Spontaneous capillary imbibition is an important fundamental phenomenon existing extensively in a variety of processes such as polymer composite manufacturing, oil recovery, soil science and hydrology, etc. In this work, analytical expressions for characterizing a spontaneous co-current imbibition process of wetting fluid into gas-saturated porous media are proposed based on the fractal characters of porous media. The mass of imbibed liquid is expressed as a function of the fractal dimensions for pores and for tortuous capillaries, the minimum and maximum hydraulic diameter of pores, and the ratio for minimum to maximum hydraulic diameters, porosity, and fluid properties, as well as the fluid-solid interaction. The imbibed weight predicted by the present model is in good agreement with the available experimental data.

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In this study, we summarized some basic characters of fractal porous media, including the fractal pore or particle size distribution, pore or particle density function, the fractal dimensions for the pore and solid phases, and their relations. The geometric porosities vs. the fractal dimensions and microstructures of porous media were reviewed and discussed in two and three dimensions. The specific surface areas of fractal porous media in two and three dimensions were derived and were expressed as a function of the fractal dimensions and microstructural parameters. The fluid velocities in fractal porous media were also derived and found to be a function of the fractal dimensions and microstructural parameters of the medium. The parameters presented are the fundamental ones and may have potential in analysis of transport properties in fractal porous media.

Nanofluids, which are produced by dispersing nanoparticles into conventional fluids, exhibit anomalously high thermal conductivity. Most experiments demonstrated that the nanolayer surrounding the solid particles and the clusters formed by nanoparticles' aggregation may play an important role in the enhancement of thermal conductivity of nanofluids. By taking into account the nanolayer and nanoparticles' aggregation, a new model for the effective thermal conductivities of nanofluids is proposed. This model is expressed as a function of the thickness of the nanolayer, the nanoparticle size, the nanoparticle volume fraction and the thermal conductivities of suspended nanoparticles and base fluid. The theoretical predictions on the effective thermal conductivities of nanofluids are shown to be in good agreement with the available experimental data.