V Direct Problem

[64.2.5.1] For strongly peaked μ⁢ϕ, 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 ε⁢ω which is easily solved. [65.1.0.1] For low frequencies ω→0, one finds

for λ⁢ϕ¯>13, and σ′⁢0=0, for λ⁢ϕ¯≤13. [65.1.0.2] Thus a percolation transition with control parameter λ⁢ϕ¯ is predicted for strongly peaked local porosity distributions. [65.1.0.3] Although the prediction of a percolation transition remains correct for arbitrary μ⁢ϕ, the control parameter is in general not λ⁢ϕ¯. [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

for λ⁢ϕ¯>13, and as

for λ⁢ϕ¯<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

where

for the real dielectric constant, and

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.

[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 μ⁢ϕ. [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

for the effective dc conductivity σ′⁢0 of the system. [65.0.3.3] Inserting equation (4.6) into (5.5),

is found. [65.0.3.4] This equation for σ′⁢0 has no positive solutions if ∫01λ⁢ϕ⁢μ⁢ϕ⁢d⁢ϕ<13. [65.0.3.5] This identifies the control parameter for the transition as

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 ε′⁢0, two equations are obtained. [65.1.1.2] If p<pc, the equation reads

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

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 σ′⁢0 will be small for p→pc 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 λ⁢ϕ⁢μ⁢ϕ. [65.1.2.4] The following three cases for λ⁢ϕ⁢μ⁢ϕ must be distinguished:

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

where σ+′ is defined by

[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 p→pc. [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 λ⁢ϕ and μ⁢ϕ 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 λ⁢ϕ⁢μ⁢ϕ∝M⁢ϕ-α and σC′⁢0;ϕ is given by eq. (4.8), then to leading order in σ′ the solution to eq. (5.6) is

where the conductivity exponent

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

[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 λ⁢ϕ and μ⁢ϕ that high porosities ϕ→1 that is relevant. [66.1.2.4] From eq. (5.8)

is obtained, where

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 p→pc. [66.1.2.7] If 1-λ⁢ϕ⁢μ⁢ϕ∝1-ϕ-β, with 0<β<1, then the superconductivity exponent becomes nonuniversal and has the value

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 λ⁢ϕ¯ 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.

[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

with the scaling function

for z≪1 and p>pc,

for z≪1 and p<pc, and

for z≫1 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 ω→0, eqs. (5.17) and (5.18) yield the well-known results

for p>pc, and σ′⁢0=0, for p≤pc for the conductivity. [66.2.1.7] For the dielectric constant, one obtains

for p<pc, and

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 λ⁢ϕ⁢μ⁢ϕ which obeys condition (5.10a). [66.2.2.4] Naturally, they are also valid whenever p→pc at finite ϕ¯.

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

for the conductivity, and

for the dielectric constant, where ω+=σ+′/ε+′. [67.1.0.4] For p≈pc one obtains

and

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

and

[67.1.0.6] These results predict a divergence of ε′⁢ω as ω→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.

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

called “Archie’s law”, which is usually written in terms of the formation factor F=σ′⁢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≈1 and m≈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 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 λ⁢ϕ and μ⁢ϕ are affected. [67.2.1.6] This implies that σ+′ and p become implicitly dependent upon ϕ¯, and consequently σ′⁢0 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 p⁢ϕ¯ should decrease as ϕ¯ is lowered. [67.2.1.11] If one assumes that σ+′⁢ϕ¯ and p⁢ϕ¯ can be expanded around ϕ¯=0 as

then eq. (5.19) implies that

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

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

[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 λ⁢ϕ and μ⁢ϕ change with ϕ¯ such that σ+′⁢ϕ¯∝ϕ¯mσ and p⁢ϕ¯∝pc+p˙⁢0⁢ϕ¯mp. [68.1.0.3] In such cases,

[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≫pc to m=1+t≈3 for p≈pc. [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 σ′⁢0;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

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

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

[68.1.1.5] The exponents mσ′ and mp′ characterize the behavior of ε-′ and p as ϕ¯→1, and β⁢ϕ¯ is the exponent governing λ⁢ϕ⁢μ⁢ϕ 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).