Properties Based on Tortuosity

Abstract

A suite of papers addressing various “types” of tortuosity is shown to be explained by the same simple spatial tortuosity function from percolation theory for which the parameters are obtained from simulation or experiment and percolation theory. Use of cluster statistics of percolation theory in the context of critical path analysis (which produced verifiable hydraulic conductivity distributions in Chap. 10) together with temporal tortuosity concepts from publications of Eugene Stanley’s group is shown to generate a distribution of solute arrival times in agreement with experiment and without appeal to adjustable parameters. The scaling of typical arrival times with system size, which resembles a power law over many decades of time, matches results from early experiments on dispersive transport in amorphous semiconductors and polymers and gives a physical basis for the observed power (ca. 1.64 in polymers and 1.87 in amorphous semiconductors). The solute velocity is thus implied also to follow a power-law, at least approximately, and the resulting time dependence is shown to map the dependence of chemical reaction rates in porous media over 10 orders of magnitude of time scale. Furthermore, the predicted range of dispersivity values is found to agree with experiment (over 2200 individual cases) over 10 orders of magnitude of length scale.

Appendix: Basic CTRW and Gaussian Results

For Gaussian dispersion:

$$\begin{aligned} \langle x \rangle \propto& t \end{aligned}$$

(11.44)

$$\begin{aligned} \sigma ( t ) \propto& t^{0.5} \end{aligned}$$

(11.45)

For CRTW:

$$ \psi (t) \propto t^{ - 1 - \beta},\quad t \to \infty $$

(11.46)

• If 0<β<1

  $$\begin{aligned} \langle x \rangle \propto& t^{\beta} \end{aligned}$$

  (11.47)
  $$\begin{aligned} \sigma ( t ) \propto& t^{\beta} \end{aligned}$$

  (11.48)

• If 1<β<2

  $$\begin{aligned} \langle x \rangle \propto& t \end{aligned}$$

  (11.49)
  $$\begin{aligned} \sigma ( t ) \propto& t^{ ( 3 - \beta ) /2} \end{aligned}$$

  (11.50)

Sahimi [144]:

• If 0<β<1

  $$ \bigl\langle x^{2} \bigr\rangle \propto t^{2\beta} $$

  (11.51)

• If 1<β<2

  $$ \bigl\langle x^{2} \bigr\rangle \propto t^{3 - \beta} $$

  (11.52)

While our results are compatible with Eq. (11.47) and Eq. (11.48) (within 5 % deviation), we do not find that the long tails of the distribution are compatible with Eq. (11.46) for values of β obtained from either Eq. (11.47) or Eq. (11.48). Here we find discrepancies.

This means that, as yet, we cannot map our treatment of dispersion onto the CTRW. If such a correspondence could be made, the calculation of dispersion for both conservative and non-conservative solutes would be greatly simplified, since the CTRW is already designed to treat either case. In other words, complications due to particle conservation can be incorporated into the CTRW even when some of the particles are adsorbed on surfaces or otherwise lost to the flow, even if subsequent desorption is also possible. Thus an important goal of subsequent research is to investigate more closely the relationship between the present treatment and the CTRW.