Normal Mode Analysis of the High Speed Channel Flow in a Bidisperse Porous Medium

The linear Darcy–Brinkman model of the high speed flow in a bidisperse porous medium proposed by Nield and Kuznetsov (Transport Phenomena in Porous Media, 2005) is revisited in this paper. For the steady unidirectional flow in a parallel plane channel the exact analytical solutions for the fluid velocities are worked out by the normal-mode reduction of the governing equations. The limiting cases of the weak and strong momentum transfer between the flows in the fracture and porous phases are discussed in detail. A comparison to the nonlinear Forchheimer extension of the model proposed recently by Nield and Kuznetsov (Transport Porous Media, 2013) shows that, in the considered parameter range, the nonlinear effect of the Forchheimer drag is negligibly small. Even the simplest zero-momentum transfer solution yields an acceptable approximation.     

Mixed Convection Boundary Layer Flow Past a Horizontal Circular Cylinder Embedded in a Bidisperse Porous Medium

Adopting a two-temperature and two-velocity model, appropriate to a bidisperse porous medium (BDPM) proposed by Nield and Kuznetsov (2008), the classical steady, mixed convection boundary layer flow about a horizontal, isothermal circular cylinder embedded in a porous medium has been theoretically studied in this article. It is shown that the boundary layer analysis leads to expressions for the flow and heat transfer characteristics in terms of an inter-phase momentum parameter, a thermal diffusivity ratio, a thermal conductivity ratio, a permeability ratio, a modified thermal capacity ratio, and a buoyancy or mixed convection parameter. The transformed partial differential equations governing the flow and heat transfer in the f-phase (the macro-pores) and the p-phase (the remainder of the structure) are solved numerically using a very efficient implicit finite-difference technique known as Keller-box method. A good agreement is observed between the present results and those known from the open literature in the special case of a traditional Darcy formulation (monodisperse system).     

Diffuse-interface Modeling of Two-phase Flow for a One-component Fluid in a Porous Medium

The diffuse-interface (DI) model for the two-phase flow of a one-component fluid in a porous medium has been presented by Papatzacos [2002, Transport Porous Media 49, 139–174] and by Papatzacos and Skjæveland [2004, SPE J. (March 2004), 47–56]. Its main characteristics are: (i) a unified treatment of two phases as manifestations of one fluid with a van der Waals type equation of state, (ii) the inclusion of wetting, and (iii) the absence of relative permeabilities. The present paper completes the presentation by including the implementation of wetting in the general case of a mixed-wet rock. As a result of this implementation, some statements are made about capillary pressure, confirming similar statements by Hassanizadeh and Gray [1993, Water Resour. Res. 29, 3389–3405]. As an application of the model, we show that relative permeabilities depend on the spatial derivatives of the saturation.     

Modeling of Two Phase Flow in a Hydrophobic Porous Medium Interacting with a Hydrophilic Structure

Transport in Porous Media - Fluid flow through layered materials with different wetting behavior is observed in a wide range of applications in biological, environmental and technical systems....     

Note to the Opposing Flow Regime at Mixed Convection Around a Heated Cylinder in a Porous Medium

Transport in Porous Media -     

Brinkman Flow of a Viscous Fluid Through a Spherical Porous Medium Embedded in Another Porous Medium

An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium is presented using the Brinkman model, assuming a uniform shear flow far away from the sphere. Semi-analytical solutions of the problem are derived and relevant quantities such as velocities and shearing stresses on the surface of the sphere are obtained. The streamlines inside and outside the sphere and the radial velocity are shown in several graphs for different values of the porous parameters \({\sigma _1 =(\mu /\tilde {\mu }) (a/\sqrt{K_1 })}\) and \({\sigma _2 =(\mu /\tilde {\mu }) (a/\sqrt{K_2 })}\) , where a is the radius of the sphere, μ is the dynamic viscosity of the fluid, \({\tilde {\mu }}\) is an effective or Brinkman viscosity, while K ₁ and K ₂ are the permeabilities of the two porous media. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase of both porous parameters σ ₁ and σ ₂.     

Modeling Fluid Flow in Fractured Porous Media with the Interfacial Conditions Between Porous Medium and Fracture

One of the most popular models that has been applied to predict the fluid velocity inside the fracture with impermeable walls is the cubic law. It highlights that the mean flux along the fracture is proportional to the cubic of fracture aperture. However, for a fractured porous medium, the normal and tangential interface conditions between the fracture and porous matrix can change the velocity profile inside the fracture. In this paper, a correction factor is introduced for flow equation along the fracture by imposing the continuity of normal and tangential components of velocity at the interface between the fracture and porous matrix. As a result, the mean velocity inside the fracture depends not only on the fracture aperture, but also on a set of non-dimensional numbers, including the matrix porosity, the ratio of intrinsic permeability of fracture to that of matrix, the wall Reynolds number, and the ratio of normal velocity on one wall to the other. Finally, the introduced correction factor is employed within the extended finite element method, which is widely used for numerical simulation of fluid flow within the fractured porous media. Several numerical results are presented for the fluid flow through a specimen containing single fracture, in order to investigate the deviation from the cubic law in different case studies.

     

Peristaltic Flow of a Maxwell Model Through Porous Boundaries in a Porous Medium

In the present investigation we have presented the peristaltic flow of a linear Maxwell model through porous boundaries in a porous medium. The governing non-dimensional partial differential are solved in wave frame by using regular perturbation method and assumed form of solution. We have discussed the problem only for free pumping case. The effects of various physical parameters involved in the problem have been investigated and shown graphically.     

Fourier Representation of Intermittent Flow in a Porous Medium

The field measurements and numerical results for intermittent flow regime in a sandy soil show that the time distributions of the soil water flux q(z, t), and the soil water content θ(z, t)at various depths are periodic in nature, where t is time and z is the depth (i.e., at the surface z = 0 and at depths z = − 5, − 10, − 15 cm, etc). The period of q(z, t) and θ(z, t) variations are generally determined by the sum of the duration of pulse and the duration between the initiation of two consecutive pulses of water at the soil surface. Fourier series models have been given for q(z, t) and θ(z, t) variations. The predicted Fourier results for these variations have been compared with the experimentally verified numerical results—designated as observed values. The results show that the amplitudes of these variations were damped exponentially with depth, and the phase shift increased linearly with depth.     

The Convective Boundary-Layer Flow on a Reacting Surface in a Porous Medium

The convective boundary-layer flow on an impermeable vertical surface in a fluid-saturated porous medium is considered where the flow results from the heat released by an exothermic catalytic reaction on the surface converting a reactive component within the convective fluid to an inert product. The reaction is modelled by first-order kinetics with an Arrhenius temperature dependence. Numerical solutions of the governing equations are obtained for a range of parameter values. These show, for large activation energies, that localized rapid changes in wall temperature and localized high reaction rates occur a little way from the leading edge. Asymptotic expansions, valid at large distances from the leading edge, are derived, the form that these expansions take is qualitatively different depending on whether or not reactant consumption is included in the model.     

Porous Medium Modeling of Air-Cooled Condensers

This article presents a porous media transport approach to model the performance of an air-cooled condenser. The finned tube bundles in the condenser are represented by a porous matrix, which is defined by its porosity, permeability, and the form drag coefficient. The porosity is equal to the tube bundle volumetric void fraction and the permeability is calculated by using the Karman–Cozney correlation. The drag coefficient is found to be a function of the porosity, with little sensitivity to the way this porosity is achieved, i.e., with different fin size or spacing. The functional form was established by analyzing a relatively wide range of tube bundle size and topologies. For each individual tube bundle configuration, the drag coefficient was selected by trial and error so as to make the pressure drop from the porous medium approach match the pressure drop calculated by the heat exchanger design software ASPEN B-JAC. The latter is a well-established commercial heat exchanger design program that calculates the pressure drop by using empirical formulae based on the tube bundle properties. A close correlation is found between the form drag coefficient and the porosity with the drag coefficient decreasing with increasing porosity. A second order polynomial is found to be adequate to represent this relationship. Heat transfer and second law (of thermodynamics) performance of the system has also been investigated. The volume-averaged thermal energy equation is able to accurately predict the hot spots. It has also been observed that the average dimensionless wall temperature is a parabolic function of the form drag coefficient. The results are found to be in good agreement with those available in the open literature.     

Darcy Flow in a Wavy Channel Filled with a Porous Medium

Flow in channels bounded by wavy or corrugated walls is of interest in both technological and geological contexts. This paper presents an analytical solution for the steady Darcy flow of an incompressible fluid through a homogeneous, isotropic porous medium filling a channel bounded by symmetric wavy walls. This packed channel may represent an idealized packed fracture, a situation which is of interest as a potential pathway for the leakage of carbon dioxide from a geological sequestration site. The channel walls change from parallel planes, to small amplitude sine waves, to large amplitude nonsinusoidal waves as certain parameters are increased. The direction of gravity is arbitrary. A plot of piezometric head against distance in the direction of mean flow changes from a straight line for parallel planes to a series of steeply sloping sections in the reaches of small aperture alternating with nearly constant sections in the large aperture bulges. Expressions are given for the stream function, specific discharge, piezometric head, and pressure.     

Poiseuille Flow of a Third Grade Fluid in a Porous Medium

Effects of porous medium have been investigated on the steady flow of a third grade fluid between two stationary porous plates. The continuity and momentum equations along with modified Darcy??s law are used for the development of mathematical problem. The governing nonlinear problem is solved by a homotopy analysis method. The dimensionless velocity and shear stresses at the plates are analyzed.     

Modeling of Heat and Mass Transfer in a Fractured Porous Medium

While fractured formations are possibly the most important contributors to the production of oil worldwide, modeling fractured formations with rigorous treatments has eluded reservoir engineers in the past. To date, one of the most commonly used fractured reservoir models remains the one that was suggested by Warren and Root nearly four decades ago. In this paper, a new model for fractures embedded in a porous medium is proposed. The model considers the Navier-Stokes equation in the fracture (channel flow) while using the Brinkman equation for the porous medium. Unlike the previous approach, the proposed model does not require the assumption of orthogonality of the fractures (sugar cube assumption) nor does it impose incorrect boundary conditions for the interface between the fracture and the porous medium. Also, the transfer coefficient between the fracture and matrix interface does not need to be specified, unlike the cases for which Darcy's law is used. In order to demonstrate the usefulness of the approach, a two-dimensional model of a fractured formation is developed and numerical simulation runs conducted.

The proposed model is derived through a series of finite element modeling runs for various cases using the Navier-Stokes equation in the channel while maintaining the Brinkman equation in the porous medium. Various cases studied include different fracture orientations, fracture frequencies, and thermal and solutal constraints. The usefulness of the proposed model in modeling complex formations is discussed. Finally, a series of numerical runs also provided validity of the proposed model for the cases in which thermal and solutal effects are important. Such a study of double diffusive phenomena, coupled with forced convection, in the context of fractured formations has not been reported before.     

Flow Through a Two-Scale Porosity,Oriented Fibre Porous Medium

A model is described for the meso- and micro-flow through an array of oriented fibre tows with meso-channels between the tows. Axial Stokes's flow was considered in the meso-channels and Darcy's law was applied within the porous fibre tows, taking into account injection pressure and capillary pressures in both types of flow. Transverse flow transfer was modelled from the leading flow front to the lagging flow and a partial-slip boundary condition was applied at the permeable boundaries of meso-channels. Flow visualisation experiments and microstructural characterisation of laminates provided appropriate experimental data for model validation. In this, the predictions for the progress of the leading meso-flow were in excellent agreement with the experimental data. Parametric studies followed including the effects of injection pressure and meso-channel size.     

Chaos in the Convective Flow of a Fluid Mixture in a Porous Medium

The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.     

Lattice Boltzmann Pore Scale Simulation of Natural Convection in a Differentially Heated Enclosure Filled with a Detached or Attached Bidisperse Porous Medium

In this research, pore scale simulation of natural convection in a differentially heated enclosure filled with a conducting bidisperse porous medium is investigated using the thermal lattice Boltzmann method. For the first time, the effect of connection of the bidisperse porous medium to the enclosure walls is studied by considering the attached geometry in addition to the detached one. Effect of most relevant parameters on the streamlines and isotherms as well as hot wall average Nusselt number is studied for two of the bidisperse porous medium configurations. It is observed that effect of geometrical and thermo-physical parameters of the bidisperse porous medium on the heat transfer characteristics is more complicated for the attached configuration. To assess the validity of the local thermal equilibrium condition in the micro-porous media, the pore scale results are used to compute the percentage of the local thermal non-equilibrium for two of the bidisperse porous medium configurations. It is concluded that for the detached configuration, the local thermal equilibrium condition is confirmed in the entire micro-porous media for the ranges of the parameters studied here. However, for the attached geometry, it is shown that departure from the local thermal equilibrium condition is observed for the higher values of the Rayleigh number, micro-porous porosity, solid–fluid thermal conductivity ratio, and the smaller values of the macro-pores volume fraction.     

Thermal Analysis of Flow in a Porous Medium Over a Permeable Stretching Wall

This work presents a similarity solution for boundary layer flow through a porous medium over a stretching porous wall. Two considered wall boundary conditions are power-law distribution of either wall temperature or heat flux which are general enough to cover the isothermal and isoflux cases. In addition to momentum, both first and second laws of thermodynamics analyses of the problem are investigated. Independent numerical simulations are also performed for verification of the proposed analytical solution. The results, from the two independent approaches, are found to be in complete agreement. A comprehensive parametric study is presented and it is shown that heat transfer and entropy generation rates increase with Reynolds number, Prandtl number, and suction to the surface.