V Direct Problem

[64.2.4.1] Consider now the direct problem with $\lambda(\phi)$ and $\mu(\phi)$ given. [64.2.4.2] The problem is to calculate $\varepsilon(\omega)$ . [64.2.4.3] The nonlinear integral equation (3.2) is too difficult for an analytical treatment, and numerical solutions must ultimately be sought. [64.2.4.4] Some analytic information can be obtained, however, by exploiting the similarity to the well-known percolation problem. [64.2.4.5] The analogy arises from the classification of local geometries into percolating and nonpercolating ones. [64.2.4.6] The first question is then whether there indeed exists a percolation threshold.

A Solution for strongly peaked local porosity distributions

[64.2.5.1] For strongly peaked $\mu(\phi)$ , eq. (3.2) can be expanded around $\overline{\phi}$ . [64.2.5.2] The lowest-order approximation (a “mean-field approximation to a mean-field approximation”) [page 65, §0] leads to a quadratic equation for $\varepsilon(\omega)$ which is easily solved. [65.1.0.1] For low frequencies $(\omega\to 0)$ , one finds

$\sigma^{\prime}(0)=\tfrac{1}{2}\sigma^{\prime}_{C}(0;\overline{\phi})[3\lambda(\overline{\phi})-1],$

(5.1)

for $\lambda(\overline{\phi})>\frac{1}{3}$ , and $\sigma^{\prime}(0)=0$ , for $\lambda(\overline{\phi})\leq\frac{1}{3}$ . [65.1.0.2] Thus a percolation transition with control parameter $\lambda(\overline{\phi})$ is predicted for strongly peaked local porosity distributions. [65.1.0.3] Although the prediction of a percolation transition remains correct for arbitrary $\mu(\phi)$ , the control parameter is in general not $\lambda(\overline{\phi})$ . [65.1.0.4] This will be seen in the next subsection. [65.1.0.5] The percolation transition leads to a divering dc dielectric constant, which is given within the present approximations as

$\varepsilon^{\prime}(0)=\tfrac{1}{2}[3\lambda(\overline{\phi}-1)]\varepsilon^{\prime}_{C}(0;\overline{\phi})+\frac{9\lambda(\overline{\phi})[\lambda(\overline{\phi})-1]\varepsilon^{\prime}_{B}(0;\overline{\phi})}{2[3\lambda(\overline{\phi})-1]},$

(5.2a)

for $\lambda(\overline{\phi})>\frac{1}{3}$ , and as

$\varepsilon^{\prime}(0)=\frac{\varepsilon^{\prime}_{B}(0;\overline{\phi})}{1-3\lambda(\overline{\phi})},$

(5.2b)

for $\lambda(\overline{\phi})<\frac{1}{3}$ . [65.2.0.1] Note that eqs. (4.9) and (5.2) identify already two possible mechanisms for dielectric enhancement, one from the necessity of a thin-plate effect for local geometries in the high-porosity limit, the secound from the percolation threshold.

[65.2.1.1] In the high-frequency limit ( $\omega\to\infty$ ), the results are

$\varepsilon^{\prime}(\infty)=\frac{B}{4}\left(1\pm\left(1+\frac{8\varepsilon^{\prime}_{C}(\infty;\overline{\phi})\varepsilon^{\prime}_{B}(\infty;\overline{\phi})}{B^{2}}\right)^{{1/2}}\right),$

(5.3a)

where

$B=[3\lambda(\overline{\phi})-1]\varepsilon^{\prime}_{C}(\infty;\overline{\phi})+\varepsilon^{\prime}_{B}(\infty;\overline{\phi})[2-3\lambda(\overline{\phi})],$

(5.3b)

for the real dielectric constant, and

$\sigma^{\prime}(\infty)=\frac{\lambda(\overline{\phi})\sigma^{\prime}_{C}(\infty;\overline{\phi})\varepsilon^{\prime}(\infty)}{\lambda(\overline{\phi})\varepsilon^{\prime}_{C}(\infty;\overline{\phi})+[1-\lambda(\overline{\phi})]\varepsilon^{\prime}_{B}(\infty;\overline{\phi})\left(\frac{\varepsilon^{\prime}_{C}(\infty;\overline{\phi})+2\varepsilon^{\prime}(\infty)}{\varepsilon^{\prime}_{B}(\infty;\overline{\phi})+2\varepsilon^{\prime}(\infty)}\right)^{2}},$

(5.4)

for real part of the conductivity. [65.0.1.2] The sign in equation (5.3a) has to be chosen such that $\sigma^{\prime}$ remains positve.

B Low-frequency limit

[65.0.2.1] In this subsection it will be shown that equation (3.2) does indeed imply the existence of a percolation threshold, also for general $\mu(\phi)$ . [65.0.2.2] But contrary to previous percolation theories for porous media, the bulk porosity $\overline{\phi}$ is not the control parameter of the transition. [65.0.2.3] Therefore, a transition can or cannot occur as $\overline{\phi}$ is varied.

[65.0.3.1] To identify the percolation transition, consider the low-frequency limit. [65.0.3.2] Expanding eq. (3.2) around $\omega=0$ leads to

$\int _{0}^{1}\frac{\sigma^{\prime}_{C}(0;\phi)-\sigma^{\prime}(0)}{\sigma^{\prime}_{C}(0;\phi)-2\sigma^{\prime}(0)}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi+\int _{0}^{1}\frac{\sigma^{\prime}_{B}(0;\phi)-\sigma^{\prime}(0)}{\sigma^{\prime}_{B}(0;\phi)-2\sigma^{\prime}(0)}[1-\lambda(\phi)]\mu(\phi)\,\mathrm{d}\phi=0,$

(5.5)

for the effective dc conductivity $\sigma^{\prime}(0)$ of the system. [65.0.3.3] Inserting equation (4.6) into (5.5),

$\sigma^{\prime}(0)\int _{0}^{1}\frac{\lambda(\phi)\mu(\phi)}{\sigma^{\prime}_{C}(0;\phi)+2\sigma^{\prime}(0)}\,\mathrm{d}\phi=-\tfrac{1}{6}+\tfrac{1}{2}\int _{0}^{1}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi$

(5.6)

is found. [65.0.3.4] This equation for $\sigma^{\prime}(0)$ has no positive solutions if $\int _{0}^{1}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi<\tfrac{1}{3}$ . [65.0.3.5] This identifies the control parameter for the transition as

$p=\int _{0}^{1}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi,$

(5.7)

and the effective-medium value $p_{c}=\frac{1}{3}$ for the percolation thershold. [65.1.0.1] The control parameter $p$ is the total fraction of percolating local geometries.

[65.1.1.1] For the effective real dielectric constant $\varepsilon^{\prime}(0)$ , two equations are obtained. [65.1.1.2] If $p<p_{c}$ , the equation reads

$[\varepsilon^{\prime}(0)]^{{-1}}\int _{0}^{1}\frac{[1-\lambda(\phi)]\mu(\phi)}{[\varepsilon^{\prime}_{B}(0;\phi)]^{{-1}}+[2\varepsilon^{\prime}(0)]^{{-1}}}\,\mathrm{d}\phi=2(p_{c}-p).$

(5.8)

[65.1.1.3] For $p>p_{c}$ the solution to equation (5.6) must be inserted into

$\varepsilon^{\prime}(0)=\frac{A+B}{C},$

(5.9)

where

	$\displaystyle A=[4\sigma^{\prime}(0)]^{{-1}}\int _{0}^{1}\varepsilon^{\prime}_{B}(0;\phi)[1-\lambda(\phi)]\mu(\phi)\,\mathrm{d}\phi,$
	$\displaystyle B=\int _{0}^{1}\frac{\varepsilon^{\prime}_{C}(0;\phi)\sigma^{\prime}(0)}{[\sigma^{\prime}_{C}(0;\phi)+2\sigma^{\prime}(0)]^{2}}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi,$
	$\displaystyle C=\int _{0}^{1}\frac{\sigma^{\prime}_{C}(0;\phi)}{[\sigma^{\prime}_{C}(0;\phi)+2\sigma^{\prime}(0)]^{2}}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi.$

[65.1.2.1] Equation (5.2) can be treated analogously to the case of ordinary percolation.[30, 31][65.1.2.2] The effective conductivity $\sigma^{\prime}(0)$ will be small for $p\to p_{c}$ or whenever $\overline{\phi}\to 0$ , i.e., in the low-porosity limit. [65.1.2.3] In this case the integral on the left-hand side of eq. (5.6) is dominated by the small $\phi$ behavior of $\lambda(\phi)\mu(\phi)$ . [65.1.2.4] The following three cases for $\lambda(\phi)\mu(\phi)$ must be distinguished:

	$\displaystyle\text{(a)}\quad\int _{0}^{1}\phi^{{-1}}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi<\infty,$		(5.10a)
	$\displaystyle\text{(b)}\quad\lambda(\phi)\mu(\phi)\to\text{const}\quad\text{for}~\phi\to 0,$		(5.10b)
	$\displaystyle\text{(c)}\quad\lambda(\phi)\mu(\phi)\propto\phi^{{-\alpha}}~\text{for}~\phi\to 0~\text{and}~0<\alpha<1.$		(5.10c)

[65.1.2.5] [page 66, §0] For case (a) the solution of eq. (5.6) is obtained as

$\sigma^{\prime}(0)\approx\sigma^{\prime}_{+}(p-p_{c}),$

(5.11)

where $\sigma^{\prime}_{+}$ is defined by

$\frac{1}{\sigma^{\prime}_{+}}=\int _{0}^{1}\frac{\lambda(\phi)\mu(\phi)}{\sigma^{\prime}_{C}(0;\phi)}\,\mathrm{d}\phi.$

(5.12)

[66.1.0.1] The integral exists because of eq. (4.8) and condition (5.10a). [66.1.0.2] Note that eq. (5.11) is valid for all $p$ for $\overline{\phi}\to 0$ and for all $\overline{\phi}$ for $p\to p_{c}$ . [66.1.0.3] The conductivity exponent $t=1$ has its expected reflective-medium value. [66.1.0.4] The result of eq. (5.11) is universal in the sense that the value of $t$ does not depend on the specific geometry contained in $\lambda(\phi)$ and $\mu(\phi)$ as long as condition (5.10a) remains fulfilled.

[66.1.1.1] The situation is very different for case (c). [66.1.1.2] If $\lambda(\phi)\mu(\phi)\propto M\phi^{{-\alpha}}$ and $\sigma^{\prime}_{C}(0;\phi)$ is given by eq. (4.8), then to leading order in $\sigma^{\prime}$ the solution to eq. (5.6) is

$\sigma^{\prime}(0)\propto C_{1}\left(\frac{1-\alpha}{M}\sin(\pi\alpha)(p-p_{c})\right)^{t},$

(5.13a)

where the conductivity exponent

$t=\frac{1}{1-\alpha}$

(5.13b)

is now no longer universal. [66.1.1.3] It depends on the behavior of $\lambda(\phi)\mu(\phi)$ for small $\phi$ . [66.1.1.4] For the marginal case $\alpha=0$ [case (b)], logarithmic corrections to eq. (5.11) are obtained, and

$\sigma^{\prime}(0)\propto(p-p_{c})\left(\log\left(\tfrac{1}{p-p_{c}}\right)\right)^{{-1}}.$

(5.14)

[66.1.2.1] For the real dielectric constant, eq. (5.8) is valid below $p_{c}$ . [66.1.2.2] This equation has the same form as eq. (5.6) for the conductivity. [66.1.2.3] However, it is now the behavior of $\lambda(\phi)$ and $\mu(\phi)$ that high porosities $\phi\to 1$ that is relevant. [66.1.2.4] From eq. (5.8)

$\varepsilon^{\prime}(0)\approx\varepsilon^{\prime}_{-}(p-p_{c})^{{-1}}$

(5.15)

is obtained, where

$\varepsilon^{\prime}_{-}=\int _{0}^{1}\varepsilon^{\prime}_{B}(0;\phi)[1-\lambda(\phi)]\mu(\phi)\,\mathrm{d}\phi,$

(5.16)

analogous to eqs. (5.11) and (5.12). [66.1.2.5] Again, the expected value $s=1$ for the superconductivity exponent is universal as long as the integral in eq. (5.16) exists. [66.1.2.6] The asymptotic solution is valid for all $p$ if $\overline{\phi}\to 1$ , and for all $\overline{\phi}$ whenever $p\to p_{c}$ . [66.1.2.7] If $[1-\lambda(\phi)]\mu(\phi)\propto(1-\phi)^{{-\beta}}$ , with $0<\beta<1$ , then the superconductivity exponent becomes nonuniversal and has the value

$s=\frac{1}{1-\beta},$

in analogy with eq. (5.13b) for the conductivity exponent $t$ .

[66.1.3.1] The central result of this section is the identification of a percolation transition underlying the random geometry of porous media. [66.2.0.1] The control parameter for the transition is neither the bulk porosity $\overline{\phi}$ nor $\lambda(\overline{\phi})$ as suggested in Sec. V.A, but the total fraction of conducting local geometries. [66.2.0.2] Another result is that the underlying transition is expected to become relevant both in the low-porosity limit ( $\overline{\phi}\to 0$ ) and in the high-porosity limit ( $\overline{\phi}\to 1$ ). [66.2.0.3] Having identified the transition using mean-field theory, the next step is to apply the results of scaling theory in the present context.

C Scaling theory

[66.2.1.1] In this section the scaling theory for the percolation transition[32, 33] is applied in the present context.[34] [66.2.1.2] Therefore, the present section goes beyond the effective-medium equation (3.2). [66.2.1.3] Scaling theory starts from the assumption that the complex dielectric constant can be written as

$\varepsilon=\varepsilon _{+}\lvert p-p_{c}\rvert^{t}f\left(\frac{\varepsilon _{-}/\varepsilon _{+}}{\lvert p-p_{c}\rvert^{{t+s}}}\right),$

(5.17)

with the scaling function

$f(z)=A^{{\mathrm{I}}}_{0}+A^{{\mathrm{I}}}_{1}z+A_{2}^{{\mathrm{I}}}z^{2}+\dots,$

(5.18a)

for $\lvert z\rvert\ll 1$ and $p>p_{c}$ ,

$f(z)=A^{{\mathrm{II}}}_{1}z+A^{{\mathrm{II}}}_{2}z^{2}+\dots,$

(5.18b)

for $\lvert z\rvert\ll 1$ and $p<p_{c}$ , and

$f(z)=A^{{\mathrm{III}}}z^{{t/(t+s)}}+\dots,$

(5.18b)

for $\lvert z\rvert\gg 1$ and all $p$ . [66.2.1.4] Here $\varepsilon _{+}$ is the complex dielectric constant of a good conductor, and $\varepsilon _{-}$ is the complex dielectric constant for the poor conductor. [66.2.1.5] $p$ is the volume fraction of good conductor, $p_{c}$ is the percolation threshold, and $t$ and $s$ are the conductivity and the superconductivity exponents. [66.2.1.6] For $\omega\to 0$ , eqs. (5.17) and (5.18) yield the well-known results

$\sigma^{\prime}(0)=A^{{\mathrm{I}}}_{0}\sigma^{\prime}_{+}\lvert p-p_{c}\rvert^{t}+\dots,$

(5.19)

for $p>p_{c}$ , and $\sigma^{\prime}(0)=0$ , for $p\leq p_{c}$ for the conductivity. [66.2.1.7] For the dielectric constant, one obtains

$\varepsilon^{\prime}(0)=A^{{\mathrm{II}}}_{1}\varepsilon^{\prime}_{-}\lvert p-p_{c}\rvert^{{-s}}+\dots,$

(5.20a)

for $p<p_{c}$ , and

$\varepsilon^{\prime}(0)=A^{{\mathrm{I}}}_{0}\varepsilon^{\prime}_{+}\lvert p-p_{c}\rvert^{t}+A^{{\mathrm{I}}}_{1}\varepsilon^{\prime}_{-}\lvert p-p_{c}\rvert^{{-s}}+\dots,$

(5.20b)

for $p>p_{c}$ .

[66.2.2.1] For one-dimensional systems, the effective-medium approximation is known to be asymptotically exact for class (a) distributions.[31][66.2.2.2] If this remains true in higher dimensions, then the scaling theory can be applied to porous media by identifying the prefactors $\sigma^{\prime}_{+}$ and $\varepsilon^{\prime}_{-}$ above as those given in eqs. (5.12) and (5.16). [66.2.2.3] The important new aspect of eqs. (5.19) and (5.20) applied to porous media is that they are universally valid in the low-porosity limit of systems having $\lambda(\phi)\mu(\phi)$ which obeys condition (5.10a). [66.2.2.4] Naturally, they are also valid whenever $p\to p_{c}$ at finite $\overline{\phi}$ .

[66.2.3.1] Consider now the case of finite frequencies $\omega\neq 0$ . [66.2.3.2] The condition $\lvert z\rvert\ll 1$ in eqs. (5.18) as always satisfied for [page 67, §0] sufficiently small $\omega$ . [67.1.0.1] On the other hand, the condition $\lvert z\rvert\gg 1$ is always if either $p\approx p_{c}$ or $\varepsilon^{\prime}_{-}/\varepsilon^{\prime}_{+}\gg 1$ . [67.1.0.2] The latter condition does not apply for the systems considered in thes paper, and thus $\lvert z\rvert\gg 1$ is always interpreted as $p\approx p_{c}$ . [67.1.0.3] Equations (5.17) and (5.18) imply, for the case $p<p_{c}$ ,

$\sigma^{\prime}(\omega)=-A^{{\mathrm{II}}}_{2}\sigma^{\prime}_{+}\lvert p-p_{c}\rvert^{{-2s-t}}\omega^{2}+\dots,$

(5.21)

for the conductivity, and

$\varepsilon^{\prime}(\omega)=A^{{\mathrm{II}}}_{1}\varepsilon^{\prime}_{-}\lvert p-p_{c}\rvert^{{-s}}+A^{{\mathrm{II}}}_{2}\varepsilon^{\prime}_{-}\lvert p-p_{c}\rvert^{{-2s-t}}\left(\frac{\varepsilon^{\prime}_{-}}{\varepsilon^{\prime}_{+}}\right)\left(\frac{\omega}{\omega _{+}}\right)^{2}+\dots,$

(5.22)

for the dielectric constant, where $\omega _{+}=\sigma^{\prime}_{+}/\varepsilon^{\prime}_{+}$ . [67.1.0.4] For $p\approx p_{c}$ one obtains

$\sigma^{\prime}(\omega)=A^{{\mathrm{III}}}\sigma^{\prime}_{+}\left(\frac{\varepsilon^{\prime}_{-}}{\varepsilon^{\prime}_{+}}\right)^{{t/(t+s)}}\left(\frac{\omega}{\omega _{+}}\right)^{{t/(t+s)}}\times\left(\cos\left(\frac{\varphi t}{t+s}\right)+\frac{\omega}{\omega _{+}}\sin\left(\frac{\varphi t}{t+s}\right)\right)+\dots$

(5.23)

and

$\varepsilon^{\prime}(\omega)=A^{{\mathrm{III}}}\varepsilon^{\prime}_{+}\left(\frac{\varepsilon^{\prime}_{-}}{\varepsilon^{\prime}_{+}}\right)^{{t/(t+s)}}\left(\frac{\omega _{+}}{\omega}\right)^{{s/(t+s)}}\times\left(\sin\left(\frac{-\varphi t}{t+s}\right)+\frac{\omega}{\omega _{+}}\cos\left(\frac{\varphi t}{t+s}\right)\right)+\dots,$

(5.24)

where $\varphi=\arg(\varepsilon _{-}/\varepsilon _{+})$ . [67.1.0.5] Finally, the case $p>p_{c}$ leads to

$\sigma^{\prime}(\omega)=\sigma^{\prime}_{+}\left(A^{{\mathrm{I}}}_{0}\lvert p-p_{c}\rvert^{t}-A^{{\mathrm{I}}}_{2}\lvert p-p_{c}\rvert^{{-2s-t}}\omega^{2}+\dots\right)$

(5.25)

and

$\varepsilon^{\prime}(\omega)=A^{{\mathrm{I}}}_{0}\varepsilon^{\prime}_{+}\lvert p-p_{c}\rvert^{t}+A^{{\mathrm{I}}}_{1}\varepsilon^{\prime}_{-}\lvert p-p_{c}\rvert^{{-s}}+A^{{\mathrm{I}}}_{2}\varepsilon^{\prime}_{-}\lvert p-p_{c}\rvert^{{-2s-t}}\left(\frac{\varepsilon^{\prime}_{-}}{\varepsilon^{\prime}_{+}}\right)\left(\frac{\omega}{\omega _{+}}\right)^{2}+\dots.$

(5.26)

[67.1.0.6] These results predict a divergence of $\varepsilon^{\prime}(\omega)$ as $\omega\to 0$ with an exponent $s/(s+t)$ whenever the control parameter $p$ approaches criticality. [67.1.0.7] Simultaneously, the conductivity will also exhibit power-law behavior with exponent $t/(t+s)$ . [67.1.0.8] Outside the critical region, the frequency dependency is quadratic.

D Archie’s law

[67.1.1.1] Most publications on the electrical properties of porous media discuss the phenomenological relationship[8] between dc conductivity and bulk porosity:

$\sigma^{\prime}(0)\propto\overline{\phi}^{m}$

(5.27)

called “Archie’s law”, which is usually written in terms of the formation factor $F=[\sigma^{\prime}(0)]^{{-1}}$ . [67.2.0.1] The widespread acceptance of eq. (5.27) as a fundamental law for the physics of porous media is rather surprising in view of the fact that most experimental data[1, 2, 3, 6, 17] rarely span more than a decade in porosity. [67.2.0.2] Correspondingly, the cementation exponent $m$ is found to scatter widely between $m\approx 1$ and $m\approx 4$ . [67.2.0.3] Having found it necessary to introduce two functions to only partially characterize the pore-space geometry, it may be understandable that the present author has strong reservations to accept $\overline{\phi}$ and $m$ as sufficient geometric information to predict $\sigma^{\prime}$ , as is done in the well-logging literature.[1] [67.2.0.4] However, he feels compelled to admit that eq. (5.27) receives a certain amount of theoretical justification from his own investigation, if it is interpreted not as a relation between geometry and electrical resistance, but as a statement about physical processes which reduce the bulk porosity. [67.2.0.5] Let me explain this in more detail.

[67.2.1.1] Sedimentary and related rocks arise from sedimentation and subsequent compactification, cementation, and other physicochemical processes. [67.2.1.2] The bulk porosity $\overline{\phi}$ changes during the sedimentation history of the rock. [67.2.1.3] The final specimen’s porosity may be primary, i.e., interparticle porosity, or secondary, i.e., resulting from dissolation of grains or cements, shrinkage, fracturing, etc. [67.2.1.4] The diagenetic processes change the local dielectric and geometric properties. [67.2.1.5] Within the present formulation, it might be assumed that primarily $\lambda(\phi)$ and $\mu(\phi)$ are affected. [67.2.1.6] This implies that $\sigma^{\prime}_{+}$ and $p$ become implicitly dependent upon $\overline{\phi}$ , and consequently $\sigma^{\prime}(0)$ will change with $\overline{\phi}$ . [67.2.1.7] To discuss these changes one needs a physical model for the changes of $\alpha$ and $\mu$ , but this is not the objective of the present investigation. [67.2.1.8] Nevertheless, it is of interest to discuss the general consequences of the scaling approach presented above. [67.2.1.9] Clearly, $\sigma^{\prime}_{+}(\overline{\phi})$ should tend to zero as $\overline{\phi}\to 0$ , and it should approach $\sigma^{\prime}_{W}$ for $\overline{\phi}\to 1$ . [67.2.1.10] It seems also plausible that $p(\overline{\phi})$ should decrease as $\overline{\phi}$ is lowered. [67.2.1.11] If one assumes that $\sigma^{\prime}_{+}(\overline{\phi})$ and $p(\overline{\phi})$ can be expanded around $\overline{\phi}=0$ as

$\displaystyle\sigma^{\prime}_{+}(\overline{\phi})$	$\displaystyle=\dot{\sigma}^{\prime}_{+}(0)\overline{\phi}+\tfrac{1}{2}\ddot{\sigma}^{\prime}_{+}(0)\overline{\phi}^{2}+\dots,$		(5.28)
$\displaystyle p(\overline{\phi})$	$\displaystyle=p(0)+\dot{p}(0)\overline{\phi}(0)\overline{\phi}+\dots,$		(5.29)

then eq. (5.19) implies that

$\sigma^{\prime}(\overline{\phi})=A^{{\mathrm{I}}}_{0}\dot{\sigma}^{\prime}_{+}(0)\overline{\phi}\big[p(0)-p_{c}+\dot{p}(0)\overline{\phi}+\dots\big]^{t}.$

(5.30)

[67.2.1.12] This already resembles eq. (5.27). [67.2.1.13] In particular, if it happens that $p(0)\approx p_{c}$ , i.e., if one approaches criticality as $\overline{\phi}\to 0$ , then eq. (5.30) yields Archie’s law with a cementation exponent,

$m=1+t,$

(5.31a)

as long as condition (5.10a) remains satisfied during the cementation process. [67.2.1.14] If the system falls under condition (5.10c), however, the cementation index becomes

$m=1+t+\frac{\alpha(\overline{\phi})}{1-\alpha(\overline{\phi})}.$

(5.31b)

[page 68, §0] [68.1.0.1] Note that $\alpha$ and thus $m$ may explicitly depend on the bulk porosity. [68.1.0.2] Even more complicated results for $m$ obtained if $\lambda(\phi)$ and $\mu(\phi)$ change with $\overline{\phi}$ such that $\sigma^{\prime}_{+}(\overline{\phi})\propto\overline{\phi}^{{m_{\sigma}}}$ and $p(\overline{\phi})\propto p_{c}+\dot{p}(0)\overline{\phi}^{{m_{p}}}$ . [68.1.0.3] In such cases,

$m=m_{\sigma}+m_{p}\left(t+\frac{\alpha(\overline{\phi})}{1-\alpha(\overline{\phi})}\right).$

(5.31b)

[68.1.0.4] The surprising result is that the simplest form for $m$ , namely, eq. (5.31a), predicts an exponent in the range from $m=2$ for $p\gg p_{c}$ to $m=1+t\approx 3$ for $p\approx p_{c}$ . [68.1.0.5] Nevertheless, the cementation exponent will in general be very different for different compaction processes, and without physical models for such processes even a nonmonotonous behavior of $\sigma^{\prime}(0;\overline{\phi})$ is possible. [68.1.0.6] The important result of this section is that it provides a general framework inside which the apparent phenomenological universality and scaling properties of Archie’s law might be understood.

[68.1.1.1] A second interesting consequence of this section is that it predicts similar scaling laws for the dielectric constant in the high-porosity limit $\overline{\phi}\to 1$ . [68.1.1.2] This is a consequence of the thin-plate effect [eq. (4.9)] and analogous assumptions about the corresponding dilution process. [68.1.1.3] More precisely, it is predicted that

$\varepsilon^{\prime}(0)\propto(1-\overline{\phi})^{{-m^{\prime}}}.$

(5.32)

[68.1.1.4] Here the “dilution exponent” $m^{\prime}$ is given in the simplest case as

$m^{\prime}=1+s,$

(5.33a)

where $s$ is the superconductivity exponent, and in the general case as

$m^{\prime}=m^{\prime}_{\sigma}+m^{\prime}_{p}\left(s+\frac{\beta(\overline{\phi})}{1-\beta(\overline{\phi})}\right).$

(5.33b)

[68.1.1.5] The exponents $m^{\prime}_{\sigma}$ and $m^{\prime}_{p}$ characterize the behavior of $\varepsilon^{\prime}_{-}$ and $p$ as $\overline{\phi}\to 1$ , and $\beta(\overline{\phi})$ is the exponent governing $\lambda(\phi)\mu(\phi)$ as $\overline{\phi}\to 1$ . [68.1.1.6] The behavior predicted by eq. (5.32) might be experimentally observable in water-filled pore casts of systems obeying (5.27).

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