Fractal Dimension For Tortuosity Research Articles

Predicting water flow in fully and partially saturated porous media: a new fractal-based permeability model

Predicting the permeability of porous media in saturated and partially saturated conditions is of crucial importance in many geo-engineering areas, from water resources to vadose zone hydrology or contaminant transport predictions. Many models have been proposed in the literature to estimate the permeability from properties of the porous media such as porosity, grain size or pore size. In this study, we develop a model of the permeability for porous media saturated by one or two fluid phases with all physically-based parameters using a fractal upscaling technique. The model is related to microstructural properties of porous media such as fractal dimension for pore space, fractal dimension for tortuosity, porosity, maximum radius, ratio of minimum pore radius and maximum pore radius, water saturation and irreducible water saturation. The model is favorably compared to existing and widely used models from the literature. Then, comparison with published experimental data for both unconsolidated and consolidated samples, we show that the proposed model estimate the permeability from the medium properties very well.

Permeability model for fractal porous media with rough surfaces

In this paper, a permeability model is derived for porous media, which are assumed to be comprised of a bundle of tortuous capillaries whose size distribution and roughness of surfaces follow the fractal scaling laws. The proposed model includes the effects of the fractal dimensions for size distributions of capillaries, for tortuosity of tortuous capillaries and for roughness of surfaces on the permeability. The analytical expression for permeability is a function of the relative roughness, the fractal dimensions for tortuosity and sizes of capillaries and for roughness of surfaces, as well as the microstructural parameters (such as the characteristic length, the maximum and minimum pore diameters and the fractal dimensions). The proposed model can properly reveal some mechanisms that affect the permeability. Every parameter in the proposed model has specific physical meaning. The ratio of the permeability ( $$ K_{\text{R}} $$ ) for rough capillaries to that (K) for smooth capillaries is found to be a function of the relative roughness and follows the quadruplicate power law, i.e., $$ K_{\text{R}} /K = (1 - \varepsilon )^{4} $$ , where $$ \varepsilon $$ is the relative roughness of surfaces of capillaries.

Fractal analysis of dimensionless capillary pressure function

The dimensionless capillary pressure function J(SW) was used to eliminate the capillary pressure difference between different strata and to describe similar lithologic reservoirs. Based on the existing function J(SW) and the fractal geometry theory, a fractal model to describe the pore structure of the lithology is presented. The proposed fractal expression for the J(SW) function reveals that when the microstructural parameters (fractal dimension for pore area, the fractal dimension for tortuosity, and the ratio of the maximum pore diameter to the minimum pore diameter) are respectively equal or close for porous samples (with different permeabilities, porosities, interfacial tensions and wetting angles), the values of the J(SW)function vs. saturation coincide each other for wetting phase. In addition, a new dimensionless function G(SW), which is based on the ratio of the average capillary diameter to the critical capillary diameter, is proposed to describe the relationships among the structural parameters of similar lithological porous media and saturation as well as the dimensionless capillary pressure function J(SW).

A Discussion of the Effect of Tortuosity on the Capillary Imbibition in Porous Media

In the past decades, there was considerable controversy over the Lucas–Washburn (LW) equation widely applied in capillary imbibition kinetics. Many experimental results showed that the time exponent of the LW equation is less than 0.5. Based on the tortuous capillary model and fractal geometry, the effect of tortuosity on the capillary imbibition in wetting porous media is discussed in this article. The average height growth of wetting liquid in porous media driven by capillary force following the \({\overline L _{\rm {s}}(t)\sim t^{1/{2D_{\rm {T}}}}}\) law is obtained (here DT is the fractal dimension for tortuosity, which represents the heterogeneity of flow in porous media). The LW law turns out to be the special case when the straight capillary tube (DT = 1) is assumed. The predictions by the present model for the time exponent for capillary imbibition in porous media are compared with available experimental data, and the present model can reproduce approximately the global trend of variation of the time exponent with porosity changing.