0$ . [70.1.2.9] Thus the general conclusion for the central pore model is that

$\lambda(\phi)\propto\phi^{\gamma},$

(6.12)

where $\gamma$ can range between $\tfrac{1}{2}$ and $\tfrac{1}{3}$ .

3 Grain consolidation model

[70.1.3.1] The grain consolidation model was proposed as a simple geometrical model for diagenesis.[11][70.1.3.2] Its main observation is the existence and smallness of the percolation threshold in regular and random bead packings when the bead radii are increased. [70.1.3.3] In fact, the model has recently been modified such that the critical porosity at which conduction ceases can be arbitrarily small.[12][70.2.0.1] For regular bead packings, this implies

$\lambda(\phi)=\begin{cases}0&\text{for}~\phi<\phi _{c}\\ 1&\text{for}~\phi>\phi _{c}\,.\end{cases}$

(6.13)

[70.2.0.2] For random packings $\lambda(\phi)$ will be smoothed out around $\phi _{c}$ . [70.2.0.3] For simplicity, in this paper eq. (6.13) will be used with $uphi_{c}=0.05$ .

[70.2.1.1] The most important aspect of $\lambda(\phi)$ is that it determines the control parameter $p$ . [70.2.1.2] According to eq. (6.12), its behavior near $\phi=0$ can influence the exponent $\alpha$ in (5.10c). [70.2.1.3] Note that for the grain consolidation model the form of $\lambda(\phi)$ always implies that condition (5.10a) is fulfilled, and universal behavior is expected. [70.2.1.4] The half-connected model in the uniformly connected model class is included to demonstrate the influence of the thin-plate effect. [70.2.1.5] The shape of $\lambda(\phi)$ in all other cases gives extremely small probability to blocking geometries with high porosities. [70.2.1.6] This is expected to be generally true for interparticle porosity. [70.2.1.7] This is expected to be generally true for interparticle porosity. [70.2.1.8] However, the secondary pore space in real rocks may contain a significant fraction of high-porosity blocking geometries.[32]

D Numerical results

[70.2.2.1] Numerical solutions to eq. (3.2) were obtained using an iterative technique. [70.2.2.2] The iteration was stopped whenever $\lvert\varepsilon _{n}(\omega)-\varepsilon _{{n-1}}(\omega)\rvert<10^{{-7}}$ . [70.2.2.3] In Figs. 2-5 selected results for $\varepsilon(\omega)$ are presented. [70.2.2.4] Figure 2 presents the uniformly connected model with $p=1$ , Figure 3 the uniformly connected model with $p=\tfrac{1}{2}$ . [70.2.2.5] Figure 4 gives the results for the central pore model with $R=0.922$ , and Fig. 5 those for the grain consolidation model with $\phi _{c}=0.05$ . [70.2.2.6] In each figure the line styles correspond to the line styles of the local porosity distributions displayed in Figure 1. [70.2.2.7] Parts (a) of each figure shows $\log _{{10}}\varepsilon^{\prime}(\omega)$ as a function of $\log _{{10}}\omega$ in the upper graph and $\log _{{10}}\sigma^{\prime}(\omega)$ in the lower graph. [70.2.2.8] In addition, the inset in the upper right-hand corner displays the local percolation probability $\lambda(\phi)$ for $0\leq\phi\leq\tfrac{1}{2}$ . [70.2.2.9] The vertical scale for the inset is always $0\leq\lambda\leq 2$ . [70.2.2.10] Part (b) display in the upper graph $-d[\log _{{10}}\varepsilon^{\prime}(\omega)]/d(\log _{{10}}\omega)$ versus $\log _{{10}}\omega$ and $d[\log _{{10}}\sigma^{\prime}(\omega)]/d(\log _{{10}}\omega)$ in the lower plot. [70.2.2.11] These quantities give a more sensitive representation of the dispersion and show at the same time the values of “local exponents”. [70.2.2.12] In all figures frequency is measured in units of the relaxation frequency of water as discussed in Section IV. [70.2.2.13] All plots are given over ten frequency decades with a resolution of five points per decade.

Figure 2: (a) Real part of the complex dielectric constant (upper graph) and real part of complex conductivity (lower graph) vs frequency in a doubly logarithmic plot. Each graph contains eight curves whose line style corresponds to the local porosity densities displayed in Figure 1. The inset shows the local percolation probability $\lambda(\phi)$ used in the calculation, in this case that of the uniformly conducting model with $p=1$ . The vertical axis of the inset ranges form $\lambda=0$ to $2$ , the horizontal axis from $\phi=0$ to $0.5$ . (b) Derivativs of the curves in (a).

Figure 3: Same as Figure 2 for the uniformly conducting model with $p=\tfrac{1}{2}$ .

Figure 4: Same as Figure 2 for the central pore model with $R=0.922$ .

Figure 5: Same as Figure 2 for the grain consolidation model with $\phi _{c}=0.05$ .

E Discussion

[70.2.3.1] It is obvious from Figs. 2-5 [especially part (b)] that the low-frequency dielectric response depends sensitively on the details of $\mu(\phi)$ and $\lambda(\phi)$ . [70.2.3.2] A general discussion is difficult, because the response is always a mixture between three basic mechanisms each of which can give significant dielectric dispersion. [page 71, §0] [71.1.0.1] The first mechanism is the dispersion resulting from the disorder in $\mu(\phi)$ itself. [71.1.0.2] The second mechanism is the despersion resulting from $p$ , i.e., from percolation geometry. [71.1.0.3] The third mechanism is the dispersion resulting from the behavior of $\lambda(\phi)$ in the $\phi\to 1$ limit, i.e., the thin-plate effect.

[71.1.1.1] The absolute dispersion for all figures is collected in Table 2. [71.1.1.2] $\Delta\varepsilon$ is defined as $\Delta\varepsilon=\varepsilon^{\prime}(0)-\varepsilon^{\prime}(\infty)$ , while $\Delta\sigma=\sigma^{\prime}(\infty)-\sigma^{\prime}(0)$ .

[71.2.1.1] Before discussing the three mechanisms, it is important to note that the bulk porosity $\overline{\phi}$ does not influence the shape of the response curves if it is changed without changing the shape of $\mu(\phi)$ . [71.2.1.2] Instead, it determines an overall frequency shift for the frequency region over which the dispersion occurs. [71.2.1.3] As $\overline{\phi}$ is lowered, this region is shifted toward lower frequencies. [71.2.1.4] This observation together with the fact that all $\mu(\phi)$ give the same bulk porosity of $0.1$ shows that the bulk porosity by itself cannot be used to characterize the dielectric response. [71.2.1.5] In particular, there is no theoretical basis for Archie’s law [eq. (5.27)] if interpreted as a relation between dc conductivity and bulk porosity (see Section V.D for a discussion).

[page 72, §0]

[page 73, §0] [73.1.1.1] A second observation is that in all figures the high-frequency real dielectric constant $\varepsilon^{\prime}(\infty)$ is not very sensitive to the details of $\mu(\phi)$ . [73.1.1.2] This is a consequence of the fact that for low $\overline{\phi}$ the local dielectric constants $\varepsilon^{\prime}_{B}$ and $\varepsilon^{\prime}_{C}$ must both approach $\varepsilon^{\prime}_{R}$ .

[73.1.2.1] The first mechanism, dispersion from the form of $\mu(\phi)$ , can be studied in pure form when $\lambda(\phi)=1$ , and the corresponding results are shown in Figure 2. [73.1.2.2] In this case there are no blocking local geometries; i.e., $p=1$ according to eq. (5.7). [73.1.2.3] If $\mu(\phi)$ is highly peaked as in curve 2, then the system is only weakly disordered, and there is almost no visible disperion with the amount of disorder contained in $\mu(\phi)$ . [73.1.2.4] In fact, distributions with power-law divergences at $\phi\to 0$ or with percolation structure generate the strongest dispersion, as can be seen from curves 3 and 5-8 in figure 2. [73.1.2.5] Table 2 shows that the dispersion varies almost three orders of magnitude between the different distributions. [73.1.2.6] Note that relatively similar local porosity distributions such as curves 6 and 8 can have very different dielectric response. [73.1.2.7] On the other hand, very different shapes for $\mu(\phi)$ can give similar $\varepsilon(\omega)$ , as demonstrated by curves 3 and 5. [73.1.2.8] This shows that the dielectric response by itself does not contain a full geometric characterization of the pore space, and it needs always to be complemented with additional physical or geometrical information. [73.1.2.9] This is not too surprising. [73.1.2.10] Indeed, it is more surprising that when the dielectric response becomes large it is also very sensitive to geometric details. [73.1.2.11] This is the case for dielectric enhancement near the percolation threshold or as a result of the thin-plate effect.

[73.1.3.1] Consider the thin-plate mechanism. [73.1.3.2] It requires the presence of blocking geometries of high porosity. [73.1.3.3] Mathematically, this means $\lambda(\phi)\neq 1$ for large $\phi$ . [73.1.3.4] As a simple illustration, Figure 3 displays the results for the uniformly connected model when $\lambda(\phi)=\tfrac{1}{2}$ . [73.1.3.5] Now $p=\tfrac{1}{2}$ , which is far away from $p_{c}$ . [73.1.3.6] Nevertheless, the dielectric dispersion is much stronger than would be obtained for solutions to the central pore model or grain consolidation model with the same $p$ . [73.1.3.7] Compare, e.g., curve 5 in figure 3 with curve 5 in figure 5. [73.1.3.8] Moreover, the dielectric dispersion becomes sensitive to the details $\mu(\phi)$ . [73.1.3.9] It is now possible to distinguish in figure 3(b) the distributions 3, 7 and 8, which have $\omega\neq 1$ from the rest for which $\omega=1$ . [73.1.3.10] In particular, curves 3 and 5, which had very similar response in figure 2, appear now very different. [73.1.3.11] The dispersion is the stronger the more weight $\mu(\phi)$ has at high $\phi$ . [73.1.3.12] This can be seen from curve 3, which shows less dispersion than curves 4 and 5, while the opposite was true for figure 2. [73.2.0.1] Similarly, $\varepsilon^{\prime}$ for curve 7 is depressed below curves 6 and 8 at intermediate frequencies. [73.2.0.2] At very low frequencies, the percolative character of distribution 7 is responsible for stronger overall dispersion than in curve 6. [73.2.0.3] The degree of asymmetry of $\mu(\phi)$ is reflected in the asymmetry of the response, as best seen in the derivatives plotted in figure 3(b).

[73.2.1.1] The percolation mechanism is responsible for strong dielectric disperion in figures 4 and 5. [73.2.1.2] There is essentially no dispersion from thin-plate mechanism in these cases because in both cases $\lambda(\phi)\approx 1$ for $\phi>0.5$ , and thus there are no local geometrics with a high dielectric constant. [73.2.1.3] Figure 4 represents the central pore model with $R=0.922$ , and results for the grain consolidation model with $\phi _{c}=0.05$ are given in figure 5. [73.2.1.4] Contrary to the situation in figures 2 and 3, $p$ is now different for each distribution. [73.2.1.5] The results of performing the integral in eq. (5.7) are listed in table 3. [73.2.1.6] Naturally, the dielectric dispersion increases strongly with $p\to p_{c}$ and this effect dominates the dispersion from $\mu$ itself. [73.2.1.7] In particular, for $p\approx p_{c}$ power-law behavior for $\varepsilon(\omega)$ as a function of frequency is obtained in agreement with the scaling theory presented in Section V. [73.2.1.8] As an example, scaling theory predicts the exponent $0.597$ for the conductivity of distribution 8 in figure 4 and $0.403$ for the real dielectric constants. [73.2.1.9] These predictions are obtained from eqs. (5.23) and (5.24) using $s=1$ and eqs. (5.13b) and (6.12) with $\gamma=0.5$ , and the exponent $\nu _{1}$ from Table 1. [73.2.1.10] Figure 4(b) shows that these values are indeed approached at low frequencies. [73.2.1.11] Similarly, scaling theory predicts the exponent $0.5$ for $\varepsilon^{\prime}$ and $\sigma^{\prime}$ corresponding to distributions $3$ , $6$ and $7$ in figure 5. [73.2.1.12] Again, these values are approached as seen fom figure 5(b), although the power-law behavior occurs over a limited frequency range because $p$ is not sufficiently close to the critical region. [73.2.1.13] Note that curve 8 in figure 5 has dropped below the percolation threshold, and thus the conductivity increases as $\omega^{2}$ for small $\omega$ .

Table 2: Numerical values for $\Delta\varepsilon=\varepsilon^{\prime}(0)-\varepsilon^{\prime}(\infty)$ and $\Delta\sigma=\sigma^{\prime}(\infty-\sigma^{\prime}(0))$ contained in the model calculations of Figures 2-5.

Curve No. 1	1	2	3	4	5	6	7	8


$\Delta\varepsilon$ (Fig. 2)	0.798	0.053	3.716	1.863	3.147	9.261	28.793	20.484
$\Delta\sigma$ (Fig. 2)	0.298	0.035	1.067	0.619	0.889	1.545	2.400	2.029
$\Delta\varepsilon$ (Fig. 3)	19.197	13.819	21.956	25.162	34.950	96.046	186.010	115.270
$\Delta\sigma$ (Fig. 3)	1.585	1.370	1.664	1.674	1.741	1.648	1.751	1.489
$\Delta\varepsilon$ (Fig. 4)	5.996	3.835	16.616	9.639	14.204	52.196	259.700	3932.522
$\Delta\sigma$ (Fig. 4)	1.105	0.801	1.974	1.455	1.708	2.176	2.877	2.283
$\Delta\varepsilon$ (Fig. 5)	2.119	0.053	371.525	5.498	11.430	291.481	277.396	37.691
$\Delta\sigma$ (Fig. 5)	0.595	0.035	3.025	1.137	1.604	2.565	3.043	2.436

[73.2.2.1] The complexity and variability of $\varepsilon(\omega)$ obtained from the simple mean-field solutions of this section correspond to the complexity and variability of possible pore-space geometries. [73.2.2.2] More approximate analytical investigations of the solutions to eq. (3.2) are necessary to identify simple parameters characterizing $\mu$ and $\lambda$ which allow a better classification of the solutions and thereby the possible geometries.

Table 3: Calculated values of $p$ corresponding to the eight different forms of $\mu(\phi)$ for the central pore model (CPM, displayed in Figure 4) and for the grain consolidation model (GCM, displayed in Figure 5). Note that $p=1$ for all cases in Figure 1 and $p=\tfrac{1}{2}$ for all cases displayed in Figure 2.

Curve No.	1	2	3	4	5	6	7	8


CPM	0.6542	0.6940	0.5420	0.5858	0.5341	0.4059	0.3643	0.3334
GCM	0.7500	1.0000	0.3399	0.6048	0.4962	0.3415	0.3376	0.2841

F Sedimentary rock

[73.2.3.1] The present paper deals only with simple homogeneous and isotropic porous media. [73.2.3.2] Real rocks are highly inhomogeneous, but they can also be discussed within the present framework. [page 74, §0] [74.1.0.1] Sedimentary rocks exhibit two main types of porosity. [74.1.0.2] Primary interparticle porosity is the porosity between the grains of the original sediment. [74.1.0.3] Often, this pore space is changed during diagenesis of the sediment. [74.1.0.4] In particular cements between the grains can exhibit a qualitatively different secondary porosity.[33] [74.1.0.5] This situation can be treated within the present formalism by replacing the dielectric constant $\varepsilon _{W}$ of the pore-filling fluid with the effective dielectric constant of the pore-filling cement. [74.1.0.6] Naturally, the results must be much more complex, as they contain additional independent geometrical information describing the cement.

Author:	R. Hilfer
published in:	Physical Review Letters: B, Vol. 44, No. 1, pages 60-75 (July, 1991)
originally submitted:	12th October 1990