Kozeny Equation Research Articles

Facies Characterization by Use of Well Logs; San Andres Formation (Permian), Hanford Field, Gaines County, Texas: ABSTRACT

Formulation of depositional models of carbonate reservoirs are now sufficiently sophisticated so that structural and stratigraphic traps of considerable complexity can be identified by use of core data. However, limited core data have, in many cases, made interpretation of the lateral extent and continuity of these beds difficult. As a result, location of wells, determination of zonal continuity, and enhanced recovery operations have presented difficulties to the petroleum geologist and engineer. In an attempt to determine if well logs, using minimal core data, could be used to define carbonate facies by depositional texture, a study of the Sligo Formation (Cretaceous), south Texas, was recently published. This study of a shelf-type stratigraphic trap concluded that high- and low-energy carbonate facies could be readily identified by use of well-log cross plots, allowing for identification of reservoir parameters of size, geometry, zonal continuity, and distribution of porosity and permeability--parameters traditionally determined by use of cores. This study of Hanford field represents a more sophisticated approach to definition of facies by use of well-log cross plots. Based on another study of Hanford field, core descriptions were correlated with well logs, and the digitized logs were then subjected to numerous cross plots. It was determined that facies could be readily disaggregated in this manner, and that permeability could be reasonably estimated using the Kozeny equation, where core data were unavailable. The fact that the San Andres Formation (Permian) has been subjected to extreme diagenesis makes this study a significant step beyond the primary textural patterns encountered in earlier work on the Sligo Formation. Its implications are important, since extensive log data, together with minimal core information, will all w the explorationist and engineer to locate and better define individual productive zones and their extent at reduced cost. End_of_Article - Last_Page 250------------

Conductivity and permeability of rocks

The electrical conductivity of salt-water-saturated rocks is modeled by a random resistance network which has a zero percolation threshold. The porosity is varied by a random bond-shrinkage mechanism. Numerical and analytical calculations of the model in different dimensions show an Archie's-law behavior: ${\ensuremath{\sigma}}_{r}=a{\ensuremath{\sigma}}_{w}{\ensuremath{\varphi}}^{m}$, where $\ensuremath{\varphi}$ is the porosity of the rock, and ${\ensuremath{\sigma}}_{r}$ and ${\ensuremath{\sigma}}_{w}$ are the conductivities of the rock and water, respectively. We find that the Archie's exponent $m$ is always greater than unity and is related to the skewness of the pore-size distribution of the rock. Applying the same model to fluid-flow permeability (${k}_{r}$) gives ${k}_{r} \ensuremath{\propto}{\ensuremath{\varphi}}^{{m}^{\ensuremath{'}}}$, where ${m}^{\ensuremath{'}}=m(m+1)$ in one dimension, and ${m}^{\ensuremath{'}}=2m$ in higher dimensions. This power-law form is consistent with the well-known Kozeny equation and has been frequently suggested by empirical studies. Experimental tests of the model are performed on artificial rocks, made by fusing small glass beads, as well as real rocks. From resistivity measurements, we demonstrate that $m$ is larger in samples with a wider fluctuation of pore sizes, which is qualitatively consistent with the model. From fluid-flow experiments on fused glass beads, we find quantitative support for the ${m}^{\ensuremath{'}}=2m$ prediction.

Fluidization characteristics of fine particles

AbstractA new voidage function is defined based on the Blake–Kozeny equation. This voidage function gives better correlations than those heretofore used in describing minimum fluidization characteristics.Due to the inter‐particle cohesive forces, the channelling and bubbling often exist in the fluidization of fine particles. In consideration of the channelling effect, a correlation of the fluidization friction factor is proposed. It is also shown in experiments that the channelling in fine particle systems is reduced when coarser particles are added.

Estimation of the components of anisotropic permeability from observations of acoustic anisotropy

Decisions on positioning of monitoring wells in the vicinity of hazardous waste sites can be significantly affected by knowledge of the natural anisotropic hydraulic conductivity. That the vertical permeability is usually less than the horizontal permeability is well known: however, the anisotrophy in the horizontal plane may influence the choice of positions for those monitoring wells that are downgradient from the waste-management area. Very little information on the horizontal anisotropy is available. Furthermore, field measurements to estimate the horizontal anisotropy are expensive because at least three interacting wells are necessary, with more than three being preferred. The Kozeny equation suggests that directional anisotropy of the electrical conductivity of a porous media should correlate closely with the anisotropy of the permeability [S. J. Pirson, Geological Well Log Analysis (Gulk Publ. Co., 1981), p. 135]. A close relationship should also exist between the acoustic anisotropy and the anisotropy of the permeability. Recent advances in techniques for measuring acoustic velocities in a borehole make it feasible to obtain estimates of the anisotropy of the permeability from observations of the anistropy of the seismic velocities. [Work supported by EPA.]

The effects of cationised ferritin and native ferritin upon the filtration coefficient of single frog capillaries. Evidence that proteins in the endothelial cell coat influence permeability

To investigate whether certain macromolecules reduce capillary permeability by binding to the surface coat of endothelial cells, the effects of cationised ferritin (CF) upon the filtration coefficient ( L p ) of individually perfused frog mesenteric capillaries were compared with those of native ferritin (NF). With perfusate CF concentrations between 0.1 g 100 ml −1 and 2.5 g 100 ml −1, L p was reduced to approximately 30% of its value for the same vessel perfused with protein-free Ringer solution. Electron micrographs of the perfused capillaries revealed that over this range of perfusate concentrations, CF was concentrated uniformly in the endothelial cell coat, occupying 8.5% of its volume. Neither the effect of cationised ferritin upon L p nor its concentration in the cell coat varied significantly over this range of perfusate concentrations. When perfusate concentration of CF was reduced to 0.01 g 100 ml −1, CF no longer reduced L p and its concentration in the cell coat fell below 2%. Native ferritin, which is excluded from the cell coat, did not reduce L p at a perfusate concentration of 0.1 g 100 ml −1. At a concentration of 2.5 g 100 ml −1, NF reduced L p in a few very permeable vessels ( L p > 60 × 10 −3 μm sec −1 cm H 2O −1) but had no significant effect on vessels with lower and more normal values of L p . The effects of CF upon L p can be described in terms of the Kozeny equation if a major proportion of the hydraulic resistance through the capillary wall is attributed to a fiber protein matrix.

Oxygen Transport in Salmon Spawning Gravels

The importance of understanding transport characteristics of flow through gravel media is discussed from the viewpoint of salmonid enhancement programs. A summary of the important features of the incubation process with respect to mass transport is provided along with applicable theories describing flow through porous media. Data obtained from experiments described herein are used to assess the accuracy of existing correlations for predicting pressure drops across gravel substrates. It is found that available hydraulic relations can be used to predict flow velocity magnitudes in gravel media with an accuracy of ± 50% over a twofold range of flow rates, providing one measurement of head loss is available at one flow rate. An adaptation of the Carman–Kozeny equation is found to be suitable for calculating the influence of fines on permeability. The importance of air entrapment on flow resistance is confirmed experimentally and modeled using available correlations. Lastly, the applications of these results for calculating oxygen transport to incubating salmon eggs and minimum water flows in hatcheries are discussed.Key words: Salmon enhancement, oxygen transport, permeability, gravel, incubation, hatcheries

Clarification of liquids using filter aids Part I. Mechanisms of filtration

Filter aids were characterized by an effective particle diameter and a pore diameter in the filter aid cake, calculated from the specific cake resistance using the Kozeny equation. The results agreed well with those from mercury porosimetry. The transition from surface filtration to depth filtration by filter-aid cakes was studied with uniform polystyrene particles as a model impurity. The critical ratio between pore diameter and impurity diameter was between 2 and 3. At conditions of non-surface filtration, an important concentration of impurity exists in the liquid flowing through the cake causing a danger of blocking. When pre-coat and body feed were used with polystyrene particles as an impurity, the blocking appeared to occur rather easily on top of the original pre-coat and on the filter medium. The type of pre-coat and the way in which it was formed were very important. In beer filtration, which is mainly a surface filtration, a small concentration of passing impurity was able to block the pre-coat layer, when the filter aid of the body feed was coarser than that of the pre-coat.

Hydraulic conductivity in surface active soils

Kozeny's equation based on a capillary tube model with constant viscosity does not give satisfactory results for fine-grained soils. When surface forces dominate over gravity forces, the pore water behaves abnormally and the physical properties of this pore water are found to be quite different from free water. Flow through saturated fine-grained soils is known to be affected by the properties and thickness of loosely and strongly bound water, whose viscosity is observed to be higher because of the modified water structure induced by clay-water interaction. An analytical solution based on a capillary tube model taking into account the changed viscosity of bound water and variation of viscosity within the bound water in relation to its thickness is attempted. The derived equation appears to be general in nature and is applicable for both surface active and inactive soils. It is shown that Kozeny's equation turns out to be the particular case of the derived equation when the thickness of the bound water is zero or when the variation of viscosity is not taken into account.

Permeability Estimates from NML Measurements

This method for estimating permeability from Nuclear Magnetism Log response incorporates the known manner in which NML response relates to pore size and to fluid and matrix properties. It also relies on the specific relations between mercury injection capillary pressure curves and permeability. Although the method as described here applies to sandstone, it is general enough for other applications. Introduction This paper presents a quantitative technique to estimate permeability from the Nuclear Magnetism Log response and from an independent determination of porosity. The introduction of new tools and porosity. The introduction of new tools and technology within the past decade has permitted lithology, fluid saturation, and porosity determinations from log response to become quite sophisticated. However, permeability estimation is still empirical and usually permeability estimation is still empirical and usually requires a detailed study of cores and logs to establish appropriate relations within a given field. The determination of permeability from logging measurements has remained elusive because the response of most logs is highly influenced by the composition, the fluid content and the electrical conductivity of the clays within the rock matrix. The Nuclear Magnetism Log (NML), however, responds only to unbound fluids within the rock pores. Since the size and configuration of that portion of the pore network containing unbound fluids also control rock permeability. a logical approach is to took to the NML as a means to deduce in-situ permeability. The NML has long been recognized as a means of conveying information related to permeability. Publications that appeared in the early 1960's, such Publications that appeared in the early 1960's, such as those by Hull and Coolidge, Brown and Ganison Coolidge and Gamson, and Wyman, all indicate that a qualitative relation between permeability and free fluid index (FFI) exists. Seevers subsequently developed an equation for sandstones, relating permeability to FFI, the observed thermal relaxation permeability to FFI, the observed thermal relaxation time of the fluid-saturated rock, and the thermal relaxation time of the saturating liquid when not confined in the rock pores. This equation required a constant of proportionality that remained constant within a given well but that varied from well to well. His method incorporated the physical model of Korringa et al. where the observed T1 was attributed to a component characteristic of bulk liquids and a shortened T1 component due to proximity to the rock matrix. It further depended upon a modification of the Kozeny equation, which related permeability to porosity, specific surface area, and tortuosity. Artus also considered an adaptation of the Kozeny equation and found that results, although encouraging, yielded calculated permeabilities that were consistently low. Timur et al. has used an adaptation of the equation proposed by Seevers to empirically relate permeability proposed by Seevers to empirically relate permeability of core samples to laboratory NMR measurements. All of these attempts to quantitatively relate permeability to NML response have depended upon some adaptation of the Kozeny equation. In addition, some are based upon a theory that does not take into account the diffusion of the protons within the liquid phase and the corresponding influence upon thermal relaxation time, T1. JPT P. 923

Hydraulic Conductivity Related to Porosity and Swelling of Soil

AbstractThe hydraulic conductivity factor in Darcy's equation for viscous flow is analyzed in terms of a modified Kozeny equation. The modifications include a correction for swelling, and the replacement of the cubic exponent of the porosity term by an exponent varying between 0 and 3. The choice depends on the interplay between surface and volume of soil pores, as determined by flocculation‐dispersion conditions. Swelling is calculated from the theory of the diffuse double layer. A theoretical analysis of experimental results on the hydraulic conductivity of two soils for percolants of variable concentration and composition showed that a single value for the electrical potential parameter sufficed to match theoretical with experimental data when the concentration of the percolant ranged from 10 to 100 meq/liter. Taking into account effects of pore size, pore density, effective porosity, and flow matrix on the hydraulic conductivity, the above has been explained in terms of expansive swelling of smaller pores at the expense of larger ones. A number of published studies have been analyzed in the light of the theory proposed. The relevance of the theory with regard to the alleged contrast between grain and pore models of soil has been considered.

13—AN AIR-FLOW METHOD FOR THE DETERMINATION OF THE FIBRE FINENESS OF JUTE AND MESTA

An account is given of air-flow experiments with paraIlel-fibre plugs of jute and mesta (kenaf). Over the working range of packing density of 0-4-0-6 g/cm3, the Kozeny equation was found to be approximately true. The dependence of κ on the porosity, ∊, was examined and κ was found to be slightly low for white jute. The effect of combing on κ was small. Random-fibre plugs were unsuitable. On the basis of these observations, a simple direct-reading fibre-fineness tester with a constant-flow aspirator, suitable for use in Indian rural areas, has been developed for jute and mesta.

13—AN AIR-FLOW METHOD FOR THE DETERMINATION OF THE FIBRE FINENESS OF JUTE AND MESTA

An account is given of air-flow experiments with paraIlel-fibre plugs of jute and mesta (kenaf). Over the working range of packing density of 0-4-0-6 g/cm3, the Kozeny equation was found to be approximately true. The dependence of κ on the porosity, ∊, was examined and κ was found to be slightly low for white jute. The effect of combing on κ was small. Random-fibre plugs were unsuitable. On the basis of these observations, a simple direct-reading fibre-fineness tester with a constant-flow aspirator, suitable for use in Indian rural areas, has been developed for jute and mesta.

A Simplified Method for Determining Specific Surface

Abstract A simplified method for evaluating specific surface of porous media is described. The theory takes as its point of departure a previous statistical method. Due to its simplicity, it is considered that the proposed method has certain advantages over methods currently used. By making some simplifications, the procedure for estimating specific surface is reduced to superposing, on an enlarged reproduction of a section of the sample to be analyzed, an evenly spaced grid and counting the number of intersections of the grid lines with the perimeter of pores. An indirect test of the theory is presented and the advantages of the method are discussed. Introduction The specific surface of a porous medium, defined as the surface area of the pores per unit bulk volume, is an important parameter with regard to flow of fluids through porous materials. In particular, it has long been recognized that specific surface is closely related to permeability. Since the internal surface of any porous material is extremely complex, the measurement of specific surface can only be accomplished by indirect means. It is considered that the best method in current use for determining specific surface is the statistical method of Chalkley, Cornfield and Park who originally applied it for estimating volume-surface ratios of morphologic components, such as cells or nuclei. This method will be outlined later. A method which has been used extensively in determining specific surface is based on the adsorption of a vapor by the internal surface of porous media. However, this method does not yield the surface pertinent to fluid flow, since it includes the tiny molecular interstices which do not influence the permeability of the porous material. This method has been employed by Brooks and Purcell to measure internal surface area of sedimentary rocks. The Kozeny equation, which relates the specific surface to the permeability of a porous medium, has found wide use in determination of specific surface. However, the Kozeny equation is not strictly correct. Large discrepancies have been found between theory and experiments. Consequently, the specific surface estimated by this method is subject to serious uncertainties. Finally, there are other methods for specific surface measurements which are based upon a variety of physical and chemical properties: the heat conduction by a gas, ionic adsorption phenomena and direct chemical reactions of the surface with other substances. This paper describes a simplified statistical method for determining specific surface of porous materials. It is considered that, due to its simplicity, it has certain advantages over the methods presently used. For its validity, it is required that the porous structure of the material under study be of a spatially random nature. This is not a serious limitation since the naturally occurring porous media have this characteristic. THEORY The method proposed in this paper takes as its point of departure the interesting result obtained by Chalkley et al. The method shows that if a line of length r is dropped repeatedly at random over a section of a porous medium. and counts are made of the number of times the two end points fail within pore areas (denoted by h for "hits") and the number of times the line intersects the perimeter of pores (denoted by c for "cuts"), then in a very large number of throws ................................(1) where S is the internal surface of the sample (surface of pores) and V, the volume of the pores (Fig. 1).

Clarification of suspensions: a study of particle deposition in granular media: Part II—A theory of clarification

The observations in Part I [1] on particle deposition have been incorporated into theoretical equations to describe pressure drop and concentration over the complete clarification run—i.e. from the initial transient to the final steady state. The equations differ from those of previous workers in having no arbitrary constants or empirical correlations—all parameters have a precise physical significance and the pressure drop is derived directly from the Kozeny equation. Experiments were performed on a model bed, a packed bed and a sintered disc filter. In all three cases, deviation from the theoretical curves was about 5 per cent maximum in those runs in which parameter values were measured directly, and about 15 per cent maximum when these parameters were estimated completely independently of the run.

Theory of Water Filtration

The Kozeny equation of laminar flow through granular materials is extended to the transition region by means of a chart of the friction factor versus the Reynolds number. A chart of the backwashing number versus the Reynolds number is developed for estimating wash rates required at impending suspension and during suspension of filter grains. The friction factor during backwash decreases greatly at porosities greater than approximately 0.6. Small floc particles deposit in a filter, as a sheath, around the sand grains. A chart for the easy computation of the deposit ratio from the sand size and hydraulic gradient is presented. The rate of removal for a given volumetric concentration of floc in the water at each depth is found to decrease greatly with depth. This decrease in effectiveness is related to the adhesiveness of the floc to the sand.

A RELATION BETWEEN PERMEABILITY AND SIZE DISTRIBUTION OF PORES

SummaryAn equation has been derived for the relation between permeability and the size distribution of the pores in isotropic material. If the mean radius of the pores in each of n equal fractions of the total pore space is represented in decreasing order of size by r1, r2., and rn cm. respectively, then permeability is given by image where is the porosity in cm.3/cm.2 of porous material, and K is in units of cm.2 By this means, permeability can be calculated from she curve relating water content to suction. This has been tested on published data for flow of air through porous stones and flow of water through saturated and unsaturated sands. Calculated values have been found to agree satisfactorily with measured values over a wide range of permeability.The Kozeny equation is discussed as a particular case of this equation

The Mechanism of Filter Pressing and Slip Casting

The literature on the mechanism of the slipcasting process is reviewed. An equation is developed to give a quantitative description of the rate at which the clay layer is deposited from the slip in the filter‐pressing and slip‐casting processes. Certain factors in the equation can be evaluated by the application of the Kozeny equation. Agreement between the calculated and observed rates of solids deposition is satisfactory and confirms that the mechanism in both processes is simple dewatering.

Permeability and the Size Distribution of Pores

MANY attempts have been made to find a relation between permeability and other measurable properties of porous materials such as particle-size and porosity. The Kozeny equation, which depends on surface area of particles and porosity, has come into rather general use in fields concerned with flow of water, oil and gases and a considerable literature has developed around it1. Its weaknesses are that it is unsuited to material having a wide range of pore-sizes, and that it makes use of an empirical factor which departs seriously from its accepted value in consolidated materials. Attention has also been given to the possibility of using pore-size instead of particle-properties, and Childs and Collis George2 have developed a permeability equation on this basis. An empirical factor is involved and this has not been widely tested for constancy in different materials.

STUDY ON THE LOOSENESS OF COTTON

To express the looseness of cotton on the basis of the porosity of sample, theoretical formula were developed as formula (2) from D'Arcy's law and Kozeny's equation on laminar air flow through the sample of cotton. In the formula (2) c; Kozeny's constant, λ; porosity of sample, s; specific surface of material, w; weight of sample, q; air flow rate passed through sample and p; pressure difference between front and back of the sample.By using the apparatus showed in Fig. I and varying the weights and porosities of samples from three diffrent kinds of cotton, some characteristic results were obtained as follows:(1) If the weight and porosity of one kind of sample are eqaul, values of q/p are constant.(2) If the weights of the same sample are equal, values of q/p are related to λx.(3) In the same kind of material, values of wq/p are correlated to λx.(4) When the porosity of one sample is equal to that of another of different kinds of materials, values of wq/p are proportional to 1/s2.(5) Irrespective of the kinds of materials, values of s2wq/p in each sample are related to λx only.(6) The curve between λx and 1/c plotted from experimental results and by the formula (1) is nearly equal to E. Lord's curve.(7) C is expressed as the function of s2wq/p.Then, the experimental result betweee C and s2wq/p is shown by formula (10). From (2) and (10), λx and the looseness of cotton Ox are shown by formula (12), (13).

Sedimentation of naturally occurring polyelectrolytes

An invariance of sedimentation rate with molecular weight was observed between certain concentrations for the polyelectrolytes sodium carrageenate and sodium alginate. This result was explained by assuming a network model which, at low concentrations, breaks up to permit sedimentation of the molecules individually. By applying the Kozeny equation to the flow through the network the effective specific volume and surface of the molecules were determined.