Kozeny-Carman Constant Research Articles

Fractal analysis of dimensionless permeability and Kozeny–Carman constant of spherical granular porous media with randomly distributed tree-like branching networks

The seepage of porous media has garnered significant interest due to its ubiquitous presence in nature, but most of the research is based on the model of a single dendritic branching network. In this study, we derive a fractal model of the dimensionless permeability and the Kozeny–Carman (KC) constant of porous media consisting of spherical particles and randomly distributed tree-like branching networks based on fractal theory. In addition, three different types of corrugated pipes are considered. Then, the relationships between the KC constant, dimensionless permeability, and other structural parameters were discussed in detail. It is worth noting that the KC constant of the porous media composed of three types of pipes decreases sharply first and then increases with the increase in the internal diameter ratio, while the dimensionless permeability has the opposite trend and conforms to the physical law. In addition, empirical constants are not included in the analytical formulas of the present model, and the physical mechanism of fluid flow in spherical granular porous media with randomly distributed tree-like branching networks is clearly revealed.

A FRACTAL-MONTE CARLO APPROACH TO SIMULATE KOZENY–CARMAN CONSTANT OF ROUGHENED FIBROUS POROUS MEDIA

The Kozeny–Carman (KC) equation is a well-known semi-empirical formula, which is used to calculate the permeability of porous media in the seepage field. The KC constant is an empirical constant in the KC equation. In this paper, based on the fractal theory, the Fractal-Monte Carlo technique is used to simulate the KC constant of the roughened fibrous porous media (RFPM) with micro-scale effects. There is no empirical constants in the proposed model, and each parameter has its physical meaning. The KC constant model of RFPM can be expressed as a function of structural parameters, including relative roughness ([Formula: see text]), porosity ([Formula: see text]), pore area fractal dimension ([Formula: see text]), tortuosity fractal dimension ([Formula: see text]), capillary diameter ([Formula: see text]) and Knudsen number ([Formula: see text]). The result shows that the KC constant increases with increases in [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. On the contrary, with increases in [Formula: see text] and [Formula: see text], the KC constant decreases. In addition, the KC constant model constructed in the paper agrees well with the existing experimental data and the model. According to the proposed Fractal-Monte Carlo technique, it is possible to better clarify the transmission physical mechanism in RFPM with micro-scale effects.

A novel method to estimate the Stress-Dependent Kozeny-Carman constant of Low-Permeability, clastic sedimentary rocks

The stress-dependent permeability of low-permeability sedimentary rocks has seized attention because of the growing need for fossil fuels, geological carbon sequestration, and radioactive waste disposal. The Kozeny-Carman equation (K-C equation) is a well-known model for predicting the permeability of sedimentary rocks. The Kozeny-Carman constant (K-C constant, ckc) in the K-C equation accounts for the shape of the pores and the complexity of the pore conduit geometry in rocks. The pore structure of rocks varies owing to compaction at different burial depths. Therefore, the ckc is stress-dependent, a key parameter for understanding the stress-dependent permeability of rocks. However, measuring the stress-dependent ckc in low-permeability rocks is challenging. The present study proposes a novel approach for measuring the stress-dependent ckc of low-permeability rocks. The key elements of the proposed approach are (1) utilizing the Klinkenberg slip factor to evaluate the pore structure of rocks under stress, and (2) measuring rock porosity and permeability simultaneously under stress. This study amended the existing apparatus to fulfill the aforementioned purpose. Silty sandstone and clayey siltstone were selected to illustrate the novel ckc measurement method. The measured ckc values were compatible with the measurement results of documented sandstones and mudstones (most stress-independent). Interestingly, the ckc increased with increasing effective confining stress but started to decrease when the applied stress exceeded a certain critical value. Finally, we provide a synthetic case demonstrating the importance of considering stress-dependent ckc for permeability prediction under stress. If we neglect the stress dependence of ckc, the prediction of permeability using the K-C equation can induce errors up to several orders of magnitude.

A NOVEL KOZENY–CARMAN CONSTANT MODEL FOR POROUS MEDIA EMBEDDED WITH TREE-LIKE BRANCHING NETWORKS WITH ROUGHENED SURFACES

Although the hydraulic features of the tree-like branching network have been widely investigated, the seepage characteristics of the networks have not been studied sufficiently. In this study, the seepage characteristics of porous media embedded with a tree-like branching network with the effects of roughness are studied based on fractal theory. Then, the Kozeny–Carman (KC) constant of the composite network is derived. The KC constant of porous media embedded with a tree-like branching network with roughened surfaces is in good agreement with the experimental data in the literature. The effects of structural parameters on seepage characteristics are also discussed. Notably, the results show that the KC constant of the composite network increases with an increasing volume porosity, and decreases with an increase in the relative roughness. Besides, the model established in this paper contains no empirical constants to ensure that each parameter has its physical significance. Thus, the proposed model can facilitate a better understanding of the seepage characteristics of fluid transport through a tree-like branching network embedded in porous media.

Modelling fluid flow in carbon fibre porous media based on X-ray microtomography and lattice Boltzmann method

Carbon fibre media possess complex internal porous structures; the microstructures are vital and can impact the efficiency of fluid flow and mass transfer. In this study, X-ray microtomography (XMT) and the lattice Boltzmann method (LBM) was employed to reconstruct the authentic three-dimensional (3D) pore structures of carbon fibre media and simulate its fluid flow. The effects of fluid properties and boundary conditions were elucidated. Results demonstrated that the flow velocity increased as the displacement differential pressure increased, but decreased as the fluid kinematic viscosity increased. The critical values of differential pressure and viscosity were 4.0106 Pa and 0.0101 m2/s, respectively. The simulation results of the average volume velocity revealed that three boundary conditions had extremely small relative errors of 0.0037 % depending on the carbon fibre material. When the Kozeny–Carman constant (KC, k) was 3.5, the relative error of permeability between the predicted values and the simulated values decreased with increasing porosity. When the porosity was 54.069 %, the relative error decreased to 4.18 %. The fractal model revealed worse results for the permeability prediction of carbon fibre porous media, and the relative error between the simulated and predicted values was 50.40 % when the porosity was 45.585 %.

Simple determination of key structural parameters for fibrous materials enabled by Ergun-Type and Kozeny-type equations

Structural parameters are critical for the final properties of fibrous materials. Three common parameters (i.e., porosity, thickness and average fiber diameter) of each fibrous material are simultaneously determined based on their air penetration behavior. The Ergun-type equation and the Kozeny-type equation, with the help of the optimization method, are applied to conduct the inverse characterization. The prediction performance of the equations is evaluated by comparing the given pressure drop and the values predicted by the Ergun-type and Kozeny-type equations. The inversely determined parameters are compared with the directly measured values. The results show that the Ergun-type equation and the Kozeny-type equation are suitable for various fibrous materials, including conventional nonwoven materials and nanofibrous membranes. The selection criteria of these two equations are provided based on the relation of pressure drop-airflow velocity and the prediction accuracy. The results also suggest that the Ergun-type equation's coefficients and the Blake–Kozeny–Carman constant of the Kozeny-type equation need to be determined for each material. The relative errors for most of the inversely determined parameters are reasonable. The findings indicate that the airflow-based method is feasible for fibrous materials.

A FRACTAL MODEL FOR KOZENY–CARMAN CONSTANT AND DIMENSIONLESS PERMEABILITY OF FIBROUS POROUS MEDIA WITH ROUGHENED SURFACES

In this paper, fluid transport through fibrous porous media is studied by the fractal theory with a focus on the effect of surface roughness of capillaries. A fractal model for Kozeny–Carman (KC) constant and dimensionless permeability of fibrous porous media with roughened surfaces is derived. The determined KC constant and dimensionless permeability of fibrous porous media with roughened surfaces are in good agreement with available experimental data and existing models reported in the literature. It is found that the KC constant of fibrous porous media with roughened surfaces increases with the increase of relative roughness, porosity, area fractal dimension of pore and tortuosity fractal dimension, respectively. Besides, it is seen that the dimensionless permeability of fibrous porous media with roughened surfaces decreases with increasing relative roughness and tortuosity fractal dimension. However, it is observed that the dimensionless permeability of fibrous porous media with roughened surfaces increases with porosity. With the proposed fractal model, the physical mechanisms of fluids transport through fibrous porous media are better elucidated.

An extended Kozeny-Carman-Klinkenberg model for gas permeability in micro/nano-porous media

Gas transport in micropores/nanopores deviates from classical continuum calculations due to nonequilibrium in gas dynamics. In such a case, transport can be classified by the Knudsen number (Kn) as the ratio of gas mean free path and characteristic flow diameter. The well-known Klinkenberg correction and its successors estimate deviation from existing permeability values as a function of Kn through a vast number of modeling attempts. However, the nonequilibrium in a porous system cannot be simply modeled using the classical definition of the Kn number calculated from Darcy’s definition of the pore size or hydraulic diameter. Instead, a proper flow dimension should consider pore connectivity in order to characterize the rarefaction level. This study performs a wide range of pore-level analysis of gas dynamics with different porosities, pore sizes, and pore throat sizes at different Kn values in the slip flow regime. First, intrinsic permeability values were calculated without any rarefaction effect and an extended Kozeny-Carman model was developed by formulating the Kozeny-Carman constant by porosity and pore to throat size ratio. Permeability increased by increasing the porosity and decreasing the pore to throat size ratio. Next, velocity slip was applied on pore surfaces to calculate apparent permeability values. Permeability increased by increasing Kn at different rates depending on the pore parameters. While the characterization by the Kn value calculated with pore height or hydraulic diameter did not display unified behavior, relating permeability values with the Kn number calculated from the equivalent height definition created a general characterization based on the porosity independent from the pore to throat size ratio. Next, we extended the Klinkenberg equation by calculating unknown Klinkenberg coefficients which were found as a simple first order function of porosity regardless of the corresponding pore connectivity. The extended model as a combination of Kozeny-Carman for intrinsic permeability and Klinkenberg for apparent permeability correction yielded successful results.

KOZENY–CARMAN CONSTANT FOR GAS FLOW THROUGH FIBROUS POROUS MEDIA BY FRACTAL-MONTE CARLO SIMULATIONS

In this paper, the Kozeny–Carman constant of fibrous porous media is simulated by the Fractal-Monte Carlo technique. The proposed probability model of the Kozeny–Carman constant is obtained based on the fractal distribution of pore size in fibrous porous media, and thus can be expressed as a function of structural parameters of fibrous porous media, including porosity, micro-pore size, fiber diameter, tortuosity fractal dimension and area fractal dimension of pores. Our results demonstrate that the Kozeny–Carman constant of fibrous porous media increases with increases in tortuosity fractal dimension and fiber diameter. Our results also illustrate a satisfying agreement of the Fractal Monte-Carlo simulations obtained by the proposed model and the existing experimental data. Therefore, the proposed Fractal-Monte Carlo technique can be used to characterize other transport properties of fluid in fibrous porous media.

A novel fractal solution for permeability and Kozeny-Carman constant of fibrous porous media made up of solid particles and porous fibers

This paper presents a novel fractal solution to investigate the permeability and the Kozeny-Carman (KC) constant of fibrous porous media made up of solid particles and porous fibers. The proposed model has been verified with satisfying agreements of the permeability and KC constant of fibrous porous media obtained by our model and those obtained by experimental data, analytical solution, and numerical simulation reported in literature. The results demonstrate that 1) an increase in particle diameter leads to an increase in the absolute permeability; 2) an increase in the tortuosity fractal dimension leads to an increase in the KC constant and decreases in the dimensionless permeability and absolute permeability; 3) an increase in the porosity results in increases in the dimensionless permeability and absolute permeability of the fibrous porous media; 4) increases in the fiber diameter yields an increase in the absolute permeability of fibrous porous media. The proposed fractal model explicitly relates the KC constant and the permeability to the microstructural parameters of the fibrous porous media, and consequently facilitating the understanding of the detailed physical mechanisms for fluids transport through fibrous porous media.

Kozeny-Carman constant of porous media: Insights from fractal-capillary imbibition theory

The Kozeny-Carman (KC) equation has been widely used in many fields including shale and tight reservoirs, soil physics and chemical engineering. However, accurate calculation of KC constant is always a challenge. This work focuses on analysis of the KC constant in capillary imbibition process based on fractal distribution characteristics of hydraulic pore diameter. The proposed model incorporates gravity over the entire time frame of imbibition process, which is expressed in terms of porosity, fractal dimension, tortuosity, maximum hydraulic pore diameter, liquid density, viscosity, surface tension and contact angle. The theoretical predictions from the developed model provide a good agreement with experimental results of emulsion imbibition, with values of fractal dimension and tortuosity from fitting of height-time data of emulsion imbibition.

An improved model for permeability estimation in low permeable porous media based on fractal geometry and modified Hagen-Poiseuille flow

Permeability is a crucial macroscopic parameter in characterizing fluid flow and mass transfer behavior in porous media. A great theoretical basis for permeability estimation has been provided by many traditional and recently presented fractal models. However, since the influence of overlying sediments, the low-permeability porous media such as natural rock and coal generally have lower permeability and porosity with abundant pores which have more complex structure and rougher surface. The fluid located at surface tends to be bound to form irreducible state. Generally, this natural phenomenon is often ignored in the conventional fractal permeability model. In this study, the improved model was theoretically deduced based on the fractal geometry theory with considering the irreducible water saturation. 133 total sandstone samples from a gas reservoir were used to verify the validity of the developed model. The mercury injection experiment and the conventional physical property test were carried out, and the thin section image of each sample was crafted. The pores were segmented by the proposed color extracted algorithm for calculating pore fractal dimension. The results show that the available fractal model overestimates permeability values. A new form of the classical Kozeny-Carman equation was also developed to accurately estimate permeability. In the low permeable porous media, the empirical Kozeny-Carman constant needs to take 3.5, instead of 5. However, no longer is typical superiority found at very low permeability (<0.5mD) in the improved fractal model and corresponding reasons are analyzed and discussed.

MODELING FOR HYDRAULIC PERMEABILITY AND KOZENY–CARMAN CONSTANT OF POROUS NANOFIBERS USING A FRACTAL APPROACH

In this study, the analytical expressions for the hydraulic permeability and Kozeny–Carman (KC) constant of porous nanofibers are derived based on fractal theory. In the present approach, the permeability is explicitly related to the porosity and the area fractal dimensions of porous nanofibers. The proposed fractal models for KC constant is also found to be a function of the microstructural parameters (porosity, area fractal dimensions). Besides, the present model clearly indicates that KC constant is not a constant and increases with porosity. However, KC constant is close to a constant value which is 18 for ϕ &gt; 0.8. Every parameter of the proposed formulas of calculating permeability and KC constant has clear physical meaning. The model predictions are compared with the existing experimental data, and fair agreement between the model predictions and experimental data is found for different porosities.

Is trabecular bone permeability governed by molecular ordering-induced fluid viscosity gain? Arguments from re-evaluation of experimental data in the framework of homogenization theory.

It is generally agreed on that trabecular bone permeability, a physiologically important quantity, is governed by the material׳s (vascular or intertrabecular) porosity as well as by the viscosity of the pore-filling fluids. Still, there is less agreement on how these two key factors govern bone permeability. In order to shed more light onto this somewhat open issue, we here develop a random homogenization scheme for upscaling Poiseuille flow in the vascular porosity, up to Darcy-type permeability of the overall porous medium "trabecular bone". The underlying representative volume element of the macroscopic bone material contains two types of phases: a spherical, impermeable extracellular bone matrix phase interacts with interpenetrating cylindrical pore channel phases that are oriented in all different space directions. This type of interaction is modeled by means of a self-consistent homogenization scheme. While the permeability of the bone matrix equals to zero, the permeability of the pore phase is found through expressing the classical Hagen-Poiseuille law for laminar flow in the format of a "micro-Darcy law". The upscaling scheme contains pore size and porosity as geometrical input variables; however, they can be related to each other, based on well-known relations between porosity and specific bone surface. As two key results, validated through comprehensive experimental data, it appears (i) that the famous Kozeny-Carman constant (which relates bone permeability to the cube of the porosity, the square of the specific surface, as well as to the bone fluid viscosity) needs to be replaced by an again porosity-dependent rational function, and (ii) that the overall bone permeability is strongly affected by the pore fluid viscosity, which, in case of polarized fluids, is strongly increased due to the presence of electrically charged pore walls.

Flow Regimes in Packed Beds of Spheres from Pre-Darcy to Turbulent

Characteristics of flow regimes in porous media, along the processes of energy dissipation in each regime, are critical for applications of such media. The current work presents new experimental data for water flow in packed steel spheres of 1- and 3-mm diameters. The porosity of the porous media was about 35 % for both cases. The extensive dataset covered a broad range of flow Reynolds number such that several important flow regimes were encountered, including the elusive pre-Darcy regime, which is rarely or never seen in porous-media literature and turbulent regime. When compared to previous information, the results of this study are seen to add to the divergence of available data on pressure drop in packed beds of spheres. The divergence was also present in the coefficients of Ergun equation and in the Kozeny–Carman constant. The porous media of the current work were seen to exhibit different values of permeability and Forchheimer coefficient in each flow regime. The current data correlated well using the friction factor based on the permeability (measured in the Darcy regime) and the Reynolds number based on the same length scale. An attempt was made to apply recent theoretical results regarding the applicability of the quadratic and cubic Forchheimer corrections in the strong and weak inertia regime.

Detailed characterisation of the flow resistance of commercial sub-2 μm reversed-phase columns

The pressure-drop characteristics of six sub-2 μm columns of three different manufacturers but with the same surface chemistry (C 18) have been studied using the recently introduced total pore blocking method for the determination of the external porosity and by using scanning electron microscopy pictures to measure the actual size of the particles. Having used the Sauter-mean particle size to correctly account for the particle size spread, and having corrected for the influence of the intra-particle porosity, it was found that all columns yielded Kozeny–Carman constant ( f KC) values close to 180, in agreement with the theory. This agreement could subsequently be used to quantify how the different system parameters such as mean particle size, packing density and intra-particle porosity (all tending to vary significantly from manufacturer to manufacturer) each contributes to the observed total bed permeability. Small (upward) deviations from the f KC = 180-value could be correlated to a larger width of the particle size distribution, and more notably to the existence of a high size ratio of the largest to the smallest particles present in the particle batches.

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

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.

Comparison between the efficiencies of columns packed with fully and partially porous C 18-bonded silica materials

The chromatographic performance of a new brand of shell particles is compared to that of a conventional brand of totally porous silica particles having a similar size. The new material (Halo, Advanced Materials Technology, Wilmington, DE) is made of 2.7 μm particles that consist in a 1.7 μm solid core covered with a 0.5 μm thick shell of porous silica. The other material consists of the porous particles of a conventional 3 μm commercial silica-B material. These two columns have the same dimensions, 150 mm × 4.6 mm. The reduced plate heights of two low molecular weight compounds, naphthalene and anthracene, two peptides (lys-bradykinin and bradykinin), and four proteins, insulin, lysozyme, β-lactoglobulin, and bovine serum albumin were measured in a wide flow rate range and analyzed on the basis of the Van Deemter equation and of modern models for its terms. The Halo column provides a smaller axial diffusion coefficient B and a smaller eddy dispersion term A than the other column, a result consistent with its lower internal porosity ( ∈ p = 0.19 versus 0.42) and with the narrower size distribution of its particles ( σ = 5% versus 13%). The two columns have similar C terms for the two low molecular weight compounds and for the two peptides. However, the C term of the proteins that are not excluded is markedly lower on the column packed with the Halo particles than on the other column. A recent theoretical analysis of the mass transfer kinetics in shell particles predicts a C term for moderately retained proteins (3 < k′ < 5) that is about 35% lower for shell than for fully porous particles while the experimental data show a value nearly 45% lower, an excellent agreement considering that the internal tortuosity of the particles might be different, affecting the ratio of the effective diffusivities ( D eff) of the proteins in the two materials. Surprisingly, the Kozeny–Carman constant of the Halo packed column is 50% larger than that of the other column, in spite of which the permeability of the Halo column is slightly larger, due to its larger external porosity.

Permeability measurements of the flow of interdendritic liquid in equiaxed aluminum-silicon alloys

A method developed recently for measuring instantaneous permeability in the mushy zone of aluminum-copper alloys is used to measure permeability in aluminum-silicon alloys. Permeability was also measured in packed glass beads using water, because measurements could be conducted in the absence of microstructural coarsening. Experimental data are compared with the Kozeny-Carman equation to determine the value of kC, the Kozeny-Carman constant. The results indicate that when kC=5.0, all the instantaneous permeability data agree well with the Kozeny-Carman equation. Factors affecting the accuracy of permeability measurements, such as surface oxidation of the sample, flow channeling during the test, microstructural coarsening, and liquid fraction reduction due to back diffusion in the solid are discussed. An experimental technique that minimizes errors due to these factors is outlined.

Numerical determination of liquid flow permeabilities for equiaxed dendritic structures

Darcy's law has been applied to the 3D finite difference numerical determination of the influence of solid fraction and geometry on the permeability of equiaxed dendritic structures. The micro-model computes the permeability for flow through a domain equivalent to the volume ultimately occupied by a single solid solution dendritic grain in an Al3Cu3Si alloy. Evolution of the dendrite shape during solidification was simulated using a novel cellular automaton-finite difference technique. Numerically determined permeabilities compare well with reported experimental data for aluminium alloys. For solid fractions in excess of ∼20%, there is also reasonable correlation with the Kozeny–Carman (KC) expression for a KC constant of unity. A significant feature of the micro-model is that it is able to account for the isolation of interdendritic liquid pools in calculating the effective values of the solid–liquid interfacial area and of the fraction liquid.