Kozeny-Carman Relation Research Articles

A percolation-threshold model for evaluating permeability in Portland pastes with high water-cement ratio

In a perspective of improved sustainability, the definition of policies for the management of the heritage of concrete buildings and infrastructures, aiming at the suitable mitigation of the harm to environment and at the reduction of societal costs, requires reliable methodologies to estimate the service-life and maintenance periodicity time for a given class of concrete structures. Concrete watertightness is a basic property for assessing durability of the built heritage to weathering exposure and is mainly dependent on permeability of the cement paste embedding the aggregates which typically have much lower permeability in an ordinary concrete. Fundamental research of the past century has shown that a reliable experimental determination of permeability in Portland concretes and pastes can be achieved in terms of intrinsic permeability by uniaxial water-permeation tests under open-flow conditions.Powers and Brownyard have early assessed in 1947 the ability of the Kozeny-Carman relation in providing predictions of intrinsic permeability of Portland cement pastes, although achieving predictivity only below a water-cement ratio (w/c) of about 0.5. Above such threshold, as they observe, their model is no longer predictive since permeation through the capillary space formed due to excess water becomes predominant over permeation through the gel-space.In the present study an enhancement of Powers and Brownyard's canonical model is proposed by addressing capillary percolation and the possibility of simultaneous flow at the gel and capillary scales. The onset of capillary percolation for w/c slightly above 0.5 is naturally predicted. Agreement of the proposed model is assessed against experimental results and discussed in relation to optimal management of existing RC constructions with a view towards sustainability.

An explicit stabilised finite element method for Navier-Stokes-Brinkman equations

We present an explicit stabilised finite element method for solving Navier-Stokes-Brinkman equations. The proposed algorithm has several advantages. First, the lower equal-order finite element space for velocity and pressure is ideal for presenting the pixel images. Stabilised finite element allows the continuity of both tangential and normal velocities at the interface between regions of different micro-permeability or at the interface free/porous domain. Second, the algorithm is fully explicit and versatile for describing complex boundary conditions. Third, the fully explicit matrix–free finite element implementation is ideal for parallelism on high-performance computers. In the last, the implicit treatment of Darcy term allowed larger time stepping and a stable computation, even if the velocity varies for several orders of magnitude in the micro-porous regions (Darcy regime).The stabilisation parameter, that may affect the velocity field, has been discussed and an optimal parameter was chosen based on the numerical examples. Velocity stability at interface between different micro-permeability has been also studied with mesh refinement. We analysed the influence of the micro-permeability field on the regime of the flow (Stokes flow, Darcy flow or a transitional regime). These benchmark tests provide guidelines for choosing the resolution of the grayscale image and its segmentation. We applied the method on real Berea Sandstone micro-CT images, and proceeded the three-phases segmentation. We studied the influence of the micro-porosity field, using the well-known Kozeny-Carman relation to derive the micro-permeability field from the micro-porosity field, on the effective permeability computed. Our analysis shows that a small fraction of micro-porosity in the rock has a significant influence on the effective permeability computed.

Mafic explosive volcanism at Llaima Volcano: 3D x-ray microtomography reconstruction of pyroclasts to constrain shallow conduit processes

Mafic volcanic activity is dominated by effusive to mildly explosive eruptions. Plinian and ignimbrite-forming mafic eruptions, while rare, are also possible; however, the conditions that promote such explosivity are still being explored. Eruption style is determined by the ability of gas to escape as magma ascends, which tends to be easier in low-viscosity, mafic magmas. If magma permeability is sufficiently high to reduce bubble overpressure during ascent, volatiles may escape from the magma, inhibiting violent explosive activity. In contrast, if the permeability is sufficiently low to retain the gas phase within the magma during ascent, bubble overpressure may drive magma fragmentation. Rapid ascent may induce disequilibrium crystallization, increasing viscosity and affecting the bubble network with consequences for permeability, and hence, explosivity. To explore the conditions that promote strongly explosive mafic volcanism, we combine microlite textural analyses with synchrotron x-ray computed microtomography of 10 pyroclasts from the 12.6 ka mafic Curacautín Ignimbrite (Llaima Volcano, Chile). We quantify microlite crystal size distributions (CSD), microlite number densities, porosity, bubble interconnectivity, bubble number density, and geometrical properties of the porous media to investigate the role of magma degassing processes at mafic explosive eruptions. We use an analytical technique to estimate permeability and tortuosity by combing the Kozeny-Carman relationship, tortuosity factor, and pyroclast vesicle textures. The groundmass of our samples is composed of up to 44% plagioclase microlites, > 85% of which are < 10 µm in length. In addition, we identify two populations of vesicles in our samples: (1) a convoluted interconnected vesicle network produced by extensive coalescence of smaller vesicles (> 99% of pore volume), and (2) a population of very small and completely isolated vesicles (< 1% of porosity). Computed permeability ranges from 3.0 × 10−13 to 6.3 × 10−12 m2, which are lower than the similarly explosive mafic eruptions of Tarawera (1886; New Zealand) and Etna (112 BC; Italy). The combination of our CSDs, microlite number densities, and 3D vesicle textures evidence rapid ascent that induced high disequilibrium conditions, promoting rapid syn-eruptive crystallization of microlites within the shallow conduit. We interpret that microlite crystallization increased viscosity while simultaneously forcing bubbles to deform as they grew together, resulting in the permeable by highly tortuous network of vesicles. Using the bubble number densities for the isolated vesicles (0.1-3−3 × 104 bubbles per mm3), we obtain a minimum average decompression rate of 1.4 MPa/s. Despite the textural evidence that the Curacautín magma reached the percolation threshold, we propose that rapid ascent suppressed outgassing and increased bubble overpressures, leading to explosive fragmentation. Further, using the porosity and permeability of our samples, we estimated that a bubble overpressure > 5 MPa could have been sufficient to fragment the Curacautín magma. Other mafic explosive eruptions report similar disequilibrium conditions induced by rapid ascent rate, implying that syn-eruptive disequilibrium conditions may control the explosivity of mafic eruptions more generally.

A study on the behaviour of offshore suction bucket foundations under cyclic axial compressive loading

ABSTRACT Suction bucket foundations of offshore wind turbines (OWTs) are subjected to cyclic loads induced by wind and waves. The behaviour of buckets embedded in saturated sandy soil subjected to cyclic axial compression was studied using a finite element (FE) model. Accumulation of permanent soil deformations and excess pore water pressure in saturated sandy soil due to cyclic axial compression were taken into account by a fully coupled two-phase model combined with a hypoplastic constitutive model with intergranular strain. The Kozeny–Carman relationship was adopted for porosity–permeability dependence. The main focus was dedicated to the changes in load transfer via components of bucket with increasing number of load cycles. Loads that led to progressive failure, attenuation and shakedown behaviour were described. Effects of soil permeability and density, load frequency and amplitude, the bucket aspect ratio on the load transfer and bearing mechanism of the bucket were presented.

Permeability of packs of polydisperse hard spheres.

The permeability of packs of spheres is important in a wide range of physical scenarios. Here, we create numerically generated random periodic domains of spheres that are polydisperse in size and use lattice-Boltzmann simulations of fluid flow to determine the permeability of the pore phase interstitial to the spheres. We control the polydispersivity of the sphere size distribution and the porosity across the full range from high porosity to a close packing of spheres. We find that all results scale with a Stokes permeability adapted for polydisperse sphere sizes. We show that our determination of the permeability of random distributions of spheres is well approximated by models for cubic arrays of spheres at porosities greater than ∼0.38, without any fitting parameters. Below this value, the Kozeny-Carman relationship provides a good approximation for dense, closely packed sphere packs across all polydispersivity.

Permeability Estimation of Regular Porous Structures: A Benchmark for Comparison of Methods

The intrinsic permeability is a crucial parameter to characterise and quantify fluid flow through porous media. However, this parameter is typically uncertain, even if the geometry of the pore structure is available. In this paper, we perform a comparative study of experimental, semi-analytical and numerical methods to calculate the permeability of a regular porous structure. In particular, we use the Kozeny–Carman relation, different homogenisation approaches (3D, 2D, very thin porous media and pseudo 2D/3D), pore-scale simulations (lattice Boltzmann method, Smoothed Particle Hydrodynamics and finite-element method) and pore-scale experiments (microfluidics). A conceptual design of a periodic porous structure with regularly positioned solid cylinders is set up as a benchmark problem and treated with all considered methods. The results are discussed with regard to the individual strengths and limitations of the used methods. The applicable homogenisation approaches as well as all considered pore-scale models prove their ability to predict the permeability of the benchmark problem. The underestimation obtained by the microfluidic experiments is analysed in detail using the lattice Boltzmann method, which makes it possible to quantify the influence of experimental setup restrictions.

Evaluation of empirical relationships to estimate the hydraulic conductivity of borehole soil samples

ABSTRACT Hydraulic conductivity is a vital hydraulic property of porous media that influences the movement of water and the transport of dissolved contaminants. The present study investigates the influence of key parameters namely, grain size and porosity on the hydraulic conductivity of the borehole soil samples. Further, the applicability of various porosity functions, nine existing empirical relationships, and the developed statistical model was assessed using experimentally measured values. Experimental investigations show that an increase in the grain size and porosity values results in an increase in the value of hydraulic conductivity. Porosity function corresponding to the Kozeny-Carman relationship provides the best fit with the measured hydraulic conductivity. The hydraulic conductivity computed using the Kozeny-Carman relationship indicates close agreement with the measured values followed by Beyer, Hazen, and Alyamani and Sen relationships while other relationships underestimate the hydraulic conductivity of the considered borehole soil samples. The values obtained using the developed model indicate better agreement as compared to those based on the empirical relationships, with the measured values of hydraulic conductivity. The reasonable values of SI, Cc, RMSE, and MAE i.e., 0.012, 0.984, 0.001, and 0.001 respectively, postulate substantial performance of the developed model in computing hydraulic conductivity.

Insight into particle detachment in clogging of porous media; a pore scale study using lattice Boltzmann method

Clogging of porous media is controlled by attachment of particles and their subsequent detachment. In this study, we explore particle transport at the pore scale by considering attachment and detachment processes. We use pore structures of three samples which have similar topological properties but different initial porosities. The main focus is on particle detachment, after they are attached to the solid grain boundaries, and to explore how particle detachment affects hydraulic properties and clogging of porous media. A new approach was proposed by combining the lattice Boltzmann method with a Lagrangian method to simulate fluid flow and particle transport, respectively. Comparisons between several simulations, with and without detachment, were used to evaluate the impact of particle detachment on fluid flow velocity distributions. The coefficients of Carman-Kozeny relation were used to relate porosity changes of each sample to its permeability alterations. Additionally, the influence of the initial porosity of the sample and flow velocity on the detachment process was analysed. The results show that compared to the impact of flow velocity, the initial porosity of the media has a greater influence on controlling the rate of detachment and change of pore structures.

Hydraulic and Mechanical Impacts of Pore Space Alterations within a Sandstone Quantified by a Flow Velocity-Dependent Precipitation Approach

Geochemical processes change the microstructure of rocks and thereby affect their physical behaviour at the macro scale. A micro-computer tomography (micro-CT) scan of a typical reservoir sandstone is used to numerically examine the impact of three spatial alteration patterns on pore morphology, permeability and elastic moduli by correlating precipitation with the local flow velocity magnitude. The results demonstrate that the location of mineral growth strongly affects the permeability decrease with variations by up to four orders in magnitude. Precipitation in regions of high flow velocities is characterised by a predominant clogging of pore throats and a drastic permeability reduction, which can be roughly described by the power law relation with an exponent of 20. A continuous alteration of the pore structure by uniform mineral growth reduces the permeability comparable to the power law with an exponent of four or the Kozeny–Carman relation. Preferential precipitation in regions of low flow velocities predominantly affects smaller throats and pores with a minor impact on the flow regime, where the permeability decrease is considerably below that calculated by the power law with an exponent of two. Despite their complete distinctive impact on hydraulics, the spatial precipitation patterns only slightly affect the increase in elastic rock properties with differences by up to 6.3% between the investigated scenarios. Hence, an adequate characterisation of the spatial precipitation pattern is crucial to quantify changes in hydraulic rock properties, whereas the present study shows that its impact on elastic rock parameters is limited. The calculated relations between porosity and permeability, as well as elastic moduli can be applied for upscaling micro-scale findings to reservoir-scale models to improve their predictive capabilities, what is of paramount importance for a sustainable utilisation of the geological subsurface.

Effect of resolution, reconstruction settings, and segmentation methods on the numerical calculation of saturated soil hydraulic conductivity from 3D computed tomography images

Over the last decade, computed tomography (CT) images have been used increasingly to gain a more complete understanding of the microscale parameters that control the soil processes. Images that are used for that purpose typically are subjected to a number of successive treatments. This article evaluates for a sandy-loam soil, the impact of two CT scan resolutions (26 and 54 μm), three noise reduction levels, four settings of conversion from a 32-bit to an 8-bit image and four segmentation methods (Hapca et al., 2013; Houston et al., 2013a; Schlüter et al., 2010; Otsu, 1979) in an incomplete factorial design, on the morphological metrics and on the numerical calculation of the saturated hydraulic conductivity, Ksat. The calculations reveal variations of up to two orders of magnitude of Ksat. Houston’s and Schlüter’s segmentations appear the most sensitive to the identification of gray voxels in the key throats controlling the water flow. When combined with high noise reduction levels they produce in some extreme cases disconnections of the percolating pores. Hapca’s segmentation produces more robust results. The Kozeny-Carman relation successfully predicted the saturated hydraulic conductivity when using the critical path diameter as the characteristic length, instead of the macroscopic hydraulic radius which appears too integrative to identify the extent of the variation of key throats.

Effects of installation on the cyclic axial behaviour of suction buckets in sandy soils

Suction buckets differ with their easy and cost-efficient installation technique from other foundation types for offshore wind turbines. For successful completion of their installation process, suction is essential, but the imposed seepage leads to the changes in states of the soil in and around the bucket. Especially, a loosening of soil inside the bucket affects the load carrying behaviour of bucket subjected to repetitive loading resulting from environmental conditions. In this study, the behaviour of buckets under cyclic axial compressive loads with considering a possible loosening and related changes in permeability of soil inside the bucket is investigated numerically. In the framework of finite element analysis, a fully coupled two-phase model and a hypoplastic constitutive model are used to describe the saturated sandy soil behaviour under repetitive loading. The porosity-permeability variation is taken into account by Kozeny–Carman relationship. Special attention is dedicated to load carrying behaviour of bucket top plate, inner and outer skirt as well as base and their changes resulting from a loosening of soil inside the bucket with variable aspect ratio. For this purpose, cyclic axial compressive loads which cause an attenuation and progressive failure of soil-bucket system response are considered. The main findings on the changes in load carrying behaviour of bucket are presented and discussed.

XCT-scan assisted flow path analysis and permeability prediction of a 3D woven fabric

Compaction response and resin flow through a fabric are key characterization parameters in Liquid Composite Molding (LCM) processes. In this study, we present a detailed flow path analysis of a complex 3D woven angle interlock fabric using an in-situ X-ray micro computed tomography (XCT) experimental set-up. The set-up is used to obtain the reinforcement compaction response and three principle permeability values at different thicknesses in a single experiment. The compaction provides real-time features of inter-tow gaps, as well as elongation/spreading of weft and warp tows associated with a through-thickness deformation of the binder yarn. The reduction of the pore network during compaction is analyzed through pore size distribution using a granulometry and percolation path analysis. The resin flow path and reinforcement permeability in principle directions are predicted through vigorous digital flow simulations using several computational unit cells extracted from the reconstructed 3D models from XCT scans. The resin flow paths and principle permeability values show a very strong dependence on the solid volume fraction of the scanned images. A relationship between the solid volume fraction and permeability is developed using a modified Kozeny-Carman relation. The predicted permeability values were found to be in good agreement with traditional benchmark experimental data, highlighting the utility of this novel technique.

Stress-Dependent Porosity and Permeability of Porous Rocks Represented by a Mechanistic Elastic Cylindrical Pore-Shell Model

The stress dependency of the porosity and permeability of porous rocks is described theoretically by representing the preferential flow paths in heterogeneous porous rocks by a bundle of tortuous cylindrical elastic tubes. A Lame-type equation is applied to relate the radial displacement of the internal wall of the cylindrical elastic tubes and the porosity to the variation of the pore fluid pressure. The variation of the permeability of porous rocks by effective stress is determined by incorporating the radial displacement of the internal wall of the cylindrical elastic tubes into the Kozeny–Carman relationship. The fully analytical solutions of the mechanistic elastic pore-shell model developed by combining the Lame and Kozeny–Carman equations are shown to lead to very accurate correlations of the stress dependency of both the porosity and the permeability of porous rocks.

A network model for the biofilm growth in porous media and its effects on permeability and porosity

Successful microbial enhanced oil recovery depends on several factors like reservoir characteristics and microbial activity. In this work, a pore network is used to study the hydrodynamic evolution over time as a result of the development of a biofilm in the pores. A new microscopic model is proposed for biofilm growth which takes into account that nutrients might not fully penetrate the biofilm. An important novelty in this model is that acknowledges the continuous spreading of the biofilm over the network. The results from the current study can be used to obtain a new relation between the porosity and permeability which might be used as an alternative to the Kozeny Carman relation.

Effect of convection heat transfer on thermal energy storage unit

Abstract Latent heat storage represents a promising technique to achieve net zero energy buildings. This work investigates the behaviour of phase change material (PCM) inside a rectangular enclosure, which represents the geometry of a latent heat storage system. The left side of the unit is exposed to a constant temperature (Th), while the other three walls are exposed to convection heat transfer boundary condition [h= 5, 10, and 15 W/(m2 K)] and different ambient temperatures (T∞ = 297◦ and 307◦K). The ambient temperatures were selected to be at/above the melting temperature of the studied PCM (coconut oil). To study the melting process of the PCM, the continuity, Navier-Stokes and energy equation were used. The Navier-Stokes equations were modified using the Carman-Kozeny relation. The finite element method was utilized to produce numerical results. The results are presented in terms of flow and thermal fields, Nusselt number (Nu), and the melt fraction (MF) of the PCM. The results show that, when T∞ = Tm, the melting rate of the PCM slows down with increasing the convection heat transfer coefficient. While the melting rate accelerates with increasing the convection heat transfer coefficient when T∞ &gt; Tm.

Stress Dependency of Permeability Represented by an Elastic Cylindrical Pore-Shell Model: Comment on Zhu et al. (Transp Porous Med (2018) 122:235–252)

Stress dependency of permeability of porous rocks is described by means of a theoretical elastic cylindrical pore-shell model. This model is developed based on a bundle of elastic capillary tubes representation of the preferential flow paths formed in heterogeneous porous rocks. The radial displacement caused in tubes by the pore fluid pressure applied over the surface of the elastic cylindrical flow tubes is expressed by a Lame-type equation. The radial displacement is incorporated into the Kozeny–Carman relationship to determine the variation of the permeability of porous rocks by variation of the pore fluid pressure. The solution of this equation yields a semi-analytical equation which provides accurate correlations of the stress dependency of the permeability data of porous rocks. The errors associated with the previous formulation of this problem by Zhu et al. (Transp Porous Med 122:235–252, 2018) are explained in view of the present formulation.

Porosity-permeability relationship in dual-porosity carbonate analogs

We have investigated the effect of micrite content and macroporosity on the porosity-permeability relationship of dual-porosity carbonates using analog samples created in the laboratory. Specifically, we control the micrite-to-coarse-grains ratio to produce samples in which the micrite content is the only parameter changing. We also introduce into the microstructures controlled volumes of an acetone-soluble solid material (camphor) at the expense of the micrite aggregates, which functions as macropores after being dissolved. With regard to the effect of micrite on the porosity-permeability relationship, our results indicated that adding micrite to samples exhibiting a grain-supported microstructure reduces porosity and permeability drastically. By increasing the micrite content up to approximately 30%, the sample becomes micrite supported, at which point adding more micrite increases the porosity but no longer affects the permeability significantly. Samples characterized by a high micrite content were found to have lower permeability at any given porosity. When macropores are introduced at the expense of micrite aggregates, permeability increases drastically with porosity. The rate of increase in permeability decreases, however, as the micrite content of the original microstructure increases. Additionally, at any given micrite content, the permeability increases as the percentage of macropores increases because such pores do contribute more significantly to fluid flow as compared with the micropores characterizing micrite aggregates. We used the varying micrite-to-coarse-grains ratio and its effect on the porosity-permeability relationship to inform the Kozeny-Carman relation for a pack of spheres. Our analysis determined that micrite affects the porosity-permeability relationship of carbonates by reducing the effective particle size and increasing the percolation porosity. Additionally, incorporating the content of micrite and macropores into the analysis of the porosity-permeability relationship increased the coefficient of determination ([Formula: see text]) from 0.24 to 0.78. We concluded that knowledge of micrite content and macroporosity is of paramount importance to interpret and model porosity-permeability relationships in carbonates.

Vibration effect on the onset of thermal convection in an inhomogeneous porous layer underlying a fluid layer

The linear stability problem for a two-layer system consisting of a horizontal pure fluid layer and an inhomogeneous porous layer saturated with the same fluid and heated from below in the presence of vertical high-frequency and small-amplitude vibrations in a gravitational field is studied in the framework of the averaging method. The problem is solved numerically with a shooting method. Bimodal neutral curves of the stability of mechanical equilibrium characterized by the absence of an averaged fluid flow in layers are obtained for the linear profile of porosity within a porous layer and permeability versus porosity through the Carman–Kozeny relation. We consider a competition between perturbations of shorter wavelength (short-wave perturbations) localized mainly in the fluid layer and those of longer wavelength (long-wave perturbations) spanning both layers near the stability threshold in the case when the intensity of vibrations and porosity gradient change. It is shown that when porosity increases with depth, instability is associated with the development of large-scale and long-wave perturbations. The stability threshold of these perturbations weakly depends on the vibration intensity in the range of the parameters considered. Perturbations penetrate into the porous medium poorly when porosity decreases with depth. Vibrations in such a situation stabilize the short-wave instability mode noticeably and lead to an increase in the wavelength of the most dangerous perturbations of equilibrium due to different contributions of inertial effects to fluid and porous layers. It has been found that when the vibration intensity and porosity gradient increase, a hard abrupt transition from short-wave to long-wave perturbations is replaced by a smooth change in the wavelength of critical perturbations related to the loss of stability of equilibrium in the examined two-layer system.

Bed of polydisperse viscous spherical drops under thermocapillary effects

Viscous flow past an ensemble of polydisperse spherical drops is investigated under thermocapillary effects. We assume that the collection of spherical drops behaves as a porous media and estimates the hydrodynamic interactions analytically via the so- called cell model that is defined around a specific representative particle. In this method, the hydrodynamic interactions are assumed to be accounted by suitable boundary conditions on a fictitious fluid envelope surrounding the representative particle. The force calculated on this representative particle will then be extended to a bed of spherical drops visualized as a Darcy porous bed. Thus, the “effective bed permeability” of such a porous bed will be computed as a function of various parameters and then will be compared with Carman–Kozeny relation. We use cell model approach to a packed bed of spherical drops of uniform size (monodisperse spherical drops) and then extend the work for a packed bed of polydisperse spherical drops, for a specific parameters. Our results show a good agreement with the Carman–Kozeny relation for the case of monodisperse spherical drops. The prediction of overall bed permeability using our present model agrees well with the Carman–Kozeny relation when the packing size distribution is narrow, whereas a small deviation can be noted when the size distribution becomes broader.

A simplified model to predict clogging of reactive barriers

A study was conducted to develop a simplified analytical model to predict the progression of a carbonate clogging front in permeable reactive barriers. Permeable reactive barriers constitute an in situ remediation technique for contaminated groundwater and are composed of reactive media (magnesium oxide in this study), whose hydraulic and chemical behaviours must be studied. Models generally used to predict their clogging are relatively complex and involve the classical advection-reaction-dispersion equation. As this equation is strongly non-linear when considering a variable porosity, a simplified model predicting the loss of hydraulic conductivity would allow an easier estimation of their longevity. In this work, the generation of precipitates is predicted using chemical reactions at equilibrium. The geometry of the precipitates is estimated applying the floating sphere model, and the hydraulic behaviour is simplified to two values of permeability in the initial and the clogged states. These values were estimated using the Kozeny-Carman relation, which predicts the hydraulic conductivity of porous media considering their porosity and specific surface. Using these hypotheses, one finds that the clogging front should advance at a constant rate given constant water flow rate. Model results were successfully compared with experimental data from three one-dimensional column filtration tests to confirm this finding.