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The utility of using electrical measurements to measure fluid permeability is unquestionable, as elecrical measurements are usually easier and faster to make, as long as the value of σo can be readily determined. For sedimentary rocks, the pore fluid is injected, so σo is predetermined, while for cement-based materials, for example, the conductivity of the pore fluid is determined by hydration chemistry [28] and is not so easily determined.
The question that this and other recent papers have focused on is the validity of the basic concept of relating electrical conductivities and fluid permeability quantities. The principal contribution of the present work is that all relevant quantities have been computed on a truly random continuum system, with a reasonably large range of pore sizes. In Ref. [5], the 2D tortuous model considered was constructed such that all pore channels had roughly the same size, so that the electrical current and fluid flow fields looked qualitatively similar. This is a special case. When there is a size distribution of necks through which the flow must go, as is typical in most real porous materials, then, as was shown in Sec. 6, the fluid velocity fields and the electric fields can sample the pore space quite differently. The results of this paper demonstrate the important effect that spatial randomness in the pore space has on the two different flow problems. The periodicity of the models considered in Refs. [5] and [8] forces the flow to be one-dimensional, so that all the flow must go through the narrowest neck. However, in a random pore structure, with a distribution of neck sizes, the flow will tend to go more through the largest necks, decreasing the importance of the narrowest necks.
In summary then, we have computed, for a 2D continuum random porous system with a reasonably wide pore size distribution, the permeability k, the electrical conductivity σ, and three pore-space-based length scales: Λ, h, and dc. We have shown that for over two decades in permeability, both Λ and dc, in conjunction with the electrical conductivity, gave reasonably constant scaling of the permeability. Study of the fluid velocity and electrical current flow fields via gray-scale images revealed significant qualitative differences between the two kinds of flow. However, a constant ratio between the effective porosities dynamically sampled by the electric and fluid flow fields, as well as the proportionality between length scales derived separately from two-point electric field (Λ) and fluid velocity (Lv) correlation functions, lend support to the idea that there may be a deeper connection between the two problems. As to the naure of this connection, we believe that further progress in this area requires more intense theoretical analysis of the basic equations of electrical and fluid flow in a porous medium, similar to work carried out recently by Torquato [6].