Spheroidal Inclusions Research Articles

Analytical models of the effective permeability of sand-shale reservoirs : Geophys J V105, N2, May 1991, P513–527

SUMMARY A study is made of the dependence of the effective permeability of sand-shale reservoirs on the geometry of the shale inclusions. Numerical simulations are described in which factors such as the orientation and the degree of anisotropy of the shales, their volume fraction and the permeability contrast ratio between the shale and the sandstone were varied. Two different reservoir models were used in the simulations: one in which the shales were in the form of random, overlapping spheroidal inclusions, and one in which the shales were generated by a geostatistical technique. The computed effective permeability data are compared with a variety of analytical models in an attempt to discover ‘rules of thumb’for the estimation of the effective permeability of sand-shale reservoirs for use in a range of practical situations. The main technical way in which this study advances on previous work is in its use of a computationally efficient, random walk algorithm for calculating the effective permeability, which has enabled larger volumes to be simulated and, hence, a thorough investigation of finite-size effects to be made. Another advantage of the work reported here is the general nature of the sand-shale geometries used in the numerical simulations and the extensive comparisons with analytical models.

Stress state of a transversely isotropic medium with an anisotropic spheroidal inclusion. Arbitrary uniform stress field at infinity

Stress state of a transversely isotropic medium with an anisotropic spheroidal inclusion. Arbitrary uniform stress field at infinity

Relaxation kinetics and relaxed stresses caused by interface diffusion around spheroidal inclusions

The kinetics of relaxation by interface diffusion around spheroidal inclusions is discussed using the Eshelby method for ellipsodal inclusions and a free energy approach. The relaxation time for the process is derived in the case where the direction of an applied uniaxial stress is parallel to the rotation axis of symmetry of the speroid. The aspect-ratio dependence of the relaxanion time is derived. For prolate inclusions with large aspect ratios, the relaxation time increases proportionally to the aspect ratio. On the other hand, for oblate inclusions with small aspect ratios, the relaxation time becines constant as the aspect ratio approaches zero. The constant value of the relaxation time depends on the elastic moduli of the inclusion but independent of the elastic moduli of the matrix. The hydrostatic stress in the inclusion after the relaxation and its aspect-ratio dependence are also discussed. Although the relaxation time for a rod-like inclusion is longer compared with that for a disk-like inclusion, the relaxed stress in the former is much smaller than that in the latter.

Analytical models of the effective permeability of sand-shale reservoirs

SUMMARY A study is made of the dependence of the effective permeability of sand-shale reservoirs on the geometry of the shale inclusions. Numerical simulations are described in which factors such as the orientation and the degree of anisotropy of the shales, their volume fraction and the permeability contrast ratio between the shale and the sandstone were varied. Two different reservoir models were used in the simulations: one in which the shales were in the form of random, overlapping spheroidal inclusions, and one in which the shales were generated by a geostatistical technique. The computed effective permeability data are compared with a variety of analytical models in an attempt to discover ‘rules of thumb’for the estimation of the effective permeability of sand-shale reservoirs for use in a range of practical situations. The main technical way in which this study advances on previous work is in its use of a computationally efficient, random walk algorithm for calculating the effective permeability, which has enabled larger volumes to be simulated and, hence, a thorough investigation of finite-size effects to be made. Another advantage of the work reported here is the general nature of the sand-shale geometries used in the numerical simulations and the extensive comparisons with analytical models.

Trapping and flow among random arrays of oriented spheroidal inclusions

The effective trapping rate k associated with diffusion-controlled reactions among random distributions of spatially correlated and uncorrelated, oriented spheroidal traps of aspect ratio ε is determined from Brownian motion simulations. Data for k are obtained for prolate cases (ε=2, 5, and 10), oblate cases (ε=0.1, 0.2, and 0.5), and spheres (ε=1) over a wide range of trap volume fractions (φ2) and satisfy recently obtained rigorous lower bounds on k for this statistically anisotropic model. The results for the trapping rate for correlated traps always bounds from above corresponding results for uncorrelated traps. Generally, the trapping rate k, for fixed φ2, increases with decreasing aspect ratio ε, showing a precipitous rise in k as the spheroids become disklike. Using a recent theorem due to Torquato [Phys. Rev. Lett. 64, 2644 (1990)], data for the trapping rate k can be employed to infer information about the fluid permeability tensor K associated with slow viscous flow through porous media composed of the same arrays of oriented spheroidal particles.

Thermal conductivity of Hawaiian basalt: A new interpretation of Robertson and Peck's data

Robertson and Peck's data published in 1974 on the thermal conductivity of Hawaiian basalts are unique in the sense that the vesicularity of the samples used for the study covers a wide range of 2 to 98%. The data consist of the measurement results of thermal conductivity on 61 samples of basalt collected from Kilauea and other volcanoes in Hawaii. The thermal conductivity was measured, by the method of thermal stacking, on samples with their pores filled first with air and then with water. Thus the data are particularly suitable for evaluating the theoretical formula describing the thermal conductivity of fluid‐saturated porous rock. In the formula for the thermal conductivity of a two‐phase material the thermal conductivity of the solid matrix is an essential parameter, which Robertson and Peck determined by the wet cell method. The result of their determination was that they could not represent the data in the whole range of sample porosities with a single theoretical formula. This paper proposes that the thermal conductivity of the solid matrix can be estimated by extrapolating the data to zero porosity. For the extrapolation, either Fricke‐Zimmerman's or Schulz's formula of thermal conductivity for a two‐phase material having spheroidal inclusions can be used. In both models, for the results of extrapolation from the air‐saturated and the water‐saturated sample thermal conductivities to be identical, the spheroid's aspect ratio is assumed to be around 1/10. This is considered to be a combined effect of the spherical pores and the microcracks in the basalt samples. With the assumed aspect ratio of the spheroidal inclusions, either of the theoretical formulae agree with the data with sufficient precision.

The effects of pore-scale fluid distribution on the physical properties of partially saturated tight sandstones

Pore-scale fluid distribution has a significant effect on the physical properties of a partially saturated porous medium. Experimental data for the dielectric response and elastic wave velocities for a tight gas sandstone undergoing a cycle of water saturation change through imbibition and drainage are analyzed. Mathematical formulations describing the internal geometrical configuration of a porous medium in terms of a rock matrix background with embedded oblate spheroidal inclusions representing the porosity are used to theoretically predict the dielectric constant and elastic wave velocities of the partially saturated sandstone. Simple geometrical models, incorporating homogeneous and heterogeneous inclusions, are used to simulate the pore-scale fluid distribution which should result from the two saturation methods employed. It is found that these simple scenarios accurately predict the functional form and magnitude of the observed saturation-induced hysteresis in the experimental data for both the dielectric constant and the elastic wave velocities.

The influence of inclusion shape on the overall elastoplastic behavior of a two-phase isotropic composite

A mean-field, approximate theory is developed for the determination of the overall stress-strain relation of a two-phase composite, consisting of randomly oriented elastic, spheroidal inclusions and a ductile matrix. The theory is intended for a low-volume concentration of inclusions. It is versatile enough to provide results under any proportionally increasing combined stresses, and yet simple enough to require no iterations. To preserve the virtue of simplicity, the simpler deformation theory is used over the incremental one. More so than the elastic behavior, the elastoplastic response of the composite is found to be extremely sensitive to the inclusion shape, with the discs providing the most remarkable reinforcement. Explicit results of the overall secant moduli are established for the three extreme inclusion shapes of disc, needle (fiber) and sphere. An interesting consequence of this analysis is that while both the particle and fiber-reinforced composites become plastically compressible, the disc-reinforced composite as the ductile matrix itself — remains plastically incompressible. When applied to a silicon-carbide /aluminum system, the theory indicates that within the aspect ratio α < 10, the prolate and oblate inclusions with inversed aspect ratios (i.e. α and 1α) have almost the same effect on the flow stress of the composite, but beyond this the discoriented inclusions begin to show a more superior influence. The theory also compares reasonably with the experimental data when the carbides exist in the form of randomly oriented platelets, with α = ¼.

Effective Dielectric function of a Composite with Aligned Spheroidal Inclusions

ABSTRACTThe effective dielectric response εM of a composite with aligned spheroidal inclusions is calculated. Using the dipolar and the mean-field approximation (MFA) an analytical expression for εM as a functional of the two-particle distribution function p(2) is obtained. It is shown that previous expressions reported in the literature correspond to different choices of p(2), thus, clarifying the origin of their discrepancies. The theory is further extended beyond the MFA by including the dipolar fluctuations through a renormalization of the polarizability tensor of the inclusions. The absorption peaks are diminished and broadened by the spatial disorder, which also yields an easily identified coupling among electromagnetic modes with perpendicular polarizations.

Effect of inclusion shape on the stiffness of nonlinear two-phase composites

constitutive relations are established for the axisymmetric deformation of an incompressible power-law matrix containing aligned, rigid spheroidal inclusions. The range of inclusion shapes considered is from thin disks with aspect ratios of 100: 1 to whiskers with aspect ratios of 50: 1. Results are presented for several matrix hardening exponents between n = 1 (linearly elastic) and n = 10. It is found that at a fixed matrix hardening exponent and a fixed volume fraction of inclusions, slender prolate inclusions give rise to a greater increase in stiffness than any other inclusion shape with the same or smaller aspect ratio. It is also found that the stiffening which results from a fixed volume fraction of inclusions is a strong function of matrix nonlinearity; the stiffness of the composite relative to the matrix material is much greater for a highly nonlinear matrix than for a linear matrix.

Effect of inclusion shape on stiffness of isotropic and transversely isotropic two-phase composites

Constitutive relations are developed for linear and non-linear incompressible, two-phase composites. The second phase is assumed to be comprised of spheroidal inclusions, and both Isotropie composites (containing randomly oriented inclusions) and transversely isotropic composites (containing aligned inclusions) are examined. Constitutive relations are established first for linearly elastic constituents and dilute concentrations of inclusions. Corresponding (approximate) constitutive relations applicable for non-dilute concentrations of inclusions are then obtained using a differential self-consistent scheme. Next, constitutive relations for non-linear matrix materials are developed using a procedure recently introduced by Ponte Castaneda [J. Mech. Phys. Solids (1991)]. The procedure uses a variational framework to exploit the constitutive relations developed for linear matrix behavior in order to obtain approximate results for non-linear matrix behavior. The results obtained are rigorous bounds on the behavior of the composites when the concentration of inclusions is sufficiently small, and are estimates for the behavior at larger concentrations of inclusions. The constitutive relations established for linear and non-linear composites are used to examine the effect of inclusion shape on the stiffness of composites, and a wide range of inclusion shapes ranging from thin disk-shaped inclusions to slender needle-like inclusions is considered.

Creep anisotropy of a metal-matrix composite containing dilute concentration of aligned spheroidal inclusions

The influence of aspect ratio of inclusions on the overall creep strain and creep anisotropy of a metal-matrix composite is investigated at the level of dilute concentration and small strain. The creep rate of the ductile matrix, as for most metals, depends nonlinearly on the stress and the primary creep, resulting from the work-hardening of creep deformation, is also considered in addition to the usual steady state, or secondary, creep. The method developed for the calculation of incremental process is based upon Eshelby's (1957) equivalence principle of an inhomogeneity-transformation problem and Mori and Tanaka's (1973) concept of mean-field stress. It is demonstrated that this simple, albeit approximate theory is capable of predicting the aspect-ratio dependence of the time-dependent creep, with a characteristic consistent with the known elastic behavior. The creep anisotropy of the composite as the shape of the aligned inclusions changes from a thin disc to a sphere, and all the way to a long fiber, is also uncovered by comparing the developments of creep strain in various loading directions.

Effective thermal conductivity of composites containing spheroidal inclusions

AbstractThe effective thermal conductivity of composite materials containing perfectly‐conducting, perfectly‐aligned, spheroidal inclusions has been computed to O(c2), where c is the volume fraction of the inclusions, by incorporating two‐particle interactions in a rigorous fashion. Relevant two‐particle problems were solved by boundary collocation and method of reflections in the near and far fields, respectively. Our results for the effective thermal conductivity are rigorous extensions of Maxwell's theory in the form of an expansion in small cℓα, where ℓ is the spheroid aspect ratio (α = 1 for oblate and α = 2 for prolate spheroids). The influence of microstructure was examined with two ad hoc model distribution functions: 1. a uniform distribution and 2. a stretched hard‐sphere pair distribution. The latter yielded superior results over a greater range of c, as judged by comparisons with Willis' (1977) bounds developed from variational methods.

Transmission electron microscopy of second-phase particles in Pb(Sc1/2Ta1/2)O3 Ferroelectric ceramics

Pb-rich second-phase particles in perovskite-structured Pb(Sc1/2Ta1/2)O3 (PST) ferroelectric ceramic relaxors have been investigated using transmission electron microscopy. The second-phase particles were observed to exhibit a range of morphologies which included spheroidal inclusions, plate-like precipitates and intergranular films. Moire fringes and lattice images have been used to identify various non-perovskite phases. It is deduced that excess PbO, which is used to assist densification through liquid phase sintering and to promote grain growth, becomes entrapped within the ceramics, thereby producing Pb-enriched phases that have adverse effects on dielectric properties.

On the application of Mori-Tanaka's theory involving transversely isotropic spheroidal inclusions

Based on the new structure recently established by Weng for the Mori-Tanaka theory, the effective elastic moduli of three types of composite containing transversely isotropic spheroidal inclusions are explicitly derived. For a multiphase composites with aligned, identically shaped inclusions, the derived moduli are believed to be generally reliable, where the three extreme cases involving circular fibers, spheres, and thin discs all lie on or within the respective Hashin-Shtrikman-Walpole bounds. For a multiphase aligned composite whose inclusion phases differ in shape, the M-T moduli tensor can lose its diagonal symmetry, which, for a hybrid composite containing fibers and another aligned spheroids, is found to be severest when the spheroids take the shape of thin discs, and tends to decrease as their aspect ratio increases. When the transversely isotropic spheroidal inclusions are randomly oriented in an isotropic matrix, the M-T moduli with spherical inclusions are shown to always lie on or within the isotropic Hashin-Shtrikman-Walpole bounds. Such a desired property however is not always assured with other inclusion shapes, where the needle and disc-like inclusions may cause the M-T moduli and Walpole's self-consistent estimates to lie outside the H-S-W bounds.

Spheroidal Cavity With Prescribed Asymmetric Displacements in Three‐Dimensional Transverse Isotropy

A general three‐dimensional analytical solution, exact within elastostatics, is presented to the problem of an infinite transversely isotropic elastic medium containing a spheroidal cavity with zero body forces, prescribed asymmetric displacements on the surface of the cavity, and vanishing stresses and displacement at infinity. The approach is applied to obtain the solution to the problem of a transversely isotropic medium containing a perfectly rigid spheroidal inclusion, where the medium is subject at infinity to uniform tension in the direction perpendicular to the axis of elastic symmetry of the material. Numerical results and the effect of anisotropy on the interfacial normal stress are discussed.

The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium

An explicit analytical solution is presented for the problem of a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium, where the inclusion is given a constant displacement in a direction perpendicular to the axis of symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. The loadfeflection relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of both the aspect ratio of the spheroid and the anisotropy on the translational stiffness.

Features of failure for ferritic cast iron with spheroidal graphite

1. The effect of spheroidal graphite inclusions on the failure of ferritic cast iron depends on the condition of the matrix: in a ductile matrix spheroidal graphite particles are centers of microcavity generation and they lead as a result of this to a reduction in the level of cast iron properties; in a brittle matrix graphite particles have a favorable tendency of retarding brittle crack development and under certain conditions they lead to a change in failure mechanism. 2. The effect of crack retardation at spheroidal graphite particles develops particularly clearly at low loading rates thus providing properties in cast iron comparable with steel.

The rotation of a rigid spheroidal inclusion embedded in a transversely isotropic medium

The exact three-dimensional elasticity solutions are given for two problems related to a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium. The first is of axisymmetric nature in which the inclusion is given a constant rotation about its axis of revolution which coincides with the axis of symmetry of the material. The second problem is asymmetric where the spheroidal inclusion is given a constant rotation about a direction that is perpendicular to the axis of elastic symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. In both cases, the moment-rotation relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of the aspect ratio of the spheroid on the rotational stiffness.

Effect of Interface Layers on Scattering of Elastic Waves

Scattering of elastic waves by spheroidal elastic inclusions has been studied in this paper. Particular attention has been focused on the effect of interface layers between the inclusions and the matrix on the scattering cross-sections. It has been assumed that properties of each layer is constant through its thickness. For spheroidal inclusion this problem cannot be solved by exact means. We have used a hybrid finite element and wave function expansion technique to analyze the problem. It is shown that solutions thus obtained for spherical inclusions and cavities agree well with analytical solutions. For spheroidal inclusions we show that when the interface layer properties are intermediate between those of the particles and the matrix the scattering cross-section increases. These results can be useful in characterizing interface layer properties.