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The present model for interfacial transport can, approximately, be extended to include fluid permeability. The correct way to calculate the permeability of a mortar would be to solve the Navier-Stokes equations for the random pore space [35,36], which would include bulk cement paste pores as well as interfacial cement paste pores. However, since we are considering mortar at the sub-millimeter scale in this paper, we can use Darcy's law , with the appropriate permeabilities, for the three component composite: sand (Ka ), interfacial zone cement paste (Ks), and bulk cement paste (Kp). Darcy's law for a system with spatially varying permeability is:
where V(r) is the macroscopic fluid velocity, K(r) is the permeability, P(r) is the pressure at a vector position r, and η is the fluid viscosity. If we identify V(r) with j(r), the electrical current density, K(r)/η with the electrical conductivity, and P(r) with the electrostatic potential, then this equation reduces to the equation for steady-state electrical current flow. The boundary conditions for the two problems are also identical, so that all the results we have obtained for electrical conductivity can be re-interpreted for fluid permeability, albeit approximately.
Within this framework, an essential step is to estimate the value of K s / Kp, the parameter analogous to σs / σp . Here we propose to use the Katz-Thompson equation , which predicts the permeability of a porous medium in terms of its electrical conductivity and a critical pore radius characteristic of the largest connected pores in the material as defined by a mercury intrusion experiment. The equation derived in Ref.  has been shown to work reasonably well on cement-based materials [38,39]. Neglecting constants of proportionality, the relevant equation is K ~ d2 / F, where F is the formation factor defined in Eq. (17) and d is the critical pore diameter. If we assume that the value of d for interfacial zone cement paste is about 10 times larger than that for the bulk cement paste, in rough agreement with the available mercury intrusion data , and take the interfacial zone conductivity to be about 10 times larger than that of the bulk cement paste, as suggested by recent experiments on synthetic interfaces , the resulting estimate is Ks / Kp approximately equals 1000. The largest value of σs / σp computed in Fig. 8 was only 50, but we can use either the fitted Pade approximant or the two effective medium theories to obtain the result K/Kp = 35 for the effective permeability ratio. Data in Ref.  indicate that the permeabilities of mortars with about 50% sand concentration are between 20 and 60 times higher than that of the bulk cement paste, in agreement with the above estimate.