Rocks with shear fractures or faults widely exist in nature such as oil/gas reservoirs, and hot dry rocks, etc. In this work, the fractal scaling law for length distribution of fractures and the ...relationship among the fractal dimension for fracture length distribution, fracture area porosity and the ratio of the maximum length to the minimum length of fractures are proposed. Then, a fractal model for permeability for fractured rocks is derived based on the fractal geometry theory and the famous cubic law for laminar flow in fractures. It is found that the analytical expression for permeability of fractured rocks is a function of the fractal dimension Df for fracture area, area porosity ϕ, fracture density D, the maximum fracture length lmax, aperture a, the facture azimuth α and facture dip angle θ. Furthermore, a novel analytical expression for the fracture density is also proposed based on the fractal geometry theory for porous media. The validity of the fractal model is verified by comparing the model predictions with the available numerical simulations.
The semi-empirical Kozeny-Carman (KC) equation is the most famous permeability-porosity relation, which is widely used in the field of flow in porous media and is the starting point for many other ...permeability models. However, this relation has many limitations from its inception, and the KC constant is an empirical parameter which was proved to be not a constant. In this paper, we briefly reviewed the KC equation, its modifications and various models for the KC constant. We then derived an analytical expression for the permeability in homogeneous porous media based on the fractal characters of porous media and capillary model. The proposed model is expressed as a function of fractal dimensions, porosity and maximum pore size. The analytical KC constant with no empirical constant is obtained from the assumption of square geometrical model. Furthermore, a distinct linear scaling law between the dimensionless permeability and porosity is found. It is also shown that our analytical permeability is more closely related to the microstructures (fractal dimensions, porosity and maximum pore size), compared to those obtained from conventional methods and models.
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
law is obtained (here
D
T
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 (
D
T
= 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.
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.
Applications of the fractal theory to analyze transport properties of porous media in science and engineering have received steady attention in the past two decades. However, the theory was rarely ...used to analyze invasion by extraneous fluids into a permeable bed where there is initially no such fluid present. Spills and leaks of non-aqueous phase liquids (NAPLs) and formation damage in drilling and completion wells are two typical examples. In this work, a fractal capillary model is proposed to analyze the depth of extraneous fluid invasion, where the tortuosity of capillaries and capillary pressure effect are taken into account. The quantitative relationship between average flow velocity and average beeline velocity are discussed based on the fractal geometry theory. Based on the proposed model, the depth of extraneous fluid invasion can be determined when the operation conditions, extraneous fluid properties and formation structure parameters are available, and the model predictions are in good agreement with the available data.
In this paper a fractal permeability model for bi-dispersed porous media is developed based on the fractal characteristics of pores in the media. The fractal permeability model is found to be a ...function of the tortuosity fractal dimension, pore area fractal dimension, sizes of particles and clusters, micro-porosity inside clusters, and the effective porosity of a medium. An analytical expression for the pore area fractal dimension is presented by approximating the unit cell by the Sierpinski-type gasket. The pore area fractal dimension and the tortuosity fractal dimension of the porous samples are determined by the box counting method. This fractal model for permeability does not contain any empirical constants. To verify the validity of the model, the predicted permeability data based on the present fractal model are compared with those of measurements. A good agreement between the fractal model prediction of permeability and experimental data is found. This verifies the validity of the present fractal permeability model for bi-dispersed porous media.
A fractal model for gas diffusivity in porous media is derived by using fractal theory and by considering rarefied gas effect in micro-/nano-channels/capillaries. The proposed gas diffusivity model ...is expressed as a function of micro-structural parameters (the fractal dimensions for pore area and for tortuosity of tortuous capillaries, porosity and pore sizes) of porous media. The effects of parameters such as porosity, microstructures of porous media and fractal dimensions on gas diffusivity are analyzed. The effective diffusivities predicted by the present fractal model are compared with the available experimental data, and a fair agreement between them is found when porosity is less than about 0.70.
► We have built a fractal model for gas diffusivity in porous media. ► The proposed gas diffusivity model is expressed as a function of micro-structural parameters of porous media. ► Effects of parameters such as porosity, microstructures of porous media and fractal dimensions on gas diffusivity are analyzed. ► Diffusivities predicted by present fractal model are compared with experimental data and a fair agreement among them is found.
Gas slip effect in porous media in the slip flow regime is very important in various scientific and engineering fields. It has been shown that gas slippage factor plays an important role in ...determination of gas apparent permeability. In this work, a novel predictive model for gas slippage factor in micro-porous media with low permeability in the slip flow regime is developed based on the fractal theory. Every parameter in the proposed model has clear physical meaning. The predictions of gas slippage factor by the proposed model show the same variation trend with the available experimental data. Based on the proposed gas slippage factor, it is found the ratio of the gas permeability to the intrinsic/liquid permeability under mean pressure has the same variation trend with empirical correlations. The effects of structural parameters of porous media on the gas slip factor are discussed in detail.
► We have built a fractal model for gas slippage factor in tight porous media. ► Gas slippage factor is a function of structural parameters of the media. ► Predicted gas slippage factor has similar trend with experimental data. ► The effects of structural parameters on gas slip are discussed in detail.
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.
Fucoidan, a sulfated polysaccharide extracted from brown seaweeds, has been shown to possess various bioactivities. In particular, low molecular weight fucoidan (LMWF) has been shown to have better ...bioactivities. In this study, a LMWF (<10 kDa) was extracted from New Zealand Undaria pinnatifida and investigated for its immune modulation effects. LMWF at a concentration range from 1 to 50 μg/mL exerted an effective immune activation in RAW264.7 macrophages. LMWF treatment promoted significant NO release, iNOS expression, and TNF-α and IL-6 secretion in a concentration-dependent manner. It also significantly stimulated the activation of NF-κB and MAPK signaling pathways, and specific inhibitors of NF-κB and MAPK pathways diminished the stimulation, confirming the activation pathways. These results indicate that LMWF possesses potential health benefits through immune-stimulation, which may lead to future pharmaceutical development.
