A fundamental component of the hydrologic cycle is the movement of fluids in the pore space of geological formations and soils. Prediction of the motion of fluids in such porous materials requires first modeling the physical properties of the medium itself, and second, invoking a capable theory to describe fluid transport in tortuous interconnected pathways. In this dissertation, for the former we use fractal geometry since most phenomena in nature are fractal, and for the latter percolation theory is applied because it has successfully described flow and transport in disordered networks and media. We propose models for the soil water retention curve and tortuosity. We also focus on modeling different kinds of transport, such as air permeability, gas and solute diffusion, unsaturated hydraulic conductivity, and dispersion.
Applications of critical path based analyses of flow and conduction properties reveals asymmetry between the saturation dependence of the air and water permeabilities as well as distinctions between the electrical and hydraulic conductivities. In particular, the saturation dependence of the hydraulic conductivity is strongly dependent on the pore size distribution, but that of the electrical conductivity is only weakly so, and the air permeability is not dependent. Gas diffusion relates more closely to the air permeability, while solute diffusion is, under a wide range of circumstances, tied directly to the electrical conductivity. Comparisons with experiment confirmed this.
Applying critical path analysis and universal scaling from percolation theory to media that could be treated within the poresolid fractal (PSF) approach, we developed unimodal and bimodal models for unsaturated hydraulic conductivity in porous media. Predictions were developed for unsaturated hydraulic conductivity using the soil water retention curve. To evaluate our unimodal model we used 104 experiments from the UNSODA database and compared with two other models. The results obtained indicated that our nonuniversal percolation based model predicted unsaturated hydraulic conductivity better than the other two models. In order to evaluate the bimodal models for soil water retention and unsaturated hydraulic conductivity curves, we compared them with 8 measured experiments collected from the UNSODA database. Although the bimodal unsaturated hydraulic conductivity model was fitted well to the experiments, we found discrepancy between measurements and predictions. We found that the predictions were relatively more successful for the first regime at large water contents than the second regime at low water contents.
The universal scaling law from percolation theory was confirmed for the saturation dependence of the air permeability. Analyzing two independent databases including 39 experiments showed that the experimental exponent was 2.028 ± 0.028 and 1.814 ± 0.386 for the first and second databases, respectively. We found the extracted exponent in the power law fit is most sensitive to the measured values of the air permeability at low values of the airfilled porosity, and in cases where these experimental values are missing, the data can yield values significantly different from 2. We also found that the threshold value of the airfilled porosity could be predicted reasonably from the wet end of the soil water retention curve.
Diffusion modeling in percolation clusters provided a theoretical framework to address gas and solute transport in porous media. Theoretically, above the percolation threshold, the saturation dependence of gas and solute diffusion should follow universal scaling from percolation theory with an exponent of 2.0. In order to evaluate our hypothesis, we used 71 and 106 gas and solute experiments, respectively, including different types of porous media available in the literature. Although our results conclusively confirmed the universality of gas diffusion, we found scatter in solute diffusion data. Nonetheless, the experimental exponent of solute diffusion was very close to 2 (1.842). We found that combining percolation and effective medium theories resulted in an accurate numerical prefactor for both gas and solute diffusion.
We also developed a saturation dependence model for dispersion. Based on concepts from critical path analysis, cluster statistics of percolation, and fractal scaling of percolation clusters we derived an expression for the characteristic velocities along different pathways through the network. We compared our theoretical framework for solute transport with two experimental databases. Our model evaluation with experiments indicated excellent results. In the first dataset, we fitted our model to the arrival time distribution calculated from the measured breakthrough curve at saturation and determined the model parameters. Then those parameters were used to predict the arrival time distribution at two other saturations, giving an excellent match with the measurements. In the second dataset, the arrival time distribution was predicted from the measured soil water retention curve. Our results indicated that we predicted the arrival time distribution very well for 5 unsaturated experiments.

Keywords: Percolation theory, Fractals, Porous media, Dispersion, Unsaturated hydraulic conductivity, Air permeability, Diffusion, Tortuosity, Saturation dependence, Poresize distribution 