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# V Direct Problem

[64.2.4.1] Consider now the direct problem with and given. [64.2.4.2] The problem is to calculate . [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 , eq. (3.2) can be expanded around . [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 which is easily solved. [65.1.0.1] For low frequencies , one finds

 (5.1)

for , and , for . [65.1.0.2] Thus a percolation transition with control parameter is predicted for strongly peaked local porosity distributions. [65.1.0.3] Although the prediction of a percolation transition remains correct for arbitrary , the control parameter is in general not . [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

 (5.2a)

for , and as

 (5.2b)

for . [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 (), the results are

 (5.3a)

where

 (5.3b)

for the real dielectric constant, and

 (5.4)

for real part of the conductivity. [65.0.1.2] The sign in equation (5.3a) has to be chosen such that 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 . [65.0.2.2] But contrary to previous percolation theories for porous media, the bulk porosity is not the control parameter of the transition. [65.0.2.3] Therefore, a transition can or cannot occur as 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 leads to

 (5.5)

for the effective dc conductivity of the system. [65.0.3.3] Inserting equation (4.6) into (5.5),

 (5.6)

is found. [65.0.3.4] This equation for has no positive solutions if . [65.0.3.5] This identifies the control parameter for the transition as

 (5.7)

and the effective-medium value for the percolation thershold. [65.1.0.1] The control parameter is the total fraction of percolating local geometries.

[65.1.1.1] For the effective real dielectric constant , two equations are obtained. [65.1.1.2] If , the equation reads

 (5.8)

[65.1.1.3] For the solution to equation (5.6) must be inserted into

 (5.9)

where

[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 will be small for or whenever , 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 behavior of . [65.1.2.4] The following three cases for must be distinguished:

 (5.10a) (5.10b) (5.10c)

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

 (5.11)

where is defined by

 (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 for and for all for . [66.1.0.3] The conductivity exponent 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 does not depend on the specific geometry contained in and 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 and is given by eq. (4.8), then to leading order in the solution to eq. (5.6) is

 (5.13a)

where the conductivity exponent

 (5.13b)

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

 (5.14)

[66.1.2.1] For the real dielectric constant, eq. (5.8) is valid below . [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 and that high porosities that is relevant. [66.1.2.4] From eq. (5.8)

 (5.15)

is obtained, where

 (5.16)

analogous to eqs. (5.11) and (5.12). [66.1.2.5] Again, the expected value 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 if , and for all whenever . [66.1.2.7] If , with , then the superconductivity exponent becomes nonuniversal and has the value

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

[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 nor 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 () and in the high-porosity limit (). [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

 (5.17)

with the scaling function

 (5.18a)

for and ,

 (5.18b)

for and , and

 (5.18b)

for and all . [66.2.1.4] Here is the complex dielectric constant of a good conductor, and is the complex dielectric constant for the poor conductor. [66.2.1.5] is the volume fraction of good conductor, is the percolation threshold, and and are the conductivity and the superconductivity exponents. [66.2.1.6] For , eqs. (5.17) and (5.18) yield the well-known results

 (5.19)

for , and , for for the conductivity. [66.2.1.7] For the dielectric constant, one obtains

 (5.20a)

for , and

 (5.20b)

for .

[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 and 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 which obeys condition (5.10a). [66.2.2.4] Naturally, they are also valid whenever at finite .

[66.2.3.1] Consider now the case of finite frequencies . [66.2.3.2] The condition in eqs. (5.18) as always satisfied for [page 67, §0]    sufficiently small . [67.1.0.1] On the other hand, the condition is always if either or . [67.1.0.2] The latter condition does not apply for the systems considered in thes paper, and thus is always interpreted as . [67.1.0.3] Equations (5.17) and (5.18) imply, for the case ,

 (5.21)

for the conductivity, and

 (5.22)

for the dielectric constant, where . [67.1.0.4] For one obtains

 (5.23)

and

 (5.24)

where . [67.1.0.5] Finally, the case leads to

 (5.25)

and

 (5.26)

[67.1.0.6] These results predict a divergence of as with an exponent whenever the control parameter approaches criticality. [67.1.0.7] Simultaneously, the conductivity will also exhibit power-law behavior with exponent . [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:

 (5.27)

called “Archie’s law”, which is usually written in terms of the formation factor . [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 is found to scatter widely between and . [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 and as sufficient geometric information to predict , 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 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 and are affected. [67.2.1.6] This implies that and become implicitly dependent upon , and consequently will change with . [67.2.1.7] To discuss these changes one needs a physical model for the changes of and , 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, should tend to zero as , and it should approach for . [67.2.1.10] It seems also plausible that should decrease as is lowered. [67.2.1.11] If one assumes that and can be expanded around as

 (5.28) (5.29)

then eq. (5.19) implies that

 (5.30)

[67.2.1.12] This already resembles eq. (5.27). [67.2.1.13] In particular, if it happens that , i.e., if one approaches criticality as , then eq. (5.30) yields Archie’s law with a cementation exponent,

 (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

 (5.31b)

[page 68, §0]    [68.1.0.1] Note that and thus may explicitly depend on the bulk porosity. [68.1.0.2] Even more complicated results for obtained if and change with such that and . [68.1.0.3] In such cases,

 (5.31b)

[68.1.0.4] The surprising result is that the simplest form for , namely, eq. (5.31a), predicts an exponent in the range from for to for . [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 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. [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

 (5.32)

[68.1.1.4] Here the “dilution exponent” is given in the simplest case as

 (5.33a)

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

 (5.33b)

[68.1.1.5] The exponents and characterize the behavior of and as , and is the exponent governing as . [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).