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V.D Permeability Length Scales

The fact that the effective permeability k¯ has dimensions of area raises the question whether k¯ has an interpretation as a length scale. The traditional answer to this question is provided by hydraulic radius theory which uses the approximate result (5.61) for the capillary tube model to postulate more generally the relation

k¯ϕ¯RH22𝒯2 (5.98)

where RH=ϕ𝕊/S𝕊 is the hydraulic radius. This generalization has been modified by incorporating the formation factor to write [327, 112]

k¯Λ2F (5.99)

where the length scale Λ=RH is still given by the hydraulic radius, and the geometrical tortuosity 𝒯2 was replaced by the electrical tortuosity defined as 𝒯el2=Fϕ¯. Because the length scale is still given by the hydraulic radius this theory is still faced with the objection that the hydraulic radius RH contains contributions from the dead ends which do not contribute to the transport.

An alternative was proposed in [318, 43]. It postulates Λ=lc where lc is a length scale related to the breakthrough pressure in mercury injection experiments. The length scale lc is well defined for network models with a broad distribution of cylindrical pores. A dynamical interpretation of Λ was proposed in [319, 320, 328] as

2Λ=Er2χrd2rEr2χrd3r (5.100)

where Er is the unknown exact solution of the microscopic dielectric problem. This “electrical length” is expected to measure, somehow, the “dynamically connected pore size [319, 328, 4]. The interpretation of Λ within local porosity theory is obtained by eliminating p-pc between the result (5.48) for the conductivity, and equation (5.95) for the permeability. This yields in general

Λ2001λϕ,S;𝕂μϕ,S;𝕂σlocϕ,SdϕdS001λϕ,S;𝕂μϕ,S;𝕂klocϕ,SdϕdS (5.101)

where σlocϕ,S and klocϕ,S are the local electrical conductivity and the local permeability. Thus Λ involves macroscopic geometrical information through μ and λ and microscopic dynamical and geometrical information through the local transport coefficients. If one assumes the hydraulic radius expressions σlocϕ,Sϕ and klocϕ,Sϕ3/S2 locally and the expression μ(ϕ,S;𝕂)δ(ϕ-ϕ)¯δ(S-S¯) valid for large measurment cells, then it follows that Λϕ¯/S¯ becomes the local hydraulic radius [170]. This expression is no longer proportional to the total internal surface but only to the average local internal surface, and thus the argument against hydraulic radius theories no longer apply.