Differential Effective Medium Model Research Articles

Effective thermal conductivity of disperse materials. I. Compliance of common models with experimental data

The thermal conductivity of packed beds of various materials (metals, ceramics, superhard materials, polymers, etc.) is measured by the steady-state heat flux method at room temperature in air. Powders and granular materials are used, with particle sizes ranging mainly from 50μm to 1.5mm, with various particle shapes (debris, pebble, cubo–octahedral single-crystals, regular spheres), the thermal conductivity of the solid ranging widely from 0.15 to 1500W/(m K). Experimental data and model calculations of the effective thermal conductivity of the bed are compared, including dependences on bed porosity and particle size. Both classical and more recent models are involved, such as the differential effective medium model and those of Raghavan–Martin, Kunii et al., Zehner–Schlunder, Gusarov et al., Hsu et al., Batchelor–O’Brein, and some others, which allow analytical calculations. The adequacy of the models is reviewed in connection with the properties of the disperse materials (particle shape etc.), based on the concepts of the absence or presence of contact thermal conductivity in the bed.

Enhanced thermal conductivity in diamond/aluminum composites with a tungsten interface nanolayer

A tungsten (W) nanolayer was first introduced onto diamond particles by a sol–gel process, and then aluminum (Al) based composites were fabricated by vacuum hot pressing using the W coated diamond (diamond@W) particles. The microstructure of the W nanolayer and its effect on the thermal properties were explored. The results showed that the W nanolayer with a dendritic morphology and a thickness of 200nm is the optimum combination to improve the interfacial bonding and minimize the thermal boundary resistance between diamond and Al. Such an observation was explained by the tunable formation of trace amount of W2C. The thermal conductivity of 50vol.% diamond@W/Al composites was 599W/mK, 21% higher than that of the composite without the W interface nanolayer. Our results were found to be in good agreement with the theoretical predictions by the combined differential effective medium (DEM) and acoustic mismatch model (AMM) schemes.

Enhanced thermal conductivity in diamond/aluminum composites: Comparison between the methods of adding Ti into Al matrix and coating Ti onto diamond surface

The enhancement of thermal conductivity (TC) in diamond/A356 aluminum alloy composites was investigated by comparing the Ti coating on diamond surface with the Ti content in Al matrix. Diamond/A356 aluminum alloy composites with Ti coating on diamond surface (Ti–diamond/aluminum) and Ti added into matrix (diamond/aluminum-Ti) were fabricated by gas pressure infiltration. The microstructure, TC and interfacial thermal conductance (ITC) of these two types of composites were compared. Unlike the homogeneous interfacial bonding for the Ti–diamond/aluminum composite, it exhibits a transition from weak to strong interface bonding as the content of Ti in the Al alloy increases. The method of coating appears more effective than the method of adding active metal into the matrix on enhancing the diamond/aluminum interface thermal conductance with the help of calculation by differential effective medium (DEM) and acoustic mismatch model (AMM).

High thermal conductivity composite of diamond particles with tungsten coating in a copper matrix for heat sink application

A composite material from particles of synthetic or natural diamond in a copper matrix, having a thermal conductivity (TC) of 500–900 W m−1 K−1, is obtained by capillary infiltration. A tungsten coating of 100–500 nm thickness is first applied to the diamond powder. The carbidization of the coating during annealing and melt infiltration is studied by X-ray diffraction. Measurements of TC λ, using the stationary heat flux method, and of thermal diffusivity a, using the flash method, agree. A longitudinal speed of sound of 8–9 km/s, an ultimate tensile strength of 150 MPa and a coefficient of thermal expansion (CTE) of ∼6 ppm/K at 25 °C are also measured for the composite. Model calculations of the composite’s CTE are carried out using the rule of mixtures and the Kerner and Turner equations. The composite’s specific electric resistance is 5–8 μΩ cm, according to calculations by the rule of mixtures, the Maxwell equation and the differential effective medium model. In comparison with other diamond–metal composites and materials with high TC intended for heat sinks, the composite developed is characterized by a combination of good thermal and mechanical properties, manufacturability and a relatively low cost.

Thermal conductivity of TiC-coated diamond/Al composites

Diamond/Al composites with TiC coating on diamond particles of 60% volume fraction were fabricated by gas pressure infiltration. The thermal conductivity of the composites with designated average particle diameters about 100 μm, 70 μm and 50 μm was investigated and compared to the Hasselman and Johnson (H–J) model and the differential effective medium (DEM) model. The results show that A356 aluminum matrix adheres on both the {100} faces and the {111} faces of diamond particles as the diamond particles are coated with TiC coating, and the TiC coating on the surface of diamond particles is slightly oxidized to form TiO2 as the packed diamond particles are preheated in the infiltration furnace. Compared to the H–J model, the DEM model gives a closer result and it fits well with the uncoated composites. The interfacial thermal conductance for coated composites is back calculated to be 3·3 × 107 W/m2/K by the DEM model, which is improved by 50% for uncoated composites. After taking the TiC interlayer with thickness of 1 μm into account, the total interfacial thermal conductance of the TiC-coated diamond/Al composite is estimated to be 2.9 × 107 W/m2/K by using the concept of interfacial thermal resistance, which is very close to that derived from the DEM model.

Impact of the cracks lost in the imaging process on computing linear elastic properties from 3D microtomographic images of Berea sandstone

With the current developments in imaging/computational techniques and resources, computational rock physics has been emerging as a new field of study. Properties of rocks are examined by carrying out extensive numerical simulations on rocks that have been digitized using high-resolution X-ray CT scans. The ultimate goal of computational rock physics is to supplement the traditional laboratory measurements, which are time consuming, with faster numerical simulations that allow the parameter space to be explored more thoroughly. We applied the finite-element method to compute the static effective elastic properties from 3D microtomographic images of Berea sandstone saturated with different fluids. From the computations, we found discrepancies between the numerical results and the laboratory measurements. The reason for such a problem is the loss of small features, such as fine cracks and micropores, in the digitized matrix during the imaging and aggregation process. We used a hybrid approach, combining the numerical computation and the effective media theories — the differential effective medium model and the Kuster-Toksöz model — to deduce the lost cracks by a very fast simulated annealing method. We analyzed the sensitivity of the inverted results — the distributions of crack aspect ratios and concentrations — to the clay content. We found that the inverted crack distribution is not so sensitive to clay content. Compared with the effect of cracks on the computed effective elastic properties, clay has only a secondary effect. Our approach can recover the lost cracks and is capable of predicting the effective elastic properties of the rocks from the microtomographic images for different fluid saturations. Compared with the traditional inversion schemes, based only on the effective media theories, this hybrid scheme has the advantage of utilizing the complex microstructures that are resolved in the imaging process, and it helps define the inversion space for crack distribution.

The theory and application of DEM-Gassmann rock physics model for complex carbonate reservoirs

The storage spaces of carbonate reservoirs in the Tarim Basin are dominated by secondary pore porosity such as dissolution caves, holes, and cracks (Song et al., 2001). Research has shown that pore shapes can significantly influence the velocities of elastic waves (Eshelby, 1957; Kuster and Toksöz, 1974; Xu and White, 1995; Yan et al., 2002; Sun et al., 2004). Empirical [Formula: see text] relationships such as Castagna's (1985) mudrock line, Han's relations (1986), and Greenberg-Castagna's (1992) relationship that ignore the effect of pore geometry are not applicable to carbonates (Xu and White, 1995; Wang et al., 2009). Cheng and Toksöz (1979) imaged various pore space structures using SEM, and proposed a pore aspect ratio spectrum that can be used to explain velocity prediction. Han (2004) measured velocities of 52 different carbonate rocks, indicating that caves influence seismic velocities due to high rigidity while micropores or cracks significantly decrease velocities of rocks due to the low rigidity. Regarding carbonate reservoirs that are dominated by secondary pore porosity, Eberli et al. (2003) discussed the velocity-porosity relationship for microporosity rocks, moldic-porosity rocks, interparticle porosity rocks, and densely cemented rocks. Wang et al. (2009) analyzed the practicality of three published models (Wyllie time-average equation, Gassmann's equation, and Kuster-Toksöz model) on velocity prediction for carbonate reservoirs in the Tarim Basin, proving that Kuster-Toksöz model is the best one. Meanwhile, they also pointed out that the Kuster-Toksöz model is a very high-frequency model, and is limited to dilute concentrations of the pores. Based on effective medium theories of Norris et al. (1985), Berryman (1992) proposed a new differential effective medium (DEM) model. The matrix in this model begins as phase 1 and is updated at each step when a new increment of phase 2 (inclusions) is added; the process continues until the expected proportion of the inclusions is reached. In this process, the former incremental inclusion (phase 2) and the matrix phase are taken as a new matrix, which makes the inclusion added at each step “dilute.” Unfortunately, DEM is also a high-frequency model. At high frequencies, there is not enough time for wave-induced pore pressure to equilibrate, so it is not applicable to consider the effect of pore fluids on elastic properties. Wu's (1992) arbitrary aspect ratio and Berryman's (1995) 3D special pores are introduced into Berryman's (1992) DEM model to calculate the elastic moduli of dry rock in this paper. Spheres, needles, and penny-shaped cracks are used to represent dissolution vugs (small-sized caves), holes (needle-shaped), and cracks that have developed in the Tarim Basin. These “special” pores are incrementally added into the matrix of carbonate rocks so that the inclusion added at each step is dilute. Then, Gassmann's equation is employed to calculate the elastic properties of the saturated rocks. This rock physics model in this paper is called the DEM-Gassmann's model.

Joint elastic-electrical properties of sediments in the Yellow Sea

We measured in the laboratory compressional wave velocity and electrical resistivity on 434 sediment samples collected from the Yellow Sea to study the joint elastic-electrical properties of marine sediments. Porosity was found to reduce both elastic velocity and electrical resistivity of the marine sediments in a non-linear fashion; velocity showed an approximate linear increase with increasing logarithm of resistivity. Various effective medium models either implemented or developed were compared with the new dataset. The model results showed that the combined self-consistent approximation and differential effective medium model using critical porosity of 0.6 and 0.5 for velocity and resistivity respectively gave a reasonable description of the joint elastic-electrical behaviors of the marine sediments. The joint elastic-electrical properties of the marine sediments established would be used to estimate resistivity from measured velocity and vice versa, and could also be suitable for detection of gas hydrate or other suitable targets from joint seismic-resistivity surveys.

3상 복합재의 등가열전도계수 예측에 대한 연구

2상 복합재에 적용되어오던 수정된 Eshelby 모델(MEM)을 두 종류의 구형 입자를 포함하는 3상 복합재로 확장하여 복합재의 등가열전도계수를 간단히 양함수 형태로 표시한다. 이의 유효성 검증을 위해 이 결과를 미소등가물모델(DEMM)로 구한 결과와 비교하고, 또 참고문헌의 2상 및 3상 복합재의 실험결과와 비교한다. 2상 복합재의 경우 MEM이 충전재의 체적비 0.5 미만에서는 DEMM보다 잘 예측한다. 3상 복합재의 경우 모재 대비 큰 열전도계수비를 갖는 충전재의 체적비가 적은 경우 MEM이 잘 예측하나, 체적비가 증가할수록 DEMM이 잘 예측한다. 이 체적비가 두 모델의 예측결과에 결정적 영향을 주는 인자임이 변수들의 영향 연구를 통해 밝혀졌으며, Molina 등이 제안한 3상 복합재에 대해 MEM과 DEMM은 동등한 예측 수준을 보였다. Effective thermal conductivity of three-phase composites, consisting of matrix and two kinds of spherical inclusions, has been derived as an explicit form by extending modified Eshelby model (MEM) for two-phase composites. The present results are compared with those by differential effective medium model (DEMM), which are also compared with the experimental results of two- and three-phase composites in the literatures to be validated. For two-phase composites, the results by MEM are better than those by DEMM for the inclusion volume fraction smaller than 0.5. Comparisons between the results by two models and experimental results have been made for three-phase composite, resulting in that MEM predicts better than DEMM for smaller volume fraction of the inclusion having larger inclusion-to-matrix thermal conductivity ratio, but DEMM predicts better as its volume fraction increases. It has been observed through parametric study that its volume fraction is the critical factor affecting the deviation of predictions by the two models. The results by them show a good agreement with the three-phase composite proposed by Molina et al..

Thermal properties of diamond/SiC/Al composites with high volume fractions

The diamond/SiC/Al composites with high volume fractions and a large ratio of diamond to SiC particle size (7.8:1) were fabricated by gas pressure infiltration. The results show that the fine SiC particles occupy efficiently the interstitial positions around coarse diamond particles; the main fracture mechanism of the composite is matrix ductile fracture, and diamond brittle fracture was observed which confirms a high interfacial bonding strength; the diamond/SiC/Al composites with 80% and 66.7% volume fraction of diamond in the reinforcement have the higher volume fraction in the reinforcement and lower coefficient of thermal expansion compared to the diamond/Al composite. Turner and Kerner models are not in good agreement with the experimental data for the composites based on reinforcement with two phases different in shape and component. When the effect of the coating layer considered, differential effective medium (DEM) model is confirmed a reliable model in designing a composite with a given thermal conductivity based on reinforcement with two phases different in size.

Joint elastic‐electrical effective medium models of reservoir sandstones

ABSTRACTImprovements in the joint inversion of seismic and marine controlled source electromagnetic data sets will require better constrained models of the joint elastic‐electrical properties of reservoir rocks. Various effective medium models were compared to a novel laboratory data set of elastic velocity and electrical resistivity (obtained on 67 reservoir sandstone samples saturated with 35 g/l brine at a differential pressure of 8 MPa) with mixed results. Hence, we developed a new three‐phase effective medium model for sandstones with pore‐filling clay minerals based on the combined self‐consistent approximation and differential effective medium model. We found that using a critical porosity of 0.5 and an aspect ratio of 1 for all three components, the proposed model gave accurate model predictions of the observed magnitudes of P‐wave velocity and electrical resistivity and of the divergent trends of clean and clay‐rich sandstones at higher porosities. Using only a few well‐constrained input parameters, the new model offers a practical way to predict in situ porosity and clay content in brine saturated sandstones from co‐located P‐wave velocity and electrical resistivity data sets.

Thermal conductivity and interfacial conductance of AlN particle reinforced metal matrix composites

Aluminum nitride (AlN) particle reinforced metal-matrix-composites produced by pressure infiltration are characterized in terms of their thermal conductivity. The composites are designed to cover a wide range of phase contrast between the dispersed particles and the matrix; this is achieved by changing the matrix conductivity using Cu, Al, Sn, and Pb as the matrix. The interface thermal conductance (hc) between AlN and the matrix metals is determined by varying the size of the AlN particles using the Hasselman–Johnson approach and the differential effective medium (DEM) model to calculate hc from measured composite conductivity values. In addition, hc is measured directly at the AlN/Al interface using the transient thermoreflectance (TTR) method on thin aluminum layers deposited on flat AlN substrates to find good agreement with the value derived directly from Al/AlN composites of variable particle size and thus confirm the approach used here to measure hc. Data from the study show that hc at AlN-metal interfaces increases with the metal/AlN Debye temperature ratio; however, the increase is much less than predicted by currently accepted models.

Thermal conductivity of the diamond-paraffin wax composite

The thermal conductivity of diamond-paraffin wax composites prepared by infiltration of a hydrocarbon binder with the thermal conductivity λm = 0.2 W m−1 K−1 into a dense bed of diamond particles (λf ∼ 1500 W m−1 K−1) with sizes of 400 and 180 μm has been investigated. The calculations using universally accepted models considering isolated inclusions in a matrix have demonstrated that the best agreement with the measured values of the thermal conductivity of the composite λ = 10–12 W m−1 K−1 is achieved with the use of the differential effective medium model, the Maxwell mean field scheme gives a very underestimated calculated value of λ, and the effective medium theory leads to a very overestimated value. An agreement between the calculation and the experiment can be provided by constructing thermal conductivity functions. The calculation of the thermal conductivity at the percolation threshold has shown that the experimental thermal conductivity of the composites is higher than this critical value. It has been established that, for the composites with closely packed diamond particles (the volume fraction is ∼0.63 for a monodisperse binder), the use of the isolated particle model (Hasselman-Johnson and differential effective medium models) for calculating the thermal conductivity is not quite correct, because the model does not take into account the percolation component of the thermal conductivity. In particular, this holds true for the calculation of the heat conductance of diamond-matrix interfaces in diamond-metal composites with a high thermal conductivity.

Elastic properties of anisotropic synthetic calcite‐muscovite aggregates

A set of synthetic aggregates, containing mixtures of calcite and muscovite, were compacted with varying uniaxial loads ranging from 20 MPa up to 400 MPa. Their elastic properties have been measured using compressional and shear waves in confining hydrostatic pressures up to 475 MPa. Measured seismic velocities are shown to depend on 1) the ratio of calcite to muscovite, 2) the uniaxial load used during sample manufacturing, and 3) the porosity amount prior to velocity measurements. The matrix framework may also affect the seismic velocities, but this effect is not easily quantifiable. In general, measured seismic velocities decrease with increasing muscovite content and porosity. Elastic properties have been calculated based on texture measurements, which were obtained by neutron diffraction. The calculated velocities are based on the calcite and muscovite single crystal elastic tensors, their orientation distribution functions, and their modal fractions. A large discrepancy is apparent between calculated and measured velocities, where the former always overestimate the actual velocities. Shear waves display less of a difference between calculated and measured values, than do compressional waves, indicating that waves propagating by shearing are less affected by sample porosity and matrix framework. A better agreement between measured and calculated seismic velocities arises when the initial porosity is taken into account, using a differential effective medium model. Seismic anisotropy is evident from both measured and calculated velocities, and is low for samples containing pure calcite, but becomes prominent as the muscovite concentration increases. The intensity of anisotropy further depends on the uniaxial load used during sample compaction and the initial porosity.

A comparison of rock physics models for fluid substitution in carbonate rocks

Rock physics models play a crucial role in seismic reservoir characterisation studies. The optimal rock physics model for a sandstone reservoir might be significantly different from that of a carbonate reservoir. There are several theories that compare the elastic properties of dry and saturated rocks. These models have mainly been explained by poroelastic theories or effective medium theories. The Gassmann’s model which is commonly used in petroleum rock physics is suitable for rocks with spherical and interconnected pores at low frequencies. These assumptions do not necessarily meet the conditions of carbonate rocks. In this work, two additional models, the differential effective medium (DEM) model and the self-consistent (SC) model have been examined for several carbonate samples. Ultrasonic 30 carbonate and 5 sandstone core samples from an oilfield in south-west Iran were measured in the laboratory. The results show that the DEM model gives the best compatibility with the dense and low porous carbonate samples. These results are confirmed by well log data from the same area.

Predicting elasticity in nonclastic rocks with a differential effective medium model

Can a theoretical inclusion model — specifically, the differential effective medium (DEM) model — match experimental velocity data in rocks that are not necessarily made of inclusions, such as clastics? It is indeed possible in some cases by using an almost constant inclusion aspect ratio (AR) within wide ranges of porosity and mineralogy. We approach this question by using empirical velocity-porosity equations as proxies for data. By finding a DEM inclusion AR to match these equations, we find that the required range of AR is narrow. Moreover, a constant AR of about 0.13 can be used to accurately match empirical relations in competent sand, shale, and quartz/calcite mixtures. This finding can be utilized practically to predict [Formula: see text] from [Formula: see text]; describe velocity-frequency dispersion between low-frequency and ultrasonic experiments; predict the dry-frame elastic properties from ultrasonic data on liquid-saturated samples where Gassmann’s fluid substitution is not applicable; predict the attenuation of P-wave velocity; and establish tight constraints for ranges of possible variation of [Formula: see text] and [Formula: see text] at a given porosity in some mineralogies. When we apply this approach to laboratory data rather than empirical equations, we confirm a positive answer to the main question, with all applications of this result still valid.

A generalized differential effective medium model of piezoelectret foams.

Piezoelectrets are closed‐cell foams consisting of a continuous polymer matrix containing preferentially oriented and electrically polarized ellipsoidal voids. These materials display low density, low characteristic impedance, thicknesses on the order of 100 μm, and potential to act as conformal transducers on complex geometries. They therefore provide an intriguing alternative to piezoelectric ceramics and piezopolymers for applications such as air‐coupled ultrasonics or underwater receive elements. Unfortunately, some fundamental attributes of piezoelectret behavior, such as their nonmonotonic change in receive sensitivity with increasing ambient pressure, remain unexplained by physical modeling or experiment. To explain this and other behavior, this work develops a fully coupled multiscale microelectromechanical model which captures the void‐scale physics and predicts macroscopically observed piezoelectric properties. The model employs volume averaged Green’s function solutions of the three‐dimensional...

A differential effective medium model for piezoelectret foams

Polymer foams with piezoelectric properties, called piezoelectrets, have recently gained interest in the acoustics and scientific community. These foams are heterogeneous materials consisting of a continuous polymer containing electrically polarized elliptical voids. The macroscopically observable piezoelectric behavior results from the unique combination of the high void-polymer elastic contrast and the void polarization. Existing modeling methods to approximate the macroscopic piezoelectric properties of these foams are elementary one-dimensional noncoupled electrostatic models. In this study, a coupled mean field microelectromechanical model has been developed as a predictive tool of the macroscopically observed piezoelectric material behavior. This technique employs Green’s function solutions of the three-dimensional (3D) stress equilibrium equations and Gauss’ Law. The result is a multiscale differential effective medium model approximating the 3D effective stiffness, dielectric permittivity, and piezoelectric coupling coefficients as a function of the constituent material properties, void shape and orientation, and deposited charge density. The model is employed to study the sensitivity of macroscopic foam behavior to various constituent material and void variables. This approach improves the approximation of the true piezoelectret behavior by capturing the influence of microscopic material structure and properties on macroscopic piezoelectric performance.

Rock physics modelling of shale diagenesis

A model for estimating the effective anisotropic properties of cemented shales is presented. The model is based on two mathematical methods for estimation of effective properties of a composite medium; a self-consistent approximation and a differential effective medium model. In combination these theories allow approximation of a shale with connected clay minerals and cement, and disconnected pores and quartz grains, which can be compared with the conditions in a real cemented shale. A strategy is also presented for estimation of stiffnesses in the transition zone from mechanical compaction to chemical compaction dominated diagenesis. Combining these theories with a shale compaction theory, enables modelling of the effective elastic stiffnesses for shales from deposition and mechanical compaction to deep burial and chemical compaction/cementing. Results from the model were compared with velocity data from three wells, showing good fit for velocity predictions, following the main velocity trends with increased temperature and depth.

Elastic properties of double‐porosity rocks using the differential effective medium model

ABSTRACTAn approach to determining the effective elastic moduli of rocks with double porosity is presented. The double‐porosity medium is considered to be a heterogeneous material composed of a homogeneous matrix with primary pores and inclusions that represent secondary pores. Fluid flows in the primary‐pore system and between primary and secondary pores are neglected because of the low permeability of the primary porosity. The prediction of the effective elastic moduli consists of two steps. Firstly, we calculate the effective elastic properties of the matrix with the primary small‐scale pores (matrix homogenization). The porous matrix is then treated as a homogeneous isotropic host in which the large‐scale secondary pores are embedded. To calculate the effective elastic moduli at each step, we use the differential effective medium (DEM) approach. The constituents of this composite medium – primary pores and secondary pores – are approximated by ellipsoidal or spheroidal inclusions with corresponding aspect ratios.We have applied this technique in order to compute the effective elastic properties for a model with randomly orientated inclusions (an isotropic medium) and aligned inclusions (a transversely isotropic medium). Using the special tensor basis, the solution of the one‐particle problem with transversely isotropic host was obtained in explicit form.The direct application of the DEM method for fluid‐saturated pores does not account for fluid displacement in pore systems, and corresponds to a model with isolated pores or the high‐frequency range of acoustic waves. For the interconnected secondary pores, we have calculated the elastic moduli for the dry inclusions and then applied Gassmann's tensor relationships. The simulation of the effective elastic characteristic demonstrated that the fluid flow between the connected secondary pores has a significant influence only in porous rocks containing cracks (flattened ellipsoids). For pore shapes that are close to spherical, the relative difference between the elastic velocities determined by the DEM method and by the DEM method with Gassmann's corrections does not exceed 2%. Examples of the calculation of elastic moduli for water‐saturated dolomite with both isolated and interconnected secondary pores are presented. The simulations were verified by comparison with published experimental data.