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# Definition and Justification of Mapping

One of the main criticisms of mercury porosimetry from its beginning has been that pores in a real porous medium are almost never circular cylinders [12]. The pore diameter d measured by mercury porosimetry is only an effective parameter that in some sense integrates over a pore's real shape. What then is the justification for using, in eq. (2), a value of dc that is based on the assumption of circular cylindrical pore geometry?

The answer to this question is as follows. Let us consider a mercury porosimetry measurement as the generator of a mapping between a real porous medium and an effective random network of randomly-connected circular cylindrical tubes. By this is meant that information about the real pore structure of a material is converted, via a mercury porosimetry measurement, into pore-size distribution information about a random network of circular cylindrical tubes. This is an experimentally-produced mapping, with some limitations. For example, a geometrical model of the effective tube network could not be constructed, based on the mercury porosimetry data, because this data does not contain complete connectivity or tube length information. However, a mapping does clearly exist, since the pore-size distribution function and the measured value of dc are generated by it, and indeed could not be interpreted without the implicit assumption of such a mapping. It is then always possible to define dc and carry out the Katz-Thompson permeability calculation on such a network [8]. Of course such mappings could be produced in other ways. A recent attempt was made, using image analysis, to define an effective tube network, and then calculate the permeability of the material by calculating the permeability of the tube network [13]. This mapping did not give permeabilities in quantitative agreement with experiment. Since the mercury porosimetry- generated mapping justifies the Katz-Thompson calculation, the question is then raised as to why this mapping is better than that of Ref. [13], for example. It is because the mercury porosimetry-generated effective network preserves the hydraulic conductivity of pores with elliptical cross-sections, as demonstrated below.

If a real pore is modelled by an elliptical cylinder with semi-major axis a, semi-minor axis b, and length lo, then the exact hydraulic conductance Kt of this cylinder is [14]

The parameter b can vary from b = 0 (crack) to b = a (circular cylinder), so that quite a wide range of pore shapes can be modelled in this way. The capillary equation can be solved for a tube with an elliptical cross-section, giving the equivalent of eq. (1) for the pressure P at which the mercury will enter the tube [15]:

By setting eq. (5) equal to eq. (1), with d replaced by de, an effective diameter, eq. (5) can be interpreted as defining a circular cylindrical tube with diameter de given by

The effective network mapping takes an elliptical cross-section pore, defined by a and b, into a circular cross-section pore with diameter de. It is obvious that pore-shape information is lost in this mapping, as many values of a and b will give the same value of de. The local property that one would most like to preserve is the value of Kt. The hydraulic conductance of the effective tube (also of length lo) is defined as [14]

Figure 1 shows Kt and Ke plotted as a function of b/a, where each quantity has been normalized to one at b/a = 1. The agreement is excellent over the whole range of 0 < b/a < 1. The electrical conductances in the conducting-fluid (conductivity o) filled pore are similarly given by t and e:

These quantities, normalized to one at b/a = 1, are also plotted in Fig. 1. The agreement between these two quantities is not quite as good as for the hydraulic conductances. However, most of the deviation occurs at the smaller values of b/a, where the conductances are individually small. It can then be concluded that it is possible to treat an mercury porosimetry measurement as defining an effective network of tubes, where the transport properties of each effective tube are closely matched to the real pore's transport properties, given that the real pore can be modelled as a tube with an elliptical cross-section.

Figure 1: True and effective electrical and hydraulic conductances for an elliptical cross-section tube (left). The second graph (right) shows a blow-up of the region near b/a = 0, for just the permeabilities, in order to show that they do indeed differ, but only for small values of b/a.

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