Terms Of Tortuosity Research Articles

Characterizing the movement patterns of minibus taxis in Kampala's paratransit system

Urban travelers in Africa depend on minibus taxis for their daily social and business commuting. This paratransit system is loosely regulated, self-organizing, and evolves organically in response to demand. Our study used floating car data to analyze and describe the movement characteristics of nine minibus taxis in Kampala, Uganda. We made three intriguing findings. Firstly, in searching for, picking up and transporting passengers, minibus taxi trajectories followed a heavy-tailed power-law distribution similar to a “Lévy walk”. Secondly, their routes' topology and shape gradually changed. Thirdly, the extraordinary winding (expressed in terms of tortuosity) of the paths suggested the extreme determination of the drivers' search for passengers. Our findings could help city planners to build on the self-organizing characteristics of the minibus taxi system, and improve the mobility of travelers, by optimizing routes and the distribution of public amenities.

Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

Endovascular treatment of the femoro-popliteal artery has recently become a valuable therapeutic option for popliteal arterial aneurysms. However, its efficacy remains controversial due to the relatively high rate of complications, such as stent occlusion as result of intra-stent thrombosis. The elucidation of the interplay among vessel geometrical features, local hemodynamics, and leg bending seems crucial to understand onset and progression of popliteal intra-stent thrombosis. To this aim, patient-specific computational fluid dynamic simulations were performed in order to assess the intra-stent hemodynamics of two patients endovascularly treated for popliteal arterial aneurysm by stent-grafts and experiencing intra-stent thrombosis. Both Newtonian and non-Newtonian blood rheological models were considered. Results were presented in terms of tortuosity, luminal area exposed to low (< 0.4 Pa) and high (> 1.5 Pa) time-averaged wall shear stress (TAWSS), area exposed to high (> 0.3) oscillatory shear index (OSI), and flow helicity. Study outcomes demonstrated that leg bending induced significant hemodynamic differences (> 50% increase) in both patients for all the considered variables, except for OSI in one of the two considered patients. In both leg configurations, stent-graft overlapping induced a severe discontinuity of the lumen diameter where the proximal stented zone is characterized by low tortuosity, low velocity, low helicity, low TAWSS, and high OSI; while the distal part has higher tortuosity, velocity, helicity, TAWSS, and lower OSI. Sensitivity study on applied boundary conditions showed that the different inlet velocity profiles for a given inlet waveform affect slightly the numerical solution; conversely, the shape and magnitude of the prescribed inlet waveform is determinant. Focusing on the comparison between the Newtonian and non-Newtonian blood models, the area with low TAWSS is greater in the Newtonian model for both patients, while no significant difference occurs between the surfaces with high TAWSS.GraphicPatient-specific computational fluid dynamic simulations were performed in order to assess the intra-stent hemodynamics of two patients endovascularly treated for popliteal arterial aneurysm and experiencing intra-stent thrombosis. Both Newtonian and non-Newtonian blood rheological models were considered. In both straight and bent leg configurations, stent-graft overlapping induced a severe discontinuity of the lumen diameter where the proximal stented zone is characterized by low tortuosity, low velocity, low helicity, low time-averaged wall shear stress (TAWSS), and high oscillatory index (OSI); while the distal part has higher tortuosity, velocity, helicity, TAWSS, and lower OSI.

The Effects of Cycling on Ionic and Electronic Conductivities of 18650 Li-Ion Cells

During cycling or other aging processes, the capacity of lithium-ion batteries decreases and it is also observed that cell impedance increases. In order to better understand these processes, we are investigating the electronic conductivity and ionic conductivity of electrodes harvested from 18650 cells that have been cycled by different amounts. Effective ionic conductivity of a porous electrode film is sometimes expressed in terms of tortuosity, which is correlated by a Bruggeman-type relationship [1]. A reasonably accurate method of measuring tortuosity is using the previously developed blocking electrolyte method [2]. As for effective electronic conductivity of electrodes, our group has developed experimental methods for accurately making this measurement as well [3]. In this work, the effects of cycling on both ionic and electronic conductivities are examined. We investigate different commercially made 18650 (cylindrical) cells, including those with iron phosphate and cobalt oxide cathodes. Each cell was charged and discharged for a different number of cycles, leading to increasing capacity losses. After certain number of cycles, each cell was cut open and the electrodes were harvested for testing. We will provide details on this experimental method as well as the conductivity results. [1] Zacharias et al, Electrochem. Soc. 160, A306 (2013). [2] Pouraghajan et al., Electrochem. Soc. 165, A2644, (2018). [3] Lanterman et al., Electrochem. Soc. 162, A2145, (2015). Figure 1

Tortuosity and connectivity evaluation by CFD simulation for morphological characterization of membranes and catalytic structures. Case study: CaF2-like structure

A systematic methodology to characterise porous structure of membranes and catalysts in terms of tortuosity and connectivity is presented. The considered case study is the CaF2-like structure of bidisperse spherical particles (i.e., inner and outer particles with different diameters), which are allowed to overlap. Consequently, the morphology of the resulting structure is shown to be completely determined by two geometrical parameters (two degrees of freedom). For this investigation, a Morphology Map is developed, it being a plot mapping void connection as a function of the characteristic geometrical parameters. Within such a map, we introduce the so-called Minimum Connection Curve, which is a characteristic curve representing the boundary between the connected region and the disconnected one (zero connectivity). It is then found that the CaF2-like structure has three specific points where there is an abrupt change in the morphology behaviour, also correlated to the normalised surface area. Then, the structure tortuosity and connectivity are systematically evaluated in a wide range of geometrical conditions of the two parameters characterising the considered structure. As a result, it is shown that it is not possible to express these parameters as a unique function of porosity. However, both tortuosity and connectivity are found to be correlated with a good approximation in a certain range of porosity values with a geometrical parameter that we called Effective Porosity, which is the porosity available for diffusion at the minimum connection conditions. Convenient empirical correlations are eventually provided to allow readers to evaluate tortuosity and connectivity in a wide range of values.

The Effects of Aging on the Tortuosity of Li-Ion Battery Electrodes

Li-ion battery performance is greatly affected by how well ions travel through the electrodes, which in turn depends on the microstructure of the porous pathways through the electrodes. During cell formation and cycling, changes in the microstructure therefore lead to changes in ionic transport. Currently what causes these changes and their magnitude is poorly understood. Changes in effective ionic conductivity and diffusivity can measured in terms of tortuosity, a dimensionless geometric factor. This report focuses on the effect of cell assembly and repeated cycling on tortuosity. Using a method previously developed by our group, namely the polarization-interrupt method [1], as well as AC impedance techniques, we were able to determine electrode tortuosity for various electrode films. There is clear evidence of changes in tortuosity during cell formation and cycling steps. These experiments also suggest that electrolyte wetting is a time-dependent process that changes the measured tortuosity. Three different stages of electrodes were compared: (1) pristine electrodes which had never been in contact with electrolyte, (2) electrodes harvested from newly assembled batteries, and (3) electrodes harvested from cycled batteries. This was done for both anodes and cathodes. In the case of the harvested electrodes, a careful washing procedure is needed to remove Li salt. To validate our results we used two different experimental methods, namely the polarization-interrupt method and a blocking-electrolyte method recently discussed by Gasteiger and coworkers [2]. This research was funded by the BMR program of the US Department of Energy. [1] Zacharias et al. , J. Electrochem. Soc. 160, A306 (2013). [2] Landesfeind et al., J. Electrochem. Soc. 163, A1373, (2016). Fig. 1 SEM/FIB cross section of Li-ion anode showing tortuous paths created by the pores in between active material. Figure 1

Analysis and Quantification of Wellbore Tortuosity

Summary Small-scale wellbore tortuosity—variations in attitude on a length scale smaller than standard survey intervals of 30 m (100 ft)—is generally neglected because of its small effect on the final position of the well and its unclear relation to traditional dogleg severity (DLS). However, it is well-known that such tortuosity may have significant influence on the drilling process and on drilling efficiency. Furthermore, it is a crucial factor for the design and installation of completions and production equipment, because a highly tortuous wellbore section, depending on borehole diameter and tortuosity amplitude and along-hole distribution, may exert strong bending forces on such equipment, or high friction on moving parts. This paper describes a novel methodology for analyzing the tortuosity of openhole wellbores, casing, drillstring, and production tubing. Several related tortuosity parameters are described, and examples of application to field data are included. The methods use high-resolution survey data [measured depth (MD), inclination, and azimuth], which may, in principle, originate from any surveying tool or service capable of providing such data. The methodology requires input data from a single survey only. On the basis of a user-defined length as single external parameter, tortuosity can be analyzed on any length scale greater than approximately 10 times the input survey-data interval, and with a maximum resolution equaling twice the data interval. The processed parameters include relative elongation of a tortuous section compared with a straight-line section, transverse displacements from the straight line, and maximum available diameter for a downhole device caused by the small-scale bendings of the section. The results can be displayed as graphs vs. MD, or as 3D-rendered views of the actual wellbore or tubing shape. Results from various field cases are included, in which the tortuosity analysis was applied to high-resolution continuous gyro survey data collected in cased wellbores. In all the field cases, the novel methods revealed sections of considerable tortuosity that were either unnoticed, or located with unacceptably low accuracy, by conventional methods. These results led to re-evaluation of the planned locations for completion and production equipment. The characterization of the wellbore in terms of tortuosity on various length scales may be of crucial importance for the functionality and lifetime of permanently installed equipment. For example, identification of highly tortuous sections will aid the placement of rod-guide wear sleeves, increasing the rod and casing life and reducing the workover frequency. Another application is the identification of low-tortuosity sections in which downhole pumps or other equipment will not be subject to excess bending. In addition, the tortuosity results may help evaluate the drilling equipment and the drilling process.

The Effect of Electrode Morphology on Solid Oxide Fuel Cell Performance

The morphology of the active electrode in a solid oxide fuel cell has a strong impact on both the transport properties and the active triple phase boundary length and distribution. The performance of the electrode emerges from the interaction of these two parameters and is impacted strongly by the size and shape of the starting powders used to make the ionically and electronically conducting phases of the electrode as well as by the manufacturing method used and the sintering process post manufacture. This paper presents a numerical study over a range of electrode morphologies chosen to represent different manufacturing processes and presents comparisons of geometric characteristics, transport properties and electrode performance. Three different morphologies are generated from three different starting powders, spheres representing non aggregated powders, agglomerates of spheres and high aspect ratio cylinders modelling splats formed in plasma sprays. Of particular interest in this study is the effect of anisotropy of the base particles on the resulting electrode. For each base particle shape multiple stochastic realizations of densely packed particle systems are generated using a collective rearrangement algorithm. The structures are then analysed geometrically with methods from spatial statistics using our own Java-based software, and meshed, solved and analysed with respect to transport and performance using a custom modified version of the open source computational fluid dynamics package, openFOAM. Detailed comparisons are presented in terms of tortuosity of paths through each phase of the material, constrictivity and chord lengths in different directions, directional transport properties in each of the three phases, distribution of the triple phase boundary lines and electrode performance

Modified Kozeny–Carmen correlation for enhanced hydraulic flow unit characterization

Although several techniques have been proposed to predict permeability using porosity–permeability relationships, the Kozeny–Carmen (K–C) correlation is the most widely acceptable methodology in the oil industry. Amaefule et al. (1993) modified that correlation introducing the concept of Reservoir Quality Index (RQI) and Flow Zone Indicator (FZI) to enhance its capability to capture the various reservoir flow behavior based on its respective characters. Yet, there are challenges in using the original correlation due to its inherent limitations and over simplified assumptions that prevent accurate Hydraulic Flow Unit (HFU) definitions. This research addresses some of those shortcomings and proposes a modified K–C correlation by handling the tortuosity term in a more robust manner. Core data from major carbonate reservoirs in Saudi Arabia is used to test the model. Additional data sets obtained from literature on sandstone reservoirs are used as well to demonstrate the global applicability of the proposed model. Results show that more permeability variations are to be expected within a given HFU. Moreover, the conventional model underestimates permeability values within a specific HFU significantly in comparison with the new model.

A fractal model for determining oxygen effective diffusivity of gas diffusion layer under the dry and wet conditions

In this paper, we propose a new fractal model to determine the oxygen effective diffusivity of the porous gas diffusion layer (GDL) in proton exchange membrane fuel cell for both the dry and wet conditions. The model considers the GDL structure in terms of tortuosity and area fractal dimensions and also takes the Knudsen effect into account, which has no empirical constant but parameters with physical meanings. The fractal model is verified by the fair agreement with the experimental data and the results obtained from existing models. It is revealed that the Knudsen effect is essential for the understanding of the oxygen transport through the GDL. Under the dry condition, the oxygen effective diffusivity increases with higher area fractal dimension and lower tortuosity fractal dimension. For the wet condition, the water condensation in the GDL of mixed wettability is considered; and it is found that the behavior of oxygen effective diffusivity with liquid saturation depends on the GDL wettability. It is further revealed that for a given liquid saturation, the oxygen effective diffusivity increases with greater area fractal dimension; with the decrease of tortuosity fractal dimension, it also increases, except for the hydrophilic case at high liquid saturation.

Oil migration in chocolate: A case of non-Fickian diffusion

Oil migration through filled chocolate products during storage periods is inevitable and is responsible for the loss of the original quality of the product. A model that predicts the extent of oil migration can help food engineers optimize the post-production period, thereby delivering an acceptable quality product to consumers. In this work, a predictive model is proposed based on an explicit formulation of the diffusion problem in terms of the molecular diffusivity and the internal microstructure of chocolate. This work overcomes the limitations of the previous models by including a moving boundary, which allows for swelling of the chocolate slab and replacing the effective diffusion coefficient with a more physically meaningful one, which includes a tortuosity term that varies with oil concentration. The model has been validated against experimental mass uptake curves.

Modeling hydrodynamics of trickle‐bed reactors at high pressure

AbstractA phenomenological hydrodynamic model of a trickle‐bed reactor developed predicts pressure gradient and liquid holdup at high pressures. The model consists of 1‐D force balance equations incorporating particle–gas drag, particle–liquid drag and gas–liquid interactions. The deviation between the measured and theoretical pressure drop is explained in terms of tortuosity and axial dispersion effects. Tortuosity encountered by the flowing liquid and gas phases has two components: (1) tortuosity angle (θ), the measure of the tortuous path the fluid would take; (2) an excess friction factor, which would approximate for resulting excess drag while the tortuous path is taken. These parameters were observed to play a major role in modeling the hydrodynamics in the trickle‐bed reactor. The model takes into account the tortousity effects in a unique way. The axial dispersion effect comes into the picture due to the back‐mixing, modeled in terms of the Peclet number (Pe) in a novel way. The overall model was tested with major experimental data for the high‐pressure concerning various types of gas–liquid systems, including hydrocarbons, aqueous solutions, coalescence inhibiting liquids, and so on, that are available in the literature, and the prediction has been found to be excellent.

Transport Properties of Porous PVDF Membranes

Polymer electrolytes based on polyvinyilidene fluoride are widely studied for their applications in lithium batteries and proton fuel cells. Semicrystalline, porous membranes are conveniently obtained by phase inversion methods. In this paper, we report on the preparation of membranes with porosity ranging from ∼7% to ∼85% and morphology which depends on the preparation conditions. The contributions to the overall electrolyte uptake given by pores filling up and swelling of the polymer amorphous phase are separated. The ionic conductivity of the membranes is discussed in terms of tortuosity and percolation through a continuous medium, that can be approximated by a fractal Bethe lattice.

Two- and three-dimensional heterogeneity in carbonate sediments using resistivity imaging

Volume heterogeneity was investigated in carbonate sediments using fine-scale electrical resistivity data, which are sensitive to porosity and to sediment macrostructure and microstructure. Variability in sediment density and porosity was assessed by X-radiographic and two-dimensional (2D) electrical resistivity measurements in 3-cm-thick slabs collected from the Dry Tortugas in the lower Florida Keys. A comparison of the 2D correlation lengths calculated from each assessment indicates that, using methods that differ in resolution, these two methods measure quite different horizontal and vertical fluctuations in sediment density and porosity. Other electrical resistivity experiments were conducted using a 2D network of resistivity electrodes on the surface of freshly collected box cores. The resulting three-dimensional (3D) resistivity data provide a unique insight into the spatial variability of sediment porosity and structure at the cm scale. Complementary X-radiograph cores provide 2D datasets of resistivity and porosity at higher resolution. These data are used to establish the porosity–resistivity–microstructure relationships for these carbonate sediments, with microstructure being described in terms of tortuosity. These relationships are used to extend our interpretation in terms of porosity and tortuosity to the corresponding 3D box core resistivity datasets. Sediment tortuosity is investigated by numerically modeling the flow of electrical currents through a range of pore morphologies in three dimensions. The results show particular sensitivity to the intra-particle porosity, particle shape and the relative sizes of pores and throats. The 3D methodology shows promise as a non-invasive measurement of buried inhomogeneities that could lead to improving models for predicting acoustic backscattering from the sediment volume.

A Closure Model for Transient Heat Conduction in Porous Media

Equations governing the transient heat conduction in porous materials consisting of solids and fluids of different thermal properties were derived with a volumetric average scheme under the assumption of nonthermal equilibrium. The derivation leads to a macroscopic two-equation system which requires the closure modeling of new unknown terms due to interfacial transport, namely, the tortuosity term and the interfacial heat transfer term. Closure relations were obtained from the microscopic equations for temperature fluctuation under quasi-steady assumption. The closure coefficients appeared in the closure relations then depend on the media geometry as well as thermal properties. To demonstrate these dependencies, the closure coefficient for the thermal tortuosity is evaluated based on the effective stagnant thermal conductivity model proposed by Hsu et al. (1995) for periodically packed cubes, and the coefficient for interfacial heat transfer based on a quasi-steady heat conduction of dispersed spheres immersed in fluids. The salient features as well as the applicability and limitation of the newly proposed transient heat conduction model were discussed.

A comparison of relative nonwetting‐phase permeability models

We compared three groups of models commonly used to describe the dependence of relative nonwetting‐phase permeability on phase saturation in porous media and tested the capability of these models to predict relative air permeabilities using two experimental data sets. The first group of models is purely empirical, involving no fitting parameters. In models of the second type, denoted here as geometry‐based models, relative nonwetting‐phase permeability is derived from the wetting‐phase retention curve on the basis of an idealized concept of pore space geometry. According to these models, nonwetting‐phase permeability should vary over the full range of saturation and should be a function of total nonwetting‐phase saturation. In models of the third type this variation is limited to a narrower range based on the concept that only part of the nonwetting phase contributes to permeability and assuming that the relationship between the “effective” saturation of this continuous nonwetting phase can be described by the same mathematical functions as between total nonwetting‐phase saturation and relative permeability in the geometry‐based models. Since this implies some kind of analogy, we refer to this group of models as “analogy‐based.” We found that matching the saturation at which nonwetting‐phase permeability emerges, i.e., the emergence point, is crucial for the description or prediction of relative nonwetting‐phase permeability in the range of high wetting‐phase saturations. Using analogy‐based models which were derived from the geometry‐based models by simply rescaling the effective saturation range to fit this endpoint, predictions were found to depend sensitively on whether the Burdine or the Mualem model was chosen to relate the permeability function to the retention curve. Differences between these two approaches were found to be primarily due to the different exponents in the tortuosity term. When we made this parameter flexible and determined it by curve‐fitting from measured data, we obtained very good fits of experimental air permeabilities.

Two- and three-dimensional heterogeneity in carbonate sediments using resistivity imaging

Improving the characterization of buried inhomogeneities will lead to better models for predicting bottom acoustic backscattering. Sediment volume heterogeneity has been investigated using fine scale electrical resistivity data which is sensitive to porosity and sediment structure, both macro and micro. 3D electrical resistivity experiments are described where a 2D network of resistivity electrodes was used on the surface of freshly collected box cores from the Dry Tortugas, Key West. The resulting 3D resistivity data provides a unique insight into the spatial variability of sediment porosity and structure at the cm scale. Complementary X-radiograph cores provided 2D datasets of resistivity and porosity, at higher resolution. They have been used to establish the porosity-resistivity-microstructure relationships for these carbonate sediments, with microstructure being described in terms of tortuosity. These relationships have been used to extend our interpretation in terms of porosity and tortuosity to the corresponding 3D box core resistivity datasets. Sediment tortuosity has been investigated by numerically modeling the flow of electrical currents through a range of pore morphologies in three dimensions. The results show particular sensitivity to the connectivity of intra-particle porosity, particle shape and the relative sizes of pores and throats.

MODELING DIFFUSION AND REACTION IN SOILS: IV. NEW MODELS FOR PREDICTING ION DIFFUSIVITY

Salt diffusivity as a function of soil-water content and soil-water characteristic curves was measured on three sieved soils of different texture. The data were used together with ion diffusivity data for sieved soils from literature to test different diffusivity models. No significant differences between salt-, counter-, and self-diffusion data were observed, suggesting an insignificant effect of diffusion type in most cases, two new models for predicting ion diffusivity, based on the Campbell soil-water retention model parameter b, were proposed : (i) a model with a b-dependent threshold water content θ th (soil-water content where the ion diffusivity approaches zero) and (ii) a power function model where the power term η, representing the liquid phase tortuosity, is a linear function of b. In both models, the diffusivity at the soil-water content equal to the total porosity was estimated from a simple impedance factor model. Both b-dependent models gave improved predictions of ion diffusivity compared with existing models. The b-dependent power function model with η = 0.3b gave better predictions than both the θ th -dependent model and a recent model for gas diffusivity in undisturbed soil (Part III of this series, Soil Sci. 161 :366-375) where the tortuosity term η was best described as a function of b -1 . For use of the new b-dependent ion diffusivity models in the absence of soil-water characteristic data, it is shown that b can be fairly accurately estimated from soil texture.

Solute rejection by porous glass membranes. II. Pore size distributions and membrane permeabilities

The pore size distributions of porous glass hyperfiltration membranes were calculated from nitrogen desorption isotherm data. Pore volume distributions of unused porous glass membranes peaked sharply between 19 and 22Å pore radius. Membranes from batches with higher solute rejection appeared to have sharp cut-offs of the pore volume distribution tail for larger pores and median pore radius under 20 A. The small differences may include “skin” effects, as it was found that etching of membrane-effluent solution interface greatly reduced solute rejection, although etching of the membrane-feed solution interface had no effect on rejection. A permeability equation with a tortuosity term was used to relate porous glass membrane permeabilities to the pore size distributions. Tortuosities of 2.6–3.4 allowed such agreement for new membranes. Membranes that lost solute rejection capability during hyperfiltration life tests with sodium chloride and urea feed solutions showed a marked broadening of the pore volume distribution and, in the case of the test with sodium chloride feed, a new pore volume distribution peak at 60–80 A radius. The addition of 1.7 gm/liter aluminum chloride to a 10 gm/liter sodium chloride feed solution minimized changes in pore size distribution during a hyperfiltration life test. Electron micrographs showed apparent flow paths of the right order of tortuosity and surface area calculated from particle dimensions were less than BET areas for unused membranes but equivalent to BET areas after hyperfiltration testing.

Calculation of Imbibition Relative Permeability for Two- and Three-Phase Flow From Rock Properties

Abstract Relative permeability functions are developed for both two- and three-phase systems with the saturation changes in the imbibition direction. An empirical relation between residual nonwetting-phase saturation after water imbibition and initial nonwetting-phase saturations is found from published data. From this empirical relation, expressions are obtained for trapped and mobile nonwetting-phase saturations which are used in connection with established theory relating relative permeability to pore-size distribution. The resulting equations yield relative permeability as a function of saturation having characteristics believed to be representative of real systems. The relative permeability of water-wet rocks for both two- and three-phase systems, with the saturation change in the imbibition direction, may be obtained by this method after properly selecting two rock properties: the residual nonwetting-phase saturation after the complete imbibition cycle, and the capillary pressure curve. Introduction Relative permeability is a function of saturation history as well as of saturation. This fact was first pointed out for two-phase flow by Geffen et al. and by Osaba et al. Hysteresis in the relative permeability-saturation relation also has been reported for three-phase systems. Since saturations may change simultaneously in two directions in a three-phase system, four possible relationships arise between relative permeability and saturation for a water-wet system. The four saturation histories of this system were given by Snell as II, ID, DI and DD. I refer to the direction of saturation change (imbibition and drainage), with the first letter of the symbol indicating the direction of change of the water phase. As used in this paper, the second letter of the symbol refers to the direction of saturation change of the gas phase, i.e., D and I indicate an increase and decrease, respectively, in gas saturation. Only a few three-phase relative permeability curves have been published. Leverett and Lewis published three-phase curves for unconsolidated sand, and Snell reported results of several English authors for both drainage and imbibition three-phase relative permeability of unconsolidated sands. Three-phase relative permeability curves for a consolidated sand were published by Caudle et al. for increasing water and gas saturations (ID). Corey et al. reported drainage (DD) three-phase relative permeability for consolidated sands. Recently, Donaldson and Dean and Sarem calculated three- phase relative permeability curves from displacement data on consolidated sands, also for saturation changes in the drainage direction. The only published three - phase relative permeability curves for consolidated sands with saturation changes in the imbibition direction (II) are those of Naar and Wygal. These curves are based on at theoretical study of the model of Wyllie and Gardner as modified by Naar and Henderson. Interest in three-phase relative permeability has increased recently due to the introduction of new recovery methods and refinements in calculation procedures brought about by the use of large-scale digital computers. The scarcity of empirical relations for three-phase flow, and the experimental difficulty encountered in obtaining such data, have made the theoretical approach to this problem attractive.