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# Two Phase Random Walk Calculations

In all random walk calculations, our objective is to estimate the mean squared displacement, < r2(t) >, of a collection of walkers as a function of time, t. At long times this quantity varies as < r2(t) > ~ Dt, where D is the effective diffusion coefficient of the pore space only. D is related to the measured diffusion coefficient of the entire sample, DT, which is referred to the entire cross-sectional area of the sample, by a factor of porosity (DT = φD). The conductivity is then given by

where σp and Dp are the pure phase coefficients of phase p.

Consider a two phase medium in which the conductivity in one phase is unity and in the other phase is h < 1. The simplest random walk is sometimes referred to as a blind walk. Within a given phase, the walker takes steps of length ε, but the time increment associated each step is inversely proportional to the conductivity. Operationally, in the high conductivity phase the walker takes steps of length ε and the clock advances by a unit increment, τ. [If the proposed step would take the walker into an insulating region (e.g. a sand grain) then the walker is returned to the attempt position but the clock is advanced by τ.] In the low conductivity phase the step length is again ε but the walker accepts each proposed step only with probability h, thus spending a certain fraction of the time simply standing still. If walker steps from the high to the low conducting phase (or vice versa) the probability for accepting the step is 2h/(1+h), as if the corresponding bond was a series connection between high and low conductivity bonds. This approach is straightforward although computationally rather inefficient because the walkers spend a certain amount of their time at rest. In Fig. 10 we compare the results of blind random walk calculations with finite element calculations for the bcc lattice described in Section 3.1, where a is of the order of the unit cell length. Generally the agreement is quite good, although the differences between the two techniques can be as large as 10 %.

Figure 10: The effective conductivity is shown for the bcc lattice of spherical grains with overlapping shells. We compare finite element and blind random walk calculations and note that better accuracy is achieved as the random walk step size is reduced (RW=random walk, FE=finite element).

In a more efficient implementation of the random walk method the walkers are referred to as being myopic. The rules in the high conductivity phase do not change. Within the low conductivity phase the walker always accepts each proposed step, but the clock is advanced by an amout 1/h. To this point it does not matter whether we are working on a lattice or in a continuum representation. The distinction between lattice and continuum systems arises only when we consider steps that take the walker from one phase to the other. Here we find that the following rules lead to quite reasonable results: (1) if the proposed step takes the walker from the high to the low conducting phase the step is accepted with probability h and the clock is advanced by an amount (1+h)/(2h) if the step is accepted and by a unit amount if the step is rejected; (2) if the proposed step takes the walker from the low to the high conducting phase the step is always accepted and the clock is again advanced by (1+h)/(2h). With these rules we have consistently obtained results that are in general agreement with blind walker calculations. In the case of the bcc lattice, a comparison of finite element, finite difference, blind walker, and myopic walker calculations is summarized in Table 2.

 Conductivity Ratio Finite Element Finite Difference Blind Walkers Myopic Walkers 1 0.355 0.349 0.340 0.350 2 0.508 0.495 0.476 (0.491) 0.486 5 0.921 0.885 0.855 (0.868) 0.866 10 1.53 1.46 1.41 (1.46) 1.42 20 2.63 2.49 2.38 (2.46) 2.45 50 5.60 5.22 5.02 (5.14) 5.13 100 10.25 9.48 9.06 (9.30) 9.40 Table 2: Comparison of finite element, finite difference, and random walk calculations for the bcc lattice described in Section 2.2. The finite element and finite difference calculations were executed on grids of the same size. The blind and myopic walker calculations were done with step size ε = 0.01 a where a is the length of the unit cell. Some additional calculations (shown in parenthesis) were done with ε = 0.005 a.

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