Kozeny Equation Research Articles

Applicability of Nuclear Magnetic Resonance Experiment in Analyzing Pore and Fluid Distribution Characteristics of Tight Sandstone: A Case Study in the Julu Area, Bohai Bay Basin, China.

Characterizing the pore and fluid distribution is critical for evaluating the reservoir potential of new areas. Nuclear magnetic resonance (NMR) is considered as an experimental method capable of full-scale characterization of pore characteristics. However, the T2 spectrum of a saturated sample is affected by a combination of sample and experimental parameters, and it is important to confirm whether the T2 spectrum fully reflects the sample pore information. Eight tight sandstone samples from the Julu area were selected for thin section identification, mercury intrusion porosimetry (MIP), NMR, NMR cryoporometry (NMRC), and centrifugation experiments to critically analyze the applicability of the NMR results. Two methods, the similarity method and Kozeny's equation method, were used to calculate surface relaxivity, a critical parameter for converting NMR T2 signals into pore information. The discussion focuses on the applicability of the calculated surface relaxivity and the phenomenon of T2 signal changes in a short relaxation range after centrifugation. The main results are as follows: The surface relaxivity values calculated using the different methods differed significantly. The surface relaxivity calculated using the same method reflected the relative magnitude of the true surface relaxivity of the samples. For the samples with large surface relaxivity, there may be partial misses of the short relaxation signal, the NMR porosity was smaller than the gas-measured porosity, there was a variation in the T2 spectrum in the short relaxation range after centrifugation, and the calculated surface relaxivity was small. The surface relaxivity calculated using Kozeny's equation was nearly accurate, but perhaps smaller than the true value. The T2 spectra mainly reflected macropore information. This study suggests that PSDs converted from T2 spectra of saturated samples should be judged with relative caution rather than solely based on the peak or range correspondence between the two curves, and the minimum centrifugal radius can be used as a constraint.

Matching method between nanoparticle displacement agent size and pore throat in low permeability reservoir

Nano-particles possess desirable attributes such as small particle size, excellent injectivity, and migration performance, making them highly compatible and adaptable for addressing the water flooding requirements of the low-permeability oil reservoir. When selecting an oil displacement agent for enhancing water flooding and improving oil recovery, factors such as injectivity and migration need to be carefully considered. In this study, through a comprehensive analysis of the mechanism and technical characteristics of nano-particle oil displacement agents, the plugging and profile control mechanisms recognized by the mainstream of nano-particles are elucidated. By examining various elements including outcrop fractures, natural micro-fractures, artificial support fractures, and dynamic monitoring data, a reevaluation of the dominant channel scale governing water drive in low permeability reservoirs is conducted, thereby defining the target entities for profile control and flooding operations. Drawing upon Darcy’s percolation law and leveraging enhanced oil recovery techniques based on the classical Kozeny equation, a profile control and flooding mechanism is proposed that focuses on increasing the specific surface area of polymer particles while simultaneously reducing reservoir permeability. This innovative approach establishes a novel matching method between nano-polymer particles and the diverse media found within the reservoir. Lastly, the application of nanoparticle flooding technology in Changqing Oilfield is presented, highlighting its practical implementation and benefits.

Contribution of Resistivity Properties in Estimating Hydraulic Conductivity in Ciremai Volcanic Deposits

The hydraulic parameters of porous media, such as porosity (φ) and hydraulic conductivity (K), are the most important factors for planning and managing water exploitation from aquifers. This study aims to estimate the hydraulic conductivity parameters using the geoelectric method on volcanic deposits on the northern slope of Mount Ciremai. For this purpose, four data types were used to estimate K and φ, including lithological profiles, water table, groundwater quality, pumping test data, and vertical electrical sounding (VES). Based on Archie's law and Kozeny's equation, we get the alpha (α) values and cementation factor (m) from which the median values of α = 1.01 and m = 1.36 represent the studied aquifer. The porosity (φ) of the aquifer varies from 0.097 to 0.187 with an average of 0.141 and is spatially related to the hydraulic conductivity (kgm), which varies from 4.97 × 10-6 to 6.75 × 10-5 m/s after the application of Kozeny's equation. The hydraulic conductivity (Kp) calculated from the pumping tests varies from 9.07 × 10-6 to 1.06 × 10-4 m/s and is strongly correlated (r = 0.87). Furthermore, a relation between resistivity and hydraulic conductivity was established for the studied aquifer to estimate these parameters in sites lacking data.

Prediction of permeability in mushy zone of Al-4.5 %Mg alloy based on CA-LBM model

Permeability is a key parameter in hot tearing prediction, and the solidification microstructure of alloys has an important influence on permeability prediction. The present paper builds a cellular automata–lattice Boltzmann method (CA-LBM) model to simulate the grain growth and liquid flow of semi-solid binary aluminum alloys. The CA model was used to simulate the microstructure of the mushy region, including the growth of dendrites and the overlap between the dendrite arms, and the LBM model was used to observe the liquid phase flow in the mushy zone. The model takes into account the complex inter-dendritic flow, including the liquid flow between the dendrite arms. The paper compared the permeability results of the current model with the Carman–Kozeny equation for two limiting cases of interfacial area concentration Sv (the dendrite structure with Sv and the globular structure with Sv) and investigated the effect of different cooling rates on permeability. Deviation from the Carman–Kozeny behavior was observed with a transition in microstructure, i.e., when the domain contained a mixture of dendritic and globular structures or when fluid flow occurred simultaneously in the intra-dendritic and extra-dendritic regions. From the analysis of the permeability results of the mushy zone under different cooling rates, the higher the cooling rate, the lower the permeability. The permeability is not only related to the solid volume fraction but also to the morphology and size of grains.

Alternative flow model of anisotropic porous media

Porous flow is typically analyzed by the Kozeny–Carman equation using geometric variables, such as hydraulic diameter and tortuosity. The hydraulic tortuosity was first described by Kozeny, and later redefined by Carman as the tortuosity square term in the Kozeny–Carman equation. However, the revised term correlating the flow velocity with path length square would be physically ambiguous. Moreover, the hydraulic diameter, which is directly correlated to the permeability and interstitial velocity in the Kozeny–Carman equation, is an isotropic constant property; thus, it should be verified whether the isotropic hydraulic diameter can reasonably correlate each anisotropic directional flow feature passing through heterogeneous complex media. Accordingly, the Kozeny–Carman equation was theoretically examined and experimentally verified in this study to obtain the proper correlation based on the definitions of truly equivalent diameter and tortuosity. Therefore, the effective variables of porous media were presented, and it confirmed that the effective diameter corresponded to the physically equivalent diameter of anisotropic porous media. Moreover, using the mass conservation relation, it was verified that Kozeny's tortuosity is exactly associated with the truly equivalent flow model, and then the Kozeny constant must be differently defined from the original Kozeny equation. Accordingly, the Kozeny–Carman equation was improved by appertaining either effective diameter or tortuosity, and the momentum conservation relation was used to verify it. The pore-scale simulations using 5-sorts of 25-series porous media models were performed to test the validity of derived effective variables and revised equations. Finally, the alternative flow model of anisotropic porous media was presented using equivalent geometric and frictional flow variables. Subsequently, their practical and more accurate estimations were achieved by introducing the concentric annulus flow model under the special hydraulic condition. The new variables and relations are expected to be usefully applied to various porous flow analyses, such as interstitial velocity estimations, geometric condition variations, flow regime changes, and anisotropic heat and multiphase flows.

Surface relaxation and permeability of cement pastes with hydrophobic agent: Combining 1H NMR and BET

The carboxylic acid ammonium salt is mixed into the concrete matrix to resist water with corrosive ions to improve concrete durability. The 1H NMR and BET experimental methods were applied to evaluate the effect of carboxylic acid ammonium salt on pore surface and structure. The results show that the surface relaxation of cement pastes containing hydrophobic agent is reduced to 60% of reference sample and excellent consistency exists between surface relaxation and water adsorption. The analytical Kozeny equation is utilized to predict the permeability of cement pastes containing different amounts of carboxylic acid ammonium salt. Results show that the reduction trend changes with carboxylic acid ammonium salt amount due to the fact that the addition of hydrophobic agent changes the pore structure. It can be concluded that the generation of the flaky crystals results in the change of pore structure and pore surface properties which can increase the water resistance ability of pore in cement paste mixing with hydrophobic agent. These results reveal the mechanism of carboxylic acid ammonium salt integral addition on hydrophobicity of cementitious materials and provide the guidance to improve the hydrophobic agent application properties.

Effective medium model for a suspension of active swimmers

Several active organisms in nature tend to reside as a community in viscous fluid media. We analyze the variation of the swimming characteristics of an active swimmer present in a dilute and disperse suspension, modeled as an effective Brinkman medium. This idealized representation of a collection of active swimmers allows one to distinguish the impact of the interior domain available to an individual swimmer as well as the contribution of its neighbors. Darcy's law along with an analytical solution enable the effective resistivity to be predicted as a function of the volume fraction, which is in close agreement with the well-known Carman–Kozeny equation. This facilitates the successive analysis of the propulsion speed, power dissipation, and swimming efficiency of the targeted swimmer—which are decisive in nutrient transport and uptake or reproduction in a collective environment—as a function of the volume fraction. A stress-jump condition is also imposed across a cell to indicate the mean effective force due to nearby swimmers. For suitable values of this stress-jump coefficient, the relative increase in migration velocity and swimming efficiency is noticeably higher at an optimum occupancy. To highlight the rheological characteristics, we have analyzed the effective viscosity of the active suspension, which varies distinctively from a random suspension and alludes to self-assembly.

Filtration and Backwashing Studies: Obafemi Awolowo University Waterworks as A Case Study

This research centres on the study of the filtration and backwashing operations of the filtration unit of the Opa Waterworks and a detailed laboratory study of the filtration and backwashing characteristics of the filter medium being used at the treatment plant. A detailed study of the filtration unit of Opa Waterworks was undertaken with particular emphasis placed on the estimation of the volume of water used during backwashing, the backwashing procedure and the average backwashing time. Specific properties of the filter medium used in the gravity filter such as porosity, density, equivalent density, specific gravity and unhindered settling velocity were investigated in the laboratory. The backwashing properties of the filter medium was also studied using the DSF (Dynamic Shape Factor) and Sphericity models and the predicted results obtained were compared to the actual laboratory results. Based on works carried out, the volume of wash-water required for the gravity filters was estimated as 14,200 litres (14.2 m3) which is about 3.12% of the total volume of the clear water tank, with an outflow rate of 0.01 m3/s. The accuracy of the Blake – Kozeny equation in predicting head loss across a filter bed using clean water runs was investigated and found to be reasonably accurate, and the prediction errors

Pore-Scale Modeling of Electrokinetics in Geomaterials

Pore-scale finite-volume continuum models of electrokinetic processes are used to predict the Debye lengths, velocity, and potential profiles for two-dimensional arrays of circles, ellipses and squares with different orientations. The pore-scale continuum model solves the coupled Navier–Stokes, Poisson, and Nernst–Planck equations to characterize the electro-osmotic pressure and streaming potentials developed on the application of an external voltage and pressure difference, respectively. This model is used to predict the macroscale permeabilities of geomaterials via the widely used Carmen–Kozeny equation and through the electrokinetic coupling coefficients. The permeability results for a two-dimensional X-ray tomography-derived sand microstructure are within the same order of magnitude as the experimentally calculated values. The effect of the particle aspect ratio and orientation on the electrokinetic coupling coefficients and subsequently the electrical and hydraulic tortuosity of the porous media has been determined. These calculations suggest a highly tortuous geomaterial can be efficient for applications like decontamination and desalination.

Convective Stability of a Net Mass Flow Through a Horizontal Porous Layer with Immobilization and Clogging

Solutal convection in a horizontal layer filled with porous media with horizontal seepage of a mixture is investigated considering the solute immobilization and clogging. A flow through porous media is modelled within the standard Darcy–Boussinesq model, and the immobilization process is described by the mobile/immobile media (MIM) approach. To describe the clogging process, the present model takes into account and the dependence of media permeability on porosity within the Carman–Kozeny equation. The presence of immobile (or adsorbed) particles of the solute decreases the porosity of media, and porous media become less permeable. The variation of porosity is modelled by a linear function of solute concentration in the immobile phase. We consider the case of high solute concentrations, in which the immobilization is described by the nonlinear MIM (mobile/immobile media) model. As a result, it was shown that the immobilization leads to the stabilization of the homogeneous filtration regime and to slowing down of the perturbation dynamics. The stability maps were plotted in a wide range of system parameters. The results showed that for some specific value of clean media porosity the system becomes most unstable and dynamics of perturbations (frequency of oscillations) is most intensive. This value corresponds to the minimal effect of porosity change to variation of permeability due to the immobilization.

Permeability of carbon refractory materials used in a blast furnace hearth

Permeability of experimental and standard micropore carbon materials, before and after the heat treatment at 1550 °C, was investigated using gas permeability measurements (GP) and mercury intrusion porosimetry (MIP). Permeability of both materials increased after the annealing. Experimental micropore carbon material was less permeable than standard one, both before and after the heat treatment. Measured gas permeabilities were compared with permeabilities determined based on MIP measurements using two different models: Hagen−Poiseuille and Carman−Kozeny equation. Using the inverse problem approach, for the first time Carman−Kozeny constant was determined for micropore carbon materials. Carman−Kozeny model gave more accurate results (i.e. close to gas permeability measurements) than Hagen−Poiseuille model. Performed investigations showed that mercury intrusion porosimetry might be alternative method to investigate infiltration resistance of low permeable materials.

The Permeability of Novel Hybrid Fiber Composite Material for Use as Diesel Particulate Filters

An investigation is presented into modeling of gas flow through particle/fiber hybrid composite materials designed for use as diesel particulate filters (DPFs). The work leads to the creation of a modified version of the Carman–Kozeny equation, specifically tailored for such structures. Details of computational fluid dynamics (CFD) simulation of gas flow through modeled structures are provided, leading to formulations to account for the loss of surface area associated with sintering and a tortuosity effect dependent on the porosity level. Based on these outcomes, a simple analytical expression is derived, and it is shown that predicted permeability values obtained with it are in good agreement with experimental data.

Pressure-Flow experiments, packing, and modeling for scale-up of a mixed mode chromatography column for biopharmaceutical manufacturing

To obtain consistent chromatographic behavior, it is important to develop resin packing methods in accordance with the characteristics of each resin. Resins, particularly those with a significant level of compressibility, require proper knowledge of the packing methodology to ensure scalable performance. The study demonstrates the applicability of pressure-flow modeling based on the Blake–Kozeny equation for cellulose based resins, using the MEP HyperCel (Pall) resin as a case study. This approach enabled the understanding of the appropriate bed compressibility and the determination of the minimum column diameter that can predict bed integrity during commercial manufacturing scale operation. Studies suggested that scale-dependent wall effects become negligible for column diameters exceeding 20 cm. Pressure-flow modeling produced a minimum compression recommendation of 0.206 for the MEP HyperCel resin. Columns with diameters up to 80 cm packed with this bed compression yielded incompressible beds with pressure-flow curves consistent with model predictions. Model parameter (particle diameter, viscosity, porosity) values were then varied to demonstrate how changing operating conditions influence model predictions. This analysis supported the successful troubleshooting of unexpected high pressures at the commercial manufacturing scale using MEP HyperCel resin, further supporting the applicability of this approach.

Meso-scale simulation of liquid feeding in an equiaxed dendritic mushy zone

A 3D meso-scale model is developed to predict the flow of liquid within a semi-solid binary Fe-C alloy with various equiaxed microstructure, ranging from dendritic to globular. The model domain consists of a set of 8000 grains given by a Voronoi tessellation. Solidification of each grain is simulated independently via a volume average approach, providing the semi-solid microstructure for the fluid flow simulation. A single domain Darcy–Brinkman model is then used to calculate the resulting pressure field.The model results are found to be in good agreement with the Carman–Kozeny equation for two limiting cases of interfacial area concentration Sv, demonstrating the model’s utility in quantifying permeability of semisolid structures where the fluid flow occurs either in the intra-dendritic (within the envelope enclosed by the dendrite) or extra-dendritic (between dendritic grains) regions. Deviation from Carman–Kozeny behaviour is observed with a transition in microstructure, i.e. when the domain contains a mixture of both dendritic and globular structures or when fluid flow occurs simultaneously in the intra-dendritic and extra-dendritic regions. A permeability-microstructure map is created as a function of grain size, secondary dendrite arm spacing, and cooling rate to show the range where the limiting values of Sv are valid and, importantly, where they are not. A comparison of the net volumetric inflow caused by shrinkage and deformation is performed, demonstrating that the shrinkage induced by the peritectic transformation is the dominant factor requiring liquid feeding. The present dendritic fluid flow model is useful in the context of multi-physics modelling of defects in peritectic steel grades and other commercially relevant alloys.

Effects of molecular weight distribution of soluble microbial products (SMPs) on membrane fouling in a membrane bioreactor (MBR): Novel mechanistic insights

While molecular weight distribution (MWD) is one of the most important properties of soluble microbial products (SMPs), mechanisms underlying effects of MWD of SMPs on membrane fouling have not well unveiled. In this study, it was found that, the supernatant of sludge suspension in a membrane bioreactor (MBR) for wastewater treatment can be fractionated into a series of SMPs samples with different molecular weight (MW) fraction. The real gel sample mainly formed by the rejected SMPs on membrane surface had a high specific filtration resistance (SFR) of 1.21 × 1016 m−1 kg−1. The SFR of SMPs samples and the model foulants of polyethylene glycol (PEG) increased with their MW. The change trend of SFR with MW cannot be sufficiently explained by three-dimensional excitation-emission matrix (EMM) and chemical compositions. Tyndall effect analysis indicated that gelating ability of SMPs and PEG in the solution increased with their MW. Scanning electron microscope (SEM) confirmed gel structure changes with the PEG MW. Accordingly, mechanisms based on Carman–Kozeny equation and Flory–Huggins lattice theory were proposed to interpret SFR of SMPs with low and high MW, respectively. Simulating these two mechanistic models on PEG samples resulted in the comparable SFR data to the experimental ones, indicating the correctness and feasibility of the proposed mechanisms. The proposed mechanisms provided in-depth understanding of membrane fouling regarding MW, facilitating to develop effective membrane fouling mitigation strategies.

Nuclear magnetic resonance surface relaxivity and its advanced application in calculating pore size distributions

Abstract Both laboratory and downhole nuclear magnetic resonance (NMR) measurements are commonly used to calculate the pore size distribution (PSD) that is important in determining the reservoir storage capacity and evaluating producibility, etc. The surface relaxivity is a key parameter to calculate the PSD using NMR transversal relaxation time (T2) distributions. Accurate surface relaxivity value would result in a reasonable PSD. In order to comprehensively study surface relaxivity, the conventional sandstone, tight sandstone, and shale samples are selected to perform experiments including porosity, permeability, NMR, mercury injection capillary pressure (MICP) and low temperature nitrogen adsorption (LTNA), etc. Three approaches, including LTNA, MICP, and Kozeny's equation, are used to determine the surface relaxivity. We find that for all samples, the surface relaxivity determined with Kozeny's equation is greater than that using LTNA and MICP data, indicating that surface relaxivity of smaller pores is less than that of bigger pores. It is also observed that the surface relaxivity of conventional sandstone is greater than that of tight sandstone and shales. The surface relaxivity of sandstone and shale samples are both affected by clay content. For shale samples, the surface relaxivity ranges from 3.41 μm/s to 9.79 μm/s and is influenced by organic matter contents as well. An equation for predicting surface relaxivity of shale is established. Finally, we propose an advanced method for calculating reasonable PSDs of sandstone and shale formation by the combination of two surface relaxivities, which could be applied to laboratory and field wells.

Permeability of micropore carbon materials based on steady-state pore-scale flow calculations in real 2D and 3D microstructures

Abstract Permeability of experimental micropore carbon material was investigated based on the solution of steady-state pore-scale flow in geometry representing real material pores in 2D and 3D. Geometries for calculations were created based on the X-ray micro-computed tomography measurements. A numerical solution of the velocity field and the Darcy equation allowed determining the permeability. Results obtained for 2D and 3D geometries were compared. A method of permeability measurements based on the Carman–Kozeny equation for real 2D and 3D microstructures was also investigated and compared with those obtained from the pore-scale flow.

Calibration of NMR porosity to estimate permeability in carbonate reservoirs

Understanding the performance of carbon dioxide (CO2) stored in geologic reservoirs requires some knowledge of the subsurface heterogeneity, where the most reliable data is extracted from wellbore measurements. This advantage is especially true for carbonate reservoirs were reactions between supercritical CO2, the reservoir brine, and carbonate minerals create new storage capacity and enhanced permeability. Techniques like NMR well logging hold promise for estimating subsurface permeability, because the total porosity and pore size can be calculated from response of 1H NMR signal. Resulting depth profiles of porosity and permeability from downhole logs are useful input for reservoir simulations and can be used to design injection protocols and estimate storage capacity.We conducted a detailed characterization study to quantify the NMR response to matrix pore space, the connectivity to larger vugs and fractures, and the resulting permeability of core samples from distinct carbonate formations. The cores used in study span 5 orders in permeability, 4 orders in mean T2, and 2%–42% in porosity. The goal of the study was to use independent measures of pore volumes, surface relaxivity, fluid-accessible surface areas, and permeability to develop carbonate permeability models.It was not possible to develop a single model that reproduced permeability for different carbonate formations, because phenomena specific to the reservoirs could not be extracted even with the state-of-the-art characterization techniques used in this study. Estimates of permeability require calibration. Best estimates for a given formation were achieved by using a simple form of the Kozeny equation (Eq. 6):

A novel method for measuring cake porosity by ions detection technique in a conventional coagulation-ultrafiltration process

Cake porosity plays an important role for evaluating membrane fouling in low-pressure membrane filtration process. Current method for calculating cake porosity is based on Carman-Kozeny equation. It would have many limitations due to the indirect analysis. CLSM has been applied to directly measure the structure of chemical particle models, but requirement that materials need to be fluorescent is difficult to achieve in conventional coagulation-ultrafiltration processes. Our study investigated a novel method using fluorescence probe to measure cake porosity for making up the defect of non-fluorescent of cake layer in conventional coagulation-ultrafiltration process. Two metal ions probes, quercetin and carbon nanodots (CNDs), were applied to detect the Al3+ ions and Fe3+ ions from Al-based salt and Fe-based salt coagulants respectively. Cake morphology could be directly observed by an established method (CLSM and image analysis technique). Using t-test to compare the obtained porosities with those calculated from specific cake resistance by Carman–Kozeny equation. Result shows that they are not significantly different at the confidence level of 95% and 98%, which proves fluorescence identification has higher accuracy. Furthermore, characters of the complexing formed by different cations and hydroxide ions also affect the cake porosity. Porosities of Al-based cake are smaller and those of Fe-based cake are larger. Effect of both fractal dimension and floc size on porosity were also investigated and concluded that role of the fractal dimension is prior to floc size.

Permeability Prediction of Iron Tailings Including Degree of Compaction and Structure

The stability of iron tailings dam is affected by the permeability of tailings. Considering the influence of it, it is necessary to analyze the permeability of tailings so as to prevent the recurrence of Brazilian iron tailings dam accidents. Nevertheless, the results of iron tailings permeability from some prediction equations (Terzaghi equation, Hazen equation, and Kozeny equation) cannot be accurate. Iron tailings are various as they can be divided into three categories: (1) silt content is less than 40%; (2) silt content is more than 40%, while clay content is less than 15%; and (3) clay content is more than 15% and less than 30%. Correspondingly, three equations are proposed to calculate the disturbed and iron undisturbed tailings permeability for the three types. And more accurate results come from it. The water‐flow paths of the iron tailings are blocked after compaction, and the critical pressure of iron tailings blockage is 200 kPa. Although the porosity is large, some of the pores are isolated from each other when the pressure is larger than 200 kPa. However, porosity becomes too large for permeability calculation after compaction and the calculated permeability gets larger as well (equations (24)–(26)). Correcting the permeability calculation equations is an absolute must. The calculated permeability by the revised equations becomes more accurate (equations (27)–(29)). In fact, the granulometric characteristics necessarily play a vital role in the evolution of the pore interconnections by blocking the water‐flow paths and modifying the morphological parameters. More research studies are required to be done in the future.