Variational theorems are established and applied to the derivation of bounds for the effective magnetic permeability of macroscopically homogeneous and isotropic multiphase materials. For reasons of mathematical analogy the results are also valid for the dielectric constant, electric conductivity, heat conductivity, and diffusivity of such materials. For the case of two‐phase materials, the bounds derived are the most restrictive ones that can be given in terms of the phase permeabilities and volume fractions. Comparison of present theoretical results with existing experimental data shows good agreement.

A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials

(1) If the hydraulic permeability of crustal material is sufficiently high, convection driven advection of heat can be an equally or even much more efficient transfer mechanism, provided sufficiently strong driving forces are supplied by forced or free convection systems. This is often the case in sedimentary basins. However, fluid driven heat advection can be important also in crystalline rocks and on a crustal scale (Etheridge et al., 1983, Torgersen, 1990, Clauser, 1992).

Thermal Conductivity of Rocks and Minerals

[1] Methane gas hydrates, crystalline inclusion compounds formed from methane and water, are found in marine continental margin and permafrost sediments worldwide. This article reviews the current understanding of phenomena involved in gas hydrate formation and the physical properties of hydrate-bearing sediments. Formation phenomena include pore-scale habit, solubility, spatial variability, and host sediment aggregate properties. Physical properties include thermal properties, permeability, electrical conductivity and permittivity, small-strain elastic P and S wave velocities, shear strength, and volume changes resulting from hydrate dissociation. The magnitudes and interdependencies of these properties are critically important for predicting and quantifying macroscale responses of hydrate-bearing sediments to changes in mechanical, thermal, or chemical boundary conditions. These predictions are vital for mitigating borehole, local, and regional slope stability hazards; optimizing recovery techniques for extracting methane from hydrate-bearing sediments or sequestering carbon dioxide in gas hydrate; and evaluating the role of gas hydrate in the global carbon cycle.

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Physical properties of hydrate-bearing sediments

The thermal conductivity of a solid or gas filled polymer is used in processing or end use application calculations. Numerous theoretical and empirical correlations are found in the literature. A careful review of these models indictes that no one correlation or technique accurately predicts the thermal conductivity of all types of composites. The investigation indicated that for solid filled composites the Lewis and Nielsen equation fitted the experimental data best for the range of fillers tested. However, for a gas filled polymer, none of the theoretical models proved adequate. The semi-empirical approach of Harding showed considerable merit.

Methods for predicting the thermal conductivity of composite systems: A review

Theoretical prediction of effective properties for multiphase material systems is very important not only to analysis and optimization of material performance, but also to new material designs. This review first examines the issues, difficulties and challenges in prediction of material behaviors by summarizing and critiquing the existing major analytical approaches dealing with material property modeling. The focus then shifts to some recent advances in numerical methodology that are able to predict more accurately and efficiently the effective physical properties of multiphase materials with complex internal microstructures. A random generation-growth algorithm is highlighted for reproducing multiphase microstructures, statistically equivalent to the actual systems, based on the geometrical and morphological information obtained from measurements and experimental estimations. Then a high-efficiency lattice Boltzmann solver for the corresponding governing equations is described which, while assuring energy conservation and the appropriate continuities at numerous interfaces in a complex system, has demonstrated its numerical power in yielding accurate solutions. Various applications are provided to validate the feasibility, effectiveness and robustness of this new methodology by comparing the predictions with existing experimental data from different transport processes, accounting for the effects due to component size, material anisotropy, internal morphology and multiphase interactions. The examples given also suggest even wider potential applicability of this methodology to other problems as long as they are governed by the similar partial differential equation(s). Thus, for given system composition and structure, this numerical methodology is in essence a model built on sound physics principles with prior validity, without resorting to ad hoc empirical treatment. Therefore, it is useful for design and optimization of new materials, beyond just predicting and analyzing the existing ones.

Predictions of effective physical properties of complex multiphase materials

The problem of determining the effective thermal conductivity of a two‐phase system, given the conductivities and volume fractions of the components, is examined. Equations are described which have been proposed as solutions to this problem, including those of Maxwell, de Vries, and Kunii and Smith, the weighted geometric mean equation, and an equation based on a three‐element resistor model found applicable to the analogous electrical conductivity problem. Experimental results are presented for five unconsolidated samples: three quartz sand packs, a glass bead pack, and a lead shot pack. The method of conductivity measurement using the transient line heat source (thermal conductivity probe) is described. Data are reported showing the variation of effective thermal conductivity with porosity, solid particle conductivity, saturating fluid conductivity, and the pressure of the saturating gas. From considerations based on the kinetic theory of gases, it is shown that the characteristic dimension of the pore ...

W. Woodside

Papers

Thermal Conductivity of Porous Media. I. Unconsolidated Sands