Tortuosity As a Function of Porosity: Comparison of Predictions with Experiments and Numerical Simulations.

For the purpose of calculating tortuosity in porous media, we apply a geometrical model based on cluster topology from percolation theory. The result is a power of porosity (less a threshold value). All model parameters e.g., fractal dimensionality and percolation threshold have physical meaning. To evaluate the model, we compare our results with both experiments (e.g., Barrande et al. 2007) and numerical simulations (e.g., Koponen et al., 1997; Matyka et al., 2008). Strictly speaking, geometrical tortuosity must be less than the hydraulic tortuosity.  However, our geometrical tortuosity model gives a very good prediction of hydraulic tortuosity, at least for uniform porous media as evaluated by the Kozeny-Carman equation. This indicates that geometrical tortuosity approaches hydraulic tortuosity as the medium becomes more uniform. We also find that Koponen et al. (1997) and Matyka et al.’s (2008) 2-D numerical results compare well with model predictions of the porosity-dependence for large porosity values (>0.50). However, as the percolation threshold is approached, large differences appear: the saturated geometrical tortuosity of our percolation-based model increases much more sharply than the others.  We interpret this discrepancy as originating in the inability of numerical simulations of finite systems to probe the divergent length scales at the percolation transition.