Percolation Theory Generates a Physically Based Description of Tortuosity in Saturated and Unsaturated Porous Media

Tortuosity is a property of porous media that is invoked and used in the literature on hydrology, soil science, physics, and engineering. It has been defined in a variety of ways, one of which is a purely geometrical concept. In this study, we focused on the geometrical tortuosity and developed a model based on percolation theory and the finite-size scaling approach. Our result, developed for porous media of any saturation, expresses the tortuosity as a power-law function of the water content, the critical water content, and the system size. The model parameters include the fractal dimension of either the backbone (the flow-carrying part of porous media) or the optimal path (for crossing the system between two opposite faces), which have clear physical meaning. The results may be combined with power laws, which percolation theory provides for the hydraulic or electrical conductivity, to develop the appropriate form that expresses the connectivity as a function of the water content of a pore space. Comparison with numerical simulations and experiments reported in the literature indicated that the model estimates the tortuosity accurately. Our model and results leads us to conclude that pore connectivity and tortuosity should be treated as two distinct properties and that significant uncertainty in results for the saturation dependence of the hydraulic conductivity is traceable to the confusion that mixes the two as a single concept.