Abstract
Most classical predictive models of relative permeability conceptualize the pores in porous media as assemblies of uniform capillary tubes with different sizes. However, this simplification may overestimate the transport capacity of porous media due to overlooking the effects of the pore nonuniformity. This study presents a simple way to quantify the effect of the nonuniformity of pore cross section on the transport characteristic of unsaturated porous media. The way is based on the index relationship between the porosity of a newly defined reference cross section and that of porous media, which satisfies the intrinsic constraints for the nonuniform porosity of cross sections in porous media. Moreover, the index factor can be captured by a newly defined parameter, called the nonuniformity factor, which is used to establish an extended Darcy’s law. Based on these, a fractal-based continuous analytical model and a fractal-based Monte Carlo model of relative permeability as well as a permeability-porosity model are established. Experimental data of five wetting-nonwetting phase systems, including the water-air, water-steam, water-nitrogen, water-oil, and oil-gas systems, are selected to assess the performance of the proposed model. The results confirm the proposed model’s capacity in capturing the transport properties of various porous media. It is found that the nonuniformity of pores can significantly increase the resistance of fluid flow and thus reduce the transport capacity of porous media.
Highlights
The relative permeability of both wetting and nonwetting phases is a significant parameter to many engineering fields, e.g., chemical engineering, environmental engineering, and oil and gas reservoir [1, 2]
Typical representatives of statistical models are the Burdine model and the Mualem model which rely on the pore size distribution function obtained by the water retention curve [16,17,18]
(22) and (23)) of the relative permeability contain three parameters, i.e., c, φ, and Df . φ is the porosity of porous media determined by experiments, c is the nonuniformity factor of pore cross section, and Df is the fractal dimension determined by the best-fitting or the box-counting method
Summary
The relative permeability of both wetting and nonwetting phases is a significant parameter to many engineering fields, e.g., chemical engineering, environmental engineering, and oil and gas reservoir [1, 2]. The empirical models express the relative permeability of a given phase as a power function of the corresponding phase saturation, which involves the best fitting of existing mathematical formulas to the available experimental data, e.g., [14, 15]. These models ignore the pore space characteristics (e.g., tortuosity, pore size distribution, and connectivity) of porous media. Since the pore size distribution of porous media is assumed to obey the fractal scaling law, the fractal permeability model can be obtained by Poisson’s law and Darcy’s law rather than the water retention curve, e.g., [22,23,24,25]
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