A mathematical model of fluid flow in tight porous media based on fractal assumptions

Natural tight reservoirs are networks with high connectivity but low porosity, in which fractal behaviors have been widely observed and proven to affect the transport property significantly. The objective of this study is to establish a mathematical model to describe fluid flow in fractal tight porous media. To address this problem, four fractal dimensions were used: the pore size fractal dimension Df, geometrical and hydraulic tortuosity fractal dimensions Dτg and Dτ, and the fractal dimension Dλ characterizing the hydraulic diameter-number distribution. The relationship among these fractal dimensions was analyzed and Df was found equal to Dλ+Dτg. Then a unified model connecting the porosity and Df is deduced for arbitrary fractal tight porous media. Based on the scaling-invariant behaviors assumed, a fractal mathematical model is developed for the permeability estimation, which is fabricated only by fundamental and well-defined physical properties of Df,Dτ, the scaling lacunarity Pλ, the range of the pore sizes, and the porosity of the fractal generator φ0. To validate the permeability model, we developed an algorithm to model fractal tight porous media according to the scaling-invariant topography of fractal objects based on Voronoi tessellations, and to simulate fluid flow in these complex networks by Lattice Boltzmann method (LBM) at pore scale. Numerical experiments indicate that the hydraulic tortuosity fractal dimension Dτ is approximately equal to 1.1. Consequently, the fractal mathematical model was quantitatively determined and its performance was verified by the LBM simulations. Finally, the fractal mathematical model was rearranged into a permeability-porosity form for practical applications.