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V.D Permeability Length Scales

The fact that the effective permeability \overline{k} has dimensions of area raises the question whether \sqrt{\overline{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

\overline{k}\propto\frac{\overline{\phi}R_{H}^{2}}{2\mathcal{T}^{2}} (5.98)

where R_{H}=\phi(\mathbb{S})/S(\mathbb{S}) is the hydraulic radius. This generalization has been modified by incorporating the formation factor to write [327, 112]

\overline{k}\propto\frac{\Lambda^{2}}{F} (5.99)

where the length scale \Lambda=R_{H} is still given by the hydraulic radius, and the geometrical tortuosity \mathcal{T}^{2} was replaced by the electrical tortuosity defined as \mathcal{T}_{{el}}^{2}=F\overline{\phi}. Because the length scale is still given by the hydraulic radius this theory is still faced with the objection that the hydraulic radius R_{H} contains contributions from the dead ends which do not contribute to the transport.

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

\frac{2}{\Lambda}=\frac{\int|{\bf E}({\bf r})|^{2}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\partial\mathbb{P}}}({\bf r})\; d^{2}{\bf r}}{\int|{\bf E}({\bf r})|^{2}\chi\rule[-4.3pt]{0.0pt}{8.6pt}_{{\mathbb{P}}}({\bf r})\; d^{3}{\bf r}} (5.100)

where {\bf E}({\bf r}) 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 \Lambda within local porosity theory is obtained by eliminating (p-p_{c}) between the result (5.48) for the conductivity, and equation (5.95) for the permeability. This yields in general

\Lambda^{2}\approx\frac{\displaystyle\int _{0}^{\infty}\int _{0}^{1}\frac{\lambda(\phi,S;\mathbb{K})\mu(\phi,S;\mathbb{K})}{\sigma\rule[-4.3pt]{0.0pt}{6.45pt}_{{loc}}(\phi,S)}\; d\phi dS}{\displaystyle\int _{0}^{\infty}\int _{0}^{1}\frac{\lambda(\phi,S;\mathbb{K})\mu(\phi,S;\mathbb{K})}{k_{{loc}}(\phi,S)}\; d\phi dS} (5.101)

where \sigma\rule[-4.3pt]{0.0pt}{6.45pt}_{{loc}}(\phi,S) and k_{{loc}}(\phi,S) are the local electrical conductivity and the local permeability. Thus \Lambda involves macroscopic geometrical information through \mu and \lambda and microscopic dynamical and geometrical information through the local transport coefficients. If one assumes the hydraulic radius expressions \sigma\rule[-4.3pt]{0.0pt}{6.45pt}_{{loc}}(\phi,S)\propto\phi and k_{{loc}}(\phi,S)\propto\phi^{3}/S^{2} locally and the expression \mu(\phi,S;\mathbb{K})\approx\delta(\phi-\overline{\phi)}\delta(S-\overline{S}) valid for large measurment cells, then it follows that \Lambda\propto\overline{\phi}/\overline{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.