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V Direct Problem

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


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


for λϕ¯>13, and as


for λϕ¯<13. [] 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 (ω), the results are




for the real dielectric constant, and


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

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


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


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


and the effective-medium value pc=13 for the percolation thershold. [] The control parameter p is the total fraction of percolating local geometries.

[] For the effective real dielectric constant ε0, two equations are obtained. [] If p<pc, the equation reads


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




[] Equation (5.2) can be treated analogously to the case of ordinary percolation.[30, 31][] The effective conductivity σ0 will be small for ppc or whenever ϕ¯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 ϕ behavior of λϕμϕ. [] The following three cases for λϕμϕ must be distinguished:


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


where σ+ is defined by


[] The integral exists because of eq. (4.8) and condition (5.10a). [] Note that eq. (5.11) is valid for all p for ϕ¯0 and for all ϕ¯ for ppc. [] 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 λϕ and μϕ as long as condition (5.10a) remains fulfilled.

[] The situation is very different for case (c). [] 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


where the conductivity exponent


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


[] For the real dielectric constant, eq. (5.8) is valid below pc. [] This equation has the same form as eq. (5.6) for the conductivity. [] However, it is now the behavior of λϕ and μϕ that high porosities ϕ1 that is relevant. [] From eq. (5.8)


is obtained, where


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 ϕ¯1, and for all ϕ¯ whenever ppc. [] If 1-λϕμϕ1-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.

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


with the scaling function


for z1 and p>pc,


for z1 and p<pc, and


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


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


for p<pc, and


for p>pc.

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

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


for the conductivity, and


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




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




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


called “Archie’s law”, which is usually written in terms of the formation factor F=σ0-1. [] 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 m1 and m4. [] 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] [] 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 ϕ¯ 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 λϕ and μϕ are affected. [] This implies that σ+ and p become implicitly dependent upon ϕ¯, and consequently σ0 will change with ϕ¯. [] 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. [] Nevertheless, it is of interest to discuss the general consequences of the scaling approach presented above. [] Clearly, σ+ϕ¯ should tend to zero as ϕ¯0, and it should approach σW for ϕ¯1. [] It seems also plausible that pϕ¯ should decrease as ϕ¯ is lowered. [] If one assumes that σ+ϕ¯ and pϕ¯ can be expanded around ϕ¯=0 as


then eq. (5.19) implies that


[] This already resembles eq. (5.27). [] 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,


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


[page 68, §0]    [] Note that α and thus m may explicitly depend on the bulk porosity. [] 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. [] In such cases,


[] 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. [] 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. [] 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 limitϕ¯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


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


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


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