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# V Direct Problem

[64.2.4.1] Consider now the direct problem with λϕ and μϕ given. [64.2.4.2] The problem is to calculate εω. [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 μϕ, eq. (3.2) can be expanded around ϕ¯. [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 εω which is easily solved. [65.1.0.1] For low frequencies ω0, one finds

 σ′⁢0=12⁢σC′⁢0;ϕ¯⁢3⁢λ⁢ϕ¯-1, (5.1)

for λϕ¯>13, and σ0=0, for λϕ¯13. [65.1.0.2] Thus a percolation transition with control parameter λϕ¯ is predicted for strongly peaked local porosity distributions. [65.1.0.3] Although the prediction of a percolation transition remains correct for arbitrary μϕ, the control parameter is in general not λϕ¯. [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

 ε′⁢0=12⁢3⁢λ⁢ϕ¯-1⁢εC′⁢0;ϕ¯+9⁢λ⁢ϕ¯⁢λ⁢ϕ¯-1⁢εB′⁢0;ϕ¯2⁢3⁢λ⁢ϕ¯-1, (5.2)

for λϕ¯>13, and as

 ε′⁢0=εB′⁢0;ϕ¯1-3⁢λ⁢ϕ¯, (5.2)

for λϕ¯<13. [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 (ω), the results are

 ε′⁢∞=B4⁢1±1+8⁢εC′⁢∞;ϕ¯⁢εB′⁢∞;ϕ¯B21/2, (5.3)

where

 B=3⁢λ⁢ϕ¯-1⁢εC′⁢∞;ϕ¯+εB′⁢∞;ϕ¯⁢2-3⁢λ⁢ϕ¯, (5.3)

for the real dielectric constant, and

 σ′⁢∞=λ⁢ϕ¯⁢σC′⁢∞;ϕ¯⁢ε′⁢∞λ⁢ϕ¯⁢εC′⁢∞;ϕ¯+1-λ⁢ϕ¯⁢εB′⁢∞;ϕ¯⁢εC′⁢∞;ϕ¯+2⁢ε′⁢∞εB′⁢∞;ϕ¯+2⁢ε′⁢∞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 σ 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 μϕ. [65.0.2.2] But contrary to previous percolation theories for porous media, the bulk porosity ϕ¯ is not the control parameter of the transition. [65.0.2.3] Therefore, a transition can or cannot occur as ϕ¯ 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 ω=0 leads to

 ∫01σC′⁢0;ϕ-σ′⁢0σC′⁢0;ϕ-2⁢σ′⁢0⁢λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ+∫01σB′⁢0;ϕ-σ′⁢0σB′⁢0;ϕ-2⁢σ′⁢0⁢1-λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ=0, (5.5)

for the effective dc conductivity σ0 of the system. [65.0.3.3] Inserting equation (4.6) into (5.5),

 σ′⁢0⁢∫01λ⁢ϕ⁢μ⁢ϕσC′⁢0;ϕ+2⁢σ′⁢0⁢d⁢ϕ=-16+12⁢∫01λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ (5.6)

is found. [65.0.3.4] This equation for σ0 has no positive solutions if 01λϕμϕdϕ<13. [65.0.3.5] This identifies the control parameter for the transition as

 p=∫01λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ, (5.7)

and the effective-medium value pc=13 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 ε0, two equations are obtained. [65.1.1.2] If p<pc, the equation reads

 ε′⁢0-1⁢∫011-λ⁢ϕ⁢μ⁢ϕεB′⁢0;ϕ-1+2⁢ε′⁢0-1⁢d⁢ϕ=2⁢pc-p. (5.8)

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

 ε′⁢0=A+BC, (5.9)

where

 A=4⁢σ′⁢0-1⁢∫01εB′⁢0;ϕ⁢1-λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ, B=∫01εC′⁢0;ϕ⁢σ′⁢0σC′⁢0;ϕ+2⁢σ′⁢02⁢λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ, C=∫01σC′⁢0;ϕσC′⁢0;ϕ+2⁢σ′⁢02⁢λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ.

[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 σ0 will be small for ppc or whenever ϕ¯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 ϕ behavior of λϕμϕ. [65.1.2.4] The following three cases for λϕμϕ must be distinguished:

 (a)⁢∫01ϕ-1⁢λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ<∞, (5.10a) (b)⁢λ⁢ϕ⁢μ⁢ϕ→const⁢for⁢ϕ→0, (5.10b) (c)⁢λ⁢ϕ⁢μ⁢ϕ∝ϕ-α⁢for⁢ϕ→0⁢and⁢0<α<1. (5.10c)

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

 σ′⁢0≈σ+′⁢p-pc, (5.11)

where σ+ is defined by

 1σ+′=∫01λ⁢ϕ⁢μ⁢ϕσC′⁢0;ϕ⁢d⁢ϕ. (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 ϕ¯0 and for all ϕ¯ for ppc. [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 λϕ and μϕ 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 λϕμϕMϕ-α and σC0;ϕ is given by eq. (4.8), then to leading order in σ the solution to eq. (5.6) is

 σ′⁢0∝C1⁢1-αM⁢sin⁡π⁢α⁢p-pct, (5.13)

where the conductivity exponent

 t=11-α (5.13)

is now no longer universal. [66.1.1.3] It depends on the behavior of λϕμϕ for small ϕ. [66.1.1.4] For the marginal case α=0 [case (b)], logarithmic corrections to eq. (5.11) are obtained, and

 σ′⁢0∝p-pc⁢log⁡1p-pc-1. (5.14)

[66.1.2.1] For the real dielectric constant, eq. (5.8) is valid below pc. [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 λϕ and μϕ that high porosities ϕ1 that is relevant. [66.1.2.4] From eq. (5.8)

 ε′⁢0≈ε-′⁢p-pc-1 (5.15)

is obtained, where

 ε-′=∫01εB′⁢0;ϕ⁢1-λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ, (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 ϕ¯1, and for all ϕ¯ whenever ppc. [66.1.2.7] If 1-λϕμϕ1-ϕ-β, with 0<β<1, then the superconductivity exponent becomes nonuniversal and has the value

 s=11-β,

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 ϕ¯ nor λϕ¯ 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 (ϕ¯0) and in the high-porosity limit (ϕ¯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

 ε=ε+⁢p-pct⁢f⁢ε-/ε+p-pct+s, (5.17)

with the scaling function

 f⁢z=A0I+A1I⁢z+A2I⁢z2+…, (5.18)

for z1 and p>pc,

 f⁢z=A1II⁢z+A2II⁢z2+…, (5.18)

for z1 and p<pc, and

 f⁢z=AIII⁢zt/t+s+…, (5.18)

for z1 and all p. [66.2.1.4] Here ε+ is the complex dielectric constant of a good conductor, and ε- is the complex dielectric constant for the poor conductor. [66.2.1.5] p is the volume fraction of good conductor, pc is the percolation threshold, and t and s are the conductivity and the superconductivity exponents. [66.2.1.6] For ω0, eqs. (5.17) and (5.18) yield the well-known results

 σ′⁢0=A0I⁢σ+′⁢p-pct+…, (5.19)

for p>pc, and σ0=0, for ppc for the conductivity. [66.2.1.7] For the dielectric constant, one obtains

 ε′⁢0=A1II⁢ε-′⁢p-pc-s+…, (5.20)

for p<pc, and

 ε′⁢0=A0I⁢ε+′⁢p-pct+A1I⁢ε-′⁢p-pc-s+…, (5.20)

for p>pc.

[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 σ+ and ε- 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 λϕμϕ which obeys condition (5.10a). [66.2.2.4] Naturally, they are also valid whenever ppc at finite ϕ¯.

[66.2.3.1] Consider now the case of finite frequencies ω0. [66.2.3.2] The condition z1 in eqs. (5.18) as always satisfied for [page 67, §0]    sufficiently small ω. [67.1.0.1] On the other hand, the condition z1 is always if either ppc or ε-/ε+1. [67.1.0.2] The latter condition does not apply for the systems considered in thes paper, and thus z1 is always interpreted as ppc. [67.1.0.3] Equations (5.17) and (5.18) imply, for the case p<pc,

 σ′⁢ω=-A2II⁢σ+′⁢p-pc-2⁢s-t⁢ω2+…, (5.21)

for the conductivity, and

 ε′⁢ω=A1II⁢ε-′⁢p-pc-s+A2II⁢ε-′⁢p-pc-2⁢s-t⁢ε-′ε+′⁢ωω+2+…, (5.22)

for the dielectric constant, where ω+=σ+/ε+. [67.1.0.4] For ppc one obtains

 σ′⁢ω=AIII⁢σ+′⁢ε-′ε+′t/t+s⁢ωω+t/t+s×cos⁡φ⁢tt+s+ωω+⁢sin⁡φ⁢tt+s+… (5.23)

and

 ε′⁢ω=AIII⁢ε+′⁢ε-′ε+′t/t+s⁢ω+ωs/t+s×sin⁡-φ⁢tt+s+ωω+⁢cos⁡φ⁢tt+s+…, (5.24)

where φ=argε-/ε+. [67.1.0.5] Finally, the case p>pc leads to

 σ′⁢ω=σ+′⁢A0I⁢p-pct-A2I⁢p-pc-2⁢s-t⁢ω2+… (5.25)

and

 ε′⁢ω=A0I⁢ε+′⁢p-pct+A1I⁢ε-′⁢p-pc-s+A2I⁢ε-′⁢p-pc-2⁢s-t⁢ε-′ε+′⁢ωω+2+…. (5.26)

[67.1.0.6] These results predict a divergence of εω as ω0 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:

 σ′⁢0∝ϕ¯m (5.27)

called “Archie’s law”, which is usually written in terms of the formation factor F=σ0-1. [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 m1 and m4. [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 ϕ¯ and m as sufficient geometric information to predict σ, 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 ϕ¯ 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 λϕ and μϕ are affected. [67.2.1.6] This implies that σ+ and p become implicitly dependent upon ϕ¯, and consequently σ0 will change with ϕ¯. [67.2.1.7] To discuss these changes one needs a physical model for the changes of α and μ, 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, σ+ϕ¯ should tend to zero as ϕ¯0, and it should approach σW for ϕ¯1. [67.2.1.10] It seems also plausible that pϕ¯ should decrease as ϕ¯ is lowered. [67.2.1.11] If one assumes that σ+ϕ¯ and pϕ¯ can be expanded around ϕ¯=0 as

 σ+′⁢ϕ¯ =σ˙+′⁢0⁢ϕ¯+12⁢σ¨+′⁢0⁢ϕ¯2+…, (5.28) p⁢ϕ¯ =p⁢0+p˙⁢0⁢ϕ¯⁢0⁢ϕ¯+…, (5.29)

then eq. (5.19) implies that

 σ′⁢ϕ¯=A0I⁢σ˙+′⁢0⁢ϕ¯⁢p⁢0-pc+p˙⁢0⁢ϕ¯+…t. (5.30)

[67.2.1.12] This already resembles eq. (5.27). [67.2.1.13] In particular, if it happens that p0pc, i.e., if one approaches criticality as ϕ¯0, then eq. (5.30) yields Archie’s law with a cementation exponent,

 m=1+t, (5.31)

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+α⁢ϕ¯1-α⁢ϕ¯. (5.31)

[page 68, §0]    [68.1.0.1] Note that α and thus m may explicitly depend on the bulk porosity. [68.1.0.2] Even more complicated results for m obtained if λϕ and μϕ change with ϕ¯ such that σ+ϕ¯ϕ¯mσ and pϕ¯pc+p˙0ϕ¯mp. [68.1.0.3] In such cases,

 m=mσ+mp⁢t+α⁢ϕ¯1-α⁢ϕ¯. (5.31)

[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 ppc to m=1+t3 for ppc. [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 σ0;ϕ¯ 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 limitϕ¯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

 ε′⁢0∝1-ϕ¯-m′. (5.32)

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

 m′=1+s, (5.33)

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

 m′=mσ′+mp′⁢s+β⁢ϕ¯1-β⁢ϕ¯. (5.33)

[68.1.1.5] The exponents mσ and mp characterize the behavior of ε- and p as ϕ¯1, and βϕ¯ is the exponent governing λϕμϕ as ϕ¯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).