## Fluid permeability

It is interesting to try to extend the modelling work to include fluid permeability. This can only be done approximately, however, via the following analogy. The correct way to calculate the permeability of a mortar would be to solve the Navier-Stokes equations [47] for the random pore space, 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 phase permeabilities, for the three component composite: sand (Ka), interfacial zone cement paste (Ks), and bulk cement paste (Kp). Darcy's law for a spatially varying permeability is:

where v is the fluid velocity, K is the permeability and P is the pressure at a position r, and η is the fluid viscosity. If we identify v with j, the electrical current density, −K/η with the electrical conductivity, and P with the electrical voltage, then this equation is identical with the equation for steady-state current flow, with the same boundary conditions, so that all the results we have obtained for electrical conductivity can be re-interpreted for permeability, albeit approximately. (A recent paper has shown that the correct equation to be used in this case is Brinkman's equation, but that Darcy's law should be a reasonably good approximation [47]).

If Darcy's law is used, the next question to be raised is: what is the effective value of Ks/Kp, the key parameter analogous to σs / σp , to be used? One way to estimate this quantity is to make use of the Katz-Thompson (K-T) equation, which predicts the permeability of a porous medium in terms of its conductivity and a critical pore radius characteristic of the largest connected pores in the material, defined by a mercury intrusion experiment [1,48]. The K-T equation has recently been shown to work reasonably well on cement-based materials [45,49,50], although using conductivities estimated from mercury injection curves did not work as well as when directly measured conductivities were used [51]. The K-T equation also seems to work better for w/c ratios greater than about 0.5 than for values less than about 0.4 [45,49]. Neglecting constants of proportionality, the K-T equation is

where σ / σo is the conductivity of the porous material relative to the conductivity σo of the conductive phase it contains, and d is the critical pore diameter. If we assume that the value of d for interfacial zone cement paste is about 10 times as large as that for the bulk cement paste, in rough agreement with the available mercury intrusion evidence [15], and take the interfacial zone conductivity to be about 10 times larger than that of the bulk cement paste, in rough agreement with experiments on flat aggregates, then we would expect that the ratio Ks/Kp would be about 1000.

The largest value of Ks/Kp or σs / σp computed in Fig. 7 was only 50, but we can use the fitted Pade approximant, which should be more accurate in this limit, to estimate the overall conductivity or permeability to be about 35 times that of the bulk cement paste. Data in Ref. [45] indicates that the permeabilities of mortars with about 50% sand are about 20-60 times higher than the equivalent cement paste mortar, again in rough agreement with the model prediction.

The above analyses of the available experimental data in light of the model predictions should serve to motivate more experimental work, on mortars with a wide range of sand fractions, including the dilute limit of a few percent sand volume fraction. Generally it appears that impedance spectroscopy should be used to measure the conductivity, as this is easier than making direct measurements of the diffusivity. Conductivity can then be related to diffusivity via the Nernst-Einstein relation [2].

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