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# Example: Random Tube Network

The mercury porosimetry-generated mapping can be tested using a simple lattice model with uniform connectivity. Consider a face-centered cubic (fcc) lattice, a regular lattice with 12 nearest-neighbor bonds per node [16]. Other regular three-dimensional lattices could be also be used, like simple cubic and body-centered cubic [16], with qualitatively similar results. Each of the 12 nearest neighbour bonds is taken to be an elliptical cylindrical tube, with semi- major axis a and semi-minor axis b. The value of a is taken to be a constant for the lattice, while b ( 0 < b < a ) is taken to be a random variable, independent and different for each tube, with probability density f(b) = 1/a.

The quantity p(b) is defined as the fraction of tubes with semi-minor axes greater than or equal to a given value b, so that p(0) = 1, and p(a) = 0. The quantity bc is defined by the relation p(bc) = pc = 0.119, the threshold for bond percolation on the fcc lattice [17]. This means that all bonds with b > bc just form a connected pathway, since they make up the required fraction pc of the lattice necessary for a continuous pathway to exist. With the above functional form for f(b), it is evident that p(b) = 1 − b/a, so that bc = a(1 − pc) = 0.881a. Using the definition of the effective diameter de in eq. (5), the value of dc for this model is then dc = 4abc / (a+bc) = 1.873a.

With the use of effective medium theory, the macroscopic quantities k and σ can each be calculated analytically, via averages over f(b), with an accuracy of a few percent, thus avoiding the need for large numerical lattice simulations [18,19]. I will first calculate them using Kt and Σt, and then compare the result for the k/σ ratio with the same calculation done using Ke and Σe for the individual conductances. This will test how much the "exact" solution is affected by using the effective tube conductances instead of the true tube conductances. The effective medium equation is [18]

where g is the conductance (electrical or hydraulic) of an individual pore, gm is the solution to the equation and represents the effective medium single-pore conductance, β = 2/(z−2), where z = 12 is the number of nearest-neighbours for the fcc lattice, and the angular brackets indicate an average over the random values of the conductances. Once gm is determined, then the macroscopic quantities k or σ are found by multiplying the appropriate gm by a constant that depends on the lattice size and geometry, and is the same for both k and σ. Eq. (10) is simple to solve numerically, with the following results:

where the superscript indicates how the calculation was carried out (EMT = effective medium theory), and the subscript indicates what expression was used for the local tube conductances ("t" means eqs. (4) and (8), "e" means eqs. (7) and (9)). Thus using the effective local conductances instead of the true local conductances resulted in only a 12% difference in the k/σ ratio. This reflects a 1% difference in the KEMT calculation and a 11.6% difference in that for ΣEMT. The error is greater for the electrical conductance, as might have been expected from Fig. 1.

The next step is to evaluate the effect of using (Ke, Σe) vs. (Kt, Σt) in a Katz- Thompson calculation of k/σ. The main point is to determine what difference the effective mercury porosimetry network approximation has on the Katz-Thompson calculation, but the deviation from the "exact" effective medium theory result (k/σ)tEMT will also be discussed.

The Katz-Thompson calculation uses the quantity p(b) to construct lower bound functions for k and σ. Being lower bounds, these functions are then maximized to give the best results, and their ratio computed. The permeability and conductivity functions are given by [8]

where W is a constant involving lattice size and geomery parameters that relates the conductance (hydraulic or electrical) of individual tubes to the overall permeability or conductivity, t = 1.9 is the critical exponent for the percolation conduction problem in three dimensions [18,19], and K(b) = Kt or Ke, Σ(b)= Σt or Σe as defined in eqs. (3), (6), (7), and (8). The effective tubes have all been chosen to have the same length lo [8].

The next step is to maximize k(b) and σ (b) independently, first using the true local conductances and then repeating the calculation with the effective quantities. When the ratios of kmax and σmax are formed, the unknown constants W and lo drop out, leaving an expression proportional to dc2. The values of b that maximize the two functions will not be the same, in general. It should be noted that, since p(b) is known exactly in this model, the maximization of k(b) and σ(b) can also be carried out exactly. The algebra is simple, and will not be given here.

Using pc = 0.119 [17], the results are

where the superscript "K-T" indicates that the calculation was done with the Katz-Thompson method [3,4]. Eq. (13) shows that, whether one does the Katz-Thompson calculation on the real lattice or on the effective mercury porosimetry-generated network, there is only a 3% difference in the result for the k/σ ratio. One should notice that (k/σ)eK-T differs from (k/σ)tEMT by only 28%, which is in good agreement considering the typical 50% or more uncertainty in experimental measurements of k/σ [3,4]. Also, the numerical prefactor, 0.0138, for (k/σ)eK-T agrees within 17% with the prefactor, 0.0114, calculated for a general pore structure by Banavar and Johnson [8] by carrying out the Katz-Thompson calculation approximately, without using any knowledge of the (generally unknown) explicit form of p(b).

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