Sie sind hier: ICP » R. Hilfer » Publikationen

V Direct Problem

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

A Solution for strongly peaked local porosity distributions

[] For strongly peaked \mu(\phi), eq. (3.2) can be expanded around \overline{\phi}. [] 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. [] 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}. [] Thus a percolation transition with control parameter \lambda(\overline{\phi}) is predicted for strongly peaked local porosity distributions. [] Although the prediction of a percolation transition remains correct for arbitrary \mu(\phi), the control parameter is in general not \lambda(\overline{\phi}). [] This will be seen in the next subsection. [] 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}. [] 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.

[] 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)


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. [] The sign in equation (5.3a) has to be chosen such that \sigma^{\prime} remains positve.

B Low-frequency limit

[] 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). [] But contrary to previous percolation theories for porous media, the bulk porosity \overline{\phi} is not the control parameter of the transition. [] Therefore, a transition can or cannot occur as \overline{\phi} is varied.

[] To identify the percolation transition, consider the low-frequency limit. [] 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. [] 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. [] This equation for \sigma^{\prime}(0) has no positive solutions if \int _{0}^{1}\lambda(\phi)\mu(\phi)\,\mathrm{d}\phi<\tfrac{1}{3}. [] 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. [] The control parameter p is the total fraction of percolating local geometries.

[] For the effective real dielectric constant \varepsilon^{\prime}(0), two equations are obtained. [] 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)

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

\varepsilon^{\prime}(0)=\frac{A+B}{C}, (5.9)


\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.

[] Equation (5.2) can be treated analogously to the case of ordinary percolation.[30, 31][] 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. [] 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). [] 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)

[] [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)

[] The integral exists because of eq. (4.8) and condition (5.10a). [] 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}. [] The conductivity exponent t=1 has its expected reflective-medium value. [] 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.

[] The situation is very different for case (c). [] 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. [] It depends on the behavior of \lambda(\phi)\mu(\phi) for small \phi. [] 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)

[] For the real dielectric constant, eq. (5.8) is valid below p_{c}. [] This equation has the same form as eq. (5.6) for the conductivity. [] However, it is now the behavior of \lambda(\phi) and \mu(\phi) that high porosities \phi\to 1 that is relevant. [] 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). [] Again, the expected value s=1 for the superconductivity exponent is universal as long as the integral in eq. (5.16) exists. [] The asymptotic solution is valid for all p if \overline{\phi}\to 1, and for all \overline{\phi} whenever p\to p_{c}. [] If [1-\lambda(\phi)]\mu(\phi)\propto(1-\phi)^{{-\beta}}, with 0<\beta<1, then the superconductivity exponent becomes nonuniversal and has the value


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

[] The central result of this section is the identification of a percolation transition underlying the random geometry of porous media. [] 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. [] 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). [] 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

[] In this section the scaling theory for the percolation transition[32, 33] is applied in the present context.[34] [] Therefore, the present section goes beyond the effective-medium equation (3.2). [] 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. [] Here \varepsilon _{+} is the complex dielectric constant of a good conductor, and \varepsilon _{-} is the complex dielectric constant for the poor conductor. [] 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. [] 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. [] 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}.

[] For one-dimensional systems, the effective-medium approximation is known to be asymptotically exact for class (a) distributions.[31][] 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). [] 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). [] Naturally, they are also valid whenever p\to p_{c} at finite \overline{\phi}.

[] Consider now the case of finite frequencies \omega\neq 0. [] The condition \lvert z\rvert\ll 1 in eqs. (5.18) as always satisfied for [page 67, §0]    sufficiently small \omega. [] 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. [] 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}. [] 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}_{+}. [] 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)


\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 _{+}). [] 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)


\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)

[] 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. [] Simultaneously, the conductivity will also exhibit power-law behavior with exponent t/(t+s). [] Outside the critical region, the frequency dependency is quadratic.

D Archie’s law

[] 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}}. [] 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. [] Correspondingly, the cementation exponent m is found to scatter widely between m\approx 1 and m\approx 4. [] 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] [] 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. [] Let me explain this in more detail.

[] Sedimentary and related rocks arise from sedimentation and subsequent compactification, cementation, and other physicochemical processes. [] The bulk porosity \overline{\phi} changes during the sedimentation history of the rock. [] 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. [] The diagenetic processes change the local dielectric and geometric properties. [] Within the present formulation, it might be assumed that primarily \lambda(\phi) and \mu(\phi) are affected. [] This implies that \sigma^{\prime}_{+} and p become implicitly dependent upon \overline{\phi}, and consequently \sigma^{\prime}(0) will change with \overline{\phi}. [] 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. [] Nevertheless, it is of interest to discuss the general consequences of the scaling approach presented above. [] 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. [] It seems also plausible that p(\overline{\phi}) should decrease as \overline{\phi} is lowered. [] 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)

[] This already resembles eq. (5.27). [] 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. [] 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]    [] Note that \alpha and thus m may explicitly depend on the bulk porosity. [] 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}}}. [] In such cases,

m=m_{\sigma}+m_{p}\left(t+\frac{\alpha(\overline{\phi})}{1-\alpha(\overline{\phi})}\right). (5.31b)

[] 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}. [] 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. [] 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.

[] 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. [] This is a consequence of the thin-plate effect [eq. (4.9)] and analogous assumptions about the corresponding dilution process. [] More precisely, it is predicted that

\varepsilon^{\prime}(0)\propto(1-\overline{\phi})^{{-m^{\prime}}}. (5.32)

[] 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)

[] 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. [] The behavior predicted by eq. (5.32) might be experimentally observable in water-filled pore casts of systems obeying (5.27).