Estimation of tortuosity coefficient under unsaturated conditions based on fractal concepts

Complex nature of porous media complicates any prediction of their hydraulic properties. To demonstrate shortcoming of hydraulic conductivity models predictions, the concept of tortuosity was introduced. Since there is no measured data of tortuosity, and tortuosity has a direct relationship to hydraulic conductivity, so in this study we aimed to develop a general mathematical relationship to determine tortuosity. An optimization code were run in MATLAB R2014a software, using Monte Carlo algorithm, aimed to minimize Root Mean Square of Logarithmic Deviation (RMSLD) between calculated hydraulic conductivity values based on Shepard (1993) and van Genuchten (1980) models, to determine tortuosity on different water contents for 69 soil samples of UNSODA database with a wide range of soil textures. Considering fractal concepts, we developed a linear equation empirically to determine hydraulic tortuosity as a function of effective saturation, pore fractal dimension, porosity, inverse of air entry pressure and soil water content, covering whole ranges of degree of saturation. Based on results, calculated values of tortuosity were greater than proposed values by Shepard about 30%. To evaluate developed equation, statistical parameters of Root Mean Square of Logarithmic Deviation (RMSLD) and Akaike’s Information Criterion (AICc) was adopted for 17 different soil samples. According to the calculated statistical parameters, using developed equation to estimate tortuosity has improved the results of Shepard’s method significantly. Totally, the results show that, despite the developed equation has a relatively complicated structure, in terms of the compromise between accuracy and complexity has an acceptable performance.