Explicit Conditions for Local Thermal Equilibrium in Porous Media Heat Conduction

Based on the traditional formulation of heat transfer in porous media it is demonstrated that Local Thermal Equilibrium (Lotheq) applies generally for any boundary conditions that are a combination of constant temperature and insulation. The resulting consequences are being analysed and discussed. Among these consequences it is shown that the linear relationship between the average temperature difference of the two phases and the heat transferred over the fluid-solid interface is inappropriate for use in connection with conditions of Lack of Local Thermal Equilibrium (La Lotheq).     

Convective Plume Paths in Anisotropic Porous Media

Free convection motions induced by point sources or horizontal line sources of heat are usually assumed to take the form of a vertically orientated plume. In this paper we consider how material anisotropy affects the path of the plume centreline and we show that it is strongly affected by both the anisotropy and the presence of impermeable bounding surfaces. The plume path is a straight line whose angle from the vertical is determined by a balance between the upward buoyancy force, the need for the plume to entrain equal amounts of fluid from the external regions either side of it, and the ease of fluid motion in the direction of the principal axis with the highest permeability.     

Flow and Transport Properties of Deforming Porous Media. I. Permeability

Transport in Porous Media - Estimating flow and transport properties of porous media that undergo deformation as a result of applying an external pressure or force is important to a wide variety of...     

Three-Dimensional Simulations for Convection Problem in Anisotropic Porous Media with Nonhomogeneous Porosity,Thermal Diffusivity,and Variable Gravity Effects

A convection problem in anisotropic and inhomogeneous porous media has been analyzed. In particular, the effect of variable permeability, thermal diffusivity, and variable gravity with respect to the vertical direction, has been studied. A linear and nonlinear stability analysis of the conduction solution has been performed. The validity of both the linear instability and global nonlinear energy stability thresholds are tested using a three- dimensional simulation. Our results show that the linear threshold accurately predicts on the onset of instability in the basic steady state. However, the required time to arrive at the basic steady state increases significantly as the Rayleigh number tends to the linear threshold.     

Flow and Transport Properties of Deforming Porous Media. II. Electrical Conductivity

In Part I of this series, we presented a new theoretical approach for computing the effective permeability of porous media that are under deformation by a hydrostatic pressure P. Beginning with the initial pore-size distribution (PSD) of a porous medium before deformation and given the Young’s modulus and Poisson’s ratio of its grains, the model used an extension of the Hertz–Mindlin theory of contact between grains to compute the new PSD that results from applying the pressure P to the medium and utilized the updated PSD in the effective-medium approximation (EMA) to estimate the effective permeability. In the present paper, we extend the theory in order to compute the electrical conductivity of the same porous media that are saturated by brine. We account for the possible contribution of surface conduction, in order to estimate the electrical conductivity of brine-saturated porous media. We then utilize the theory to update the PSD and, hence, the pore-conductance distribution, which is then used in the EMA to predict the pressure dependence of the electrical conductivity. Comparison between the predictions and experimental data for twenty-six sandstones indicates agreement between the two that ranges from excellent to good.

     

Transient Heat Transport in Gas Flow Through Granular Porous Media

Models are developed to describe the time-evolution of gas and solid temperature profiles in a class of granular porous media; this time-dependence being the result of a heat source supplying energy uniformly to the gas as it flows into the medium. The solid-phase is treated as a fixed, axi-symmetric bed of randomly packed spheres of uniform size and material properties. An analytic expression for the locally averaged surface temperature of the solid phase is formulated in terms of the time history of the corresponding local gas temperature. This approach avoids the need to assume locally uniform temperatures within the solid phase, and provides a coupled model for the thermal energy transport in the system. A numerical scheme for treating the resulting transport equations is outlined, and results presented. A quasi-steady approximation is proposed, and this approximation is assessed by reference to numerical results obtained from the numerical scheme. One application of the work is to fixed-bed catalytic reactors and absorbers, and results are presented indicating how the regeneration times of such systems depend on operating parameters.     

Miscible Displacements of Reactive and Anisotropic Dispersive Flows in Porous Media

The viscous fingering instability of miscible reactive–dispersive flows in a homogeneous porous media is investigated through nonlinear numerical simulations. In particular, the role of velocity-dependent transverse and longitudinal dispersions as well as the type and rate of auto-catalytic chemical reactions is analyzed. It is found that for a third-order auto-catalytic reaction, the higher the reaction rate, the more complex the finger structures. Furthermore, major differences between the flow development of third-order and second-order autocatalytic reactions are reported. In addition, the anisotropy and velocity dependence of the dispersion tensor are found to have a more profound effect on the fingering instability in the case of reactive flows than in the non-reactive ones. The qualitative characterization of the finger structures is explained by examining the flow velocity field and further quantified through an analysis of the average concentration and relative contact area.     

A Note on Local Thermal Non-equilibrium in Porous Media and Heat Flux Bifurcation Phenomenon in Porous Media

This work address a number of fundamental issues and concepts related to local thermal non-equilibrium and the heat flux bifurcation phenomenon in porous media. Different types of heat flux bifurcation phenomenon are discussed in relation to previous works by the authors.     

About the Porous Media Flow Through Circular and Elliptical Anisotropic Inhomogeneities

In this article, we describe horizontal groundwater flow due to a uniform flow at infinity around a cylindrical or elliptical inhomogeneity, where the permeability inside the inhomogeneity is anisotropic and different from the isotropic domain outside the inhomogeneity. The orientation of the uniform flow with respect to the orientation of the ellipse is arbitrary as well as the orientation of the anisotropy inside the ellipse. We derive an expression for the ratio of the flow through the ellipse with respect to the flow in the homogeneous case.     

Modeling and Simulation of Local Thermal Non-equilibrium Effects in Porous Media with Small Thermal Conductivity

The two-equation model in porous media can describe the local thermal non-equilibrium (LTNE) effects between fluid and solid at REV scale, with the temperature differences in a solid particle neglected. A multi-scale model has been proposed in this study. In the model, the temperature differences in a solid particle are considered by the coupling of the fluid energy equation at REV scale with the heat conduction equation of a solid particle at pore scale. The experiments were conducted to verify the model and numerical strategy. The multi-scale model is more suitable than the two-equation model to predict the LTNE effects in porous media with small thermal conductivity. The effects of particle diameter, mass flow rate, and solid material on the LTNE effects have been investigated numerically when cryogenic nitrogen flows through the porous bed with small thermal conductivity. The results indicate that the temperature difference between solid center and fluid has the same trend at different particle diameters and mass flow rates, while the time to reach the local thermal equilibrium is affected by solid diameter dramatically. The results also show that the temperature difference between solid center and surface is much greater than that between solid surface and fluid. The values of \( \rho {\text{c}} \) for different materials have important influence on the time to reach the local thermal equilibrium between solid and fluid.     

Effective Diffusivities of Point-Like Molecules in Isotropic Porous Media by Monte Carlo Simulation

Monte Carlo simulations of point-like molecules in random and structured media are used to determine and characterize the effective diffusion coefficients of the molecules in the media. Simulations were carried out in 2D and 3D media. Monte Carlo simulation results in 2D and 3D media are compared with those obtained by analytical techniques. Simulation results indicate that for the structured, isotropic media the effective diffusivities can be characterized according to percolation thresholds in addition to porosity. The effective diffusivities in two isotropic media with the same porosity but different percolation thresholds can differ significantly. The effects of dimensionality on the effective diffusivities can also be significant. It is shown that in general the effective diffusion coefficients obtained from 2D simulation are not a good approximation to those of 3D, especially when the percolation thresholds of the 2D media and the 3D media are very different.     

Effect of Rotation on Thermal Convection in an Anisotropic Porous Medium with Temperature-dependent Viscosity

A linear stability analysis is performed for mono-diffusive convection in an anisotropic rotating porous medium with temperature-dependent viscosity. The Galerkin variant of the weighted residual technique is used to obtain the eigen value of the problem. The effect of Taylor–Vadasz number and the other parameters of the problem are considered for stationary convection in the absence or presence of rotation. Oscillatory convection seems highly improbable. Some new results on the parameters’ influence on convection in the presence of rotation, for both high and low rotation rates, are presented.     

Bridging Effective Stress and Soil Water Retention Equations in Deforming Unsaturated Porous Media: A Thermodynamic Approach

The finite deformation of an unsaturated porous medium is analysed from first principles of mixture theory. An expression for Bishop’s effective stress is derived from (1) the deformation-dependent Brooks and Corey’s water retention curve and (2) the restrictions on the constitutive relationships of an unsaturated medium subject to finite deformation. The resulting expression for the effective stress parameter \(\chi \) is reasonably consistent with experimental data from the literature. Hence, it is shown that Bishop’s equation is constitutively linked to water retention curves in deforming media.     

Experimental Investigation of Transient Thermal Convection in Porous Media

A laboratory experiment of transient thermal convection in a 1-m-high cell was conducted to compare the length and time scales of plume development to theory. The temperature field was resolved to less than 1 mm and was measured by dissolving a solution of thermochromic crystals into the water–glycerin working fluid. The time-dependent experiment was run by applying heat at the bottom boundary that eventually was \(6\,^\circ \) C above the background temperature of the fluid. After development of a thermal boundary layer, the instability became visible at 26 min, with the development of 11, 3–4 cm width plumes growing from the boundary layer. The initially rapid growth rate reached a limiting velocity of approximately 0.5 cm min \(^{-1}\) , and then decelerated throughout the experiment. Plumes interacted primarily by merging together; by the end of the experiment only three plumes were present. The Nusselt number at the onset of convection was 10, although it dropped to 4 after 45 min, which would be expected of a barely unstable system.     

An Analysis of Thermal Radiation in Porous Media Under Local Thermal Non-equilibrium

Transport in Porous Media - The present work investigates the thermal radiation transport inside porous media under local thermal non-equilibrium conditions. Two different geometrical situations, a...     

Viscous Fingering Instability in Porous Media: Effect of Anisotropic Velocity-Dependent Dispersion Tensor

The viscous fingering of miscible flow displacements in a homogeneous porous media is examined to determine the effects of an anisotropic dispersion tensor on the development of the instability. In particular, the role of velocity-dependent transverse and longitudinal dispersions is investigated through linear stability analysis and nonlinear simulations. It is found that an isotropic velocity-dependent dispersion tensor does not affect substantially the development of the instability and effectively has the same effect as molecular diffusion. On the other hand, an anisotropic velocity-dependent dispersion tensor results in different instability characteristics and more intricate finger structures. It is shown that anisotropic dispersion has profound effects on the development of the fingers and on the mechanisms of interactions between neighboring fingers. The development of the new finger structures is explained by examining the velocity field and characterized qualitatively through a spectral analysis of the average concentration and an analysis of the variations of the sweep efficiency and relative contact area.     

Effects of Material Symmetry on the Coefficients of Transport in Anisotropic Porous Media

The objective of this article is to highlight certain features of a number of coefficients that appear in models of phenomena of transport in anisotropic porous media, especially the coefficient of dispersion the second-rank tensor D _ij , and the dispersivity coefficient, the fourth-rank tensor a _ijkl , that appear in models of solute transport. Although we shall focus on the transport of mass of a dissolved chemical species in a fluid phase that occupies the void space, or part of it, the same discussion is also applicable to transport coefficients that appear in models that describe the advective mass flux of a fluid and the diffusive transport of other extensive quantities, like heat. The case of coupled processes, e.g. the simultaneous transport of heat and mass of a chemical species, are also considered. The entire discussion will be at the macroscopic level, at which a porous medium domain is visualized as a homogenized continuum.     

A Two-Equation Model for Heat Conduction in Porous Media (I: Theory)

A two-equation model is presented which describes the conservation of heat in each phase of a porous medium in which diffusion is the predominant means of heat transfer, and of which the phases are not in thermal equilibrium with each other. The model is derived using the method of local volume averaging. This formulation, together with the introduction of characteristic temperature distributions, yields the definition of an effective and a coupled thermal conductivity tensor.     

A Two-Equation Model for Heat Conduction in Porous Media (II: Application)

The formulism of a two-equation model for heat conduction in porous media, developed in a previous paper, is applied to the case of steady state one-dimensional heat transfer in a porous medium that is made up of geometrically similar units of similar size and of ordered spatial distribution. For this case study, the model-predicted locally volume-averaged temperature distributions for the solid and the fluid phase are compared to a numerical solution at a microscopic level, showing excellent agreement.