Developing a new form of permeability and Kozeny-Carman constant for homogeneous porous media by means of fractal geometry

The concept of tortuosity is used to characterize the structure of porous media, to estimate their electrical and hydraulic conductivity, and to study the travel time and length for tracer dispersion, but different types of tortuosity—geometric, hydraulic, electrical, and diffusive—have been used essentially interchangeably in the literature. Here, we critically review the tortuosity models developed empirically, analytically, and numerically for flow in both saturated and unsaturated porous media. We emphasize that the proposed tortuosity models are distinct and thus may not be used interchangeably. Given the variety of models that have been developed, and the sharp differences between some of them, no consensus has emerged unifying the models in a coherent way. Related treatments of tortuosity are found in the literature on porous catalysts. In such materials, nonlinear reactions ordinarily accompany transport, and the effective diffusivity within the pore space in the presence of the reactions is distinct from the one in their absence. Thus, because tortuosity may be defined as the ratio of the effective diffusivities in the bulk material and within the pore space, a careful treatment of tortuosity may need to distinguish between transport with and without reactions. This complication is ultimately relevant to soils as well, because bioremediation and biodegradation in soils are always accompanied by nonlinear reactions. Common models of tortuosity include both logarithmic functions and power laws. In many cases, the differences between the logarithmic and power-law phenomenologies are not great, but power laws can usually be reconciled with percolation concepts. Invoking percolation theory provides both insight into the origin of the power functions and a framework for understanding differences between tortuosity models.

Tortuosity in Porous Media: A Critical Review

Spontaneous imbibition of wetting liquids in porous media is a ubiquitous natural phenomenon which has received much attention in a wide variety of fields over several decades. Many traditional and recently presented capillary-driven flow models are derived based on Hagen–Poiseuille (H–P) flow in cylindrical capillaries. However, some limitations of these models have motivated modifications by taking into account different geometrical factors. In this work, a more generalized spontaneous imbibition model is developed by considering the different sizes and shapes of pores, the tortuosity of imbibition streamlines in random porous media, and the initial wetting-phase saturation. The interrelationships of accumulated imbibition weight, imbibition rate and gas recovery and the properties of the porous media, wetting liquids, and their interactions are derived analytically. A theoretical analysis and comparison denote that the presented equations can generalize several traditional and newly developed models fr...

https://eps.utk.edu/faculty/perfect/Papers/Cai%20et%20al.%20(2014).pdf

Generalized Modeling of Spontaneous Imbibition Based on Hagen− Poiseuille Flow in Tortuous Capillaries with Variably Shaped Apertures

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.

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

The flow in porous media has received a great deal of attention due to its importance and many unresolved problems in science and engineering such as geophysics, soil science, underground water resources, petroleum engineering, fibrous composite manufacturing, biophysics (tissues and organs), etc. It has been shown that natural and some synthetic porous media are fractals, and these media may be called fractal porous media. The flow and transport properties such as flow resistance and permeability for fractal porous media have steadily attracted much attention in the past decades. This review article intends to summarize the theories, methods, mathematical models, achievements, and open questions in the area of flow in fractal porous media by applying the fractal geometry theory and technique. The emphases are placed on the theoretical analysis based on the fractal geometry applied to fractal porous media. This review article shows that fractal geometry and technique have the potentials in analysis of flow and transport properties in fractal porous media. A few remarks are made with respect to the theoretical studies that should further be made in this area in the future. This article contains 220 references.

Analysis of Flow in Fractal Porous Media

A simple geometry model for tortuosity of flow path in porous media is proposed based on the assumption that some particles in a porous medium are unrestrictedly overlapped and the others are not. The proposed model is expressed as a function of porosity and there is no empirical constant in this model. The model predictions are compared with those from available correlations obtained numerically and experimentally, both of which are in agreement with each other. The present model can also give the tortuosity with a good approximation near the percolation threshold. The validity of the present tortuosity model is thus verified.

https://iopscience.iop.org/article/10.1088/0256-307X/21/8/044/pdf

A Geometry Model for Tortuosity of Flow Path in Porous Media

We report a model for the fractal dimension Ds of rough surfaces based on the fractal distribution of roughness elements on surfaces and the fractal character of surface profiles. The proposed model for the fractal dimension Ds is expressed as a function of the fractal dimensions D for conic roughness diameter/height and Dp for surface profile, maximum roughness base diameter λmax, the ratio β of conic roughness height to its base radius as well as the ratio λminλmax of the minimum to the maximal base diameter.

https://iopscience.iop.org/article/10.1088/0256-307X/26/11/116101/pdf

A Model for Fractal Dimension of Rough Surfaces

A quasi-analytical model, i.e. the fractal model, for the transverse thermal dispersion conductivity in porous media is presented based on the fractal characteristics of tortuous flow paths/streamlines in porous media. The fractal dimension of tortuous flow paths, the spatial deviation velocity and the transverse thermal dispersion conductivity are derived. The proposed model is expressed as functions of the fractal dimension of tortuous flow paths/streamlines, Peclet number, porosity and structural parameters. The present results are compared with those from the existing correlation, and good agreement is found between the present model predictions and those from the existing correlation.

LI Jian-Hua

Papers

A Geometry Model for Tortuosity of Flow Path in Porous Media

A Model for Fractal Dimension of Rough Surfaces

A Fractal Model for the Transverse Thermal Dispersion Conductivity in Porous Media