Darcy Regime Research Articles

Permeability up-scaling using Haar Wavelets

In the context of flow in porous media, up-scaling is the coarsening of a geological model and it is at the core of water resources research and reservoir simulation. An ideal up-scaling procedure preserves heterogeneities at different length-scales but reduces the computational costs required by dynamic simulations. A number of up-scaling procedures have been proposed. We present a block renormalization algorithm using Haar wavelets which provide a representation of data based on averages and fluctuations.In this work, absolute permeability will be discussed for single-phase incompressible creeping flow in the Darcy regime, leading to a finite difference diffusion type equation for pressure. By transforming the terms in the flow equation, given by Darcy’s law, and assuming that the change in scale does not imply a change in governing physical principles, a new equation is obtained, identical in form to the original. Haar wavelets allow us to relate the pressures to their averages and apply the transformation to the entire equation, exploiting their orthonormal property, thus providing values for the coarse permeabilities.Focusing on the mean-field approximation leads to an up-scaling where the solution to the coarse scale problem well approximates the averaged fine scale pressure profile.

Sudden or smooth transitions in porous media natural convection

In porous media isothermal flow a transition from the Darcy regime, via an inertia dominated regime, towards turbulence is anticipated. In porous medium natural convection the transition to turbulence follows a different route. The first transition from a motionless-conduction regime to steady natural convection is followed by a direct second transition to a non-steady (time dependent) and non-periodic regime (referred to as weak turbulent), prior to the amplitude of the convection reaching such large values as to involve inertial, non-Darcy effects. The latter is due to an additional non-linear interaction that appears in natural convection as a result of the coupling between the equations governing the fluid flow and the energy equation. The present paper deals with identifying whether the transitions are sudden or possibly smooth. The latter is accomplished by using a truncated Galerkin representation of the natural convection problem in a porous layer heated from below (an extended Darcy model) leading to the familiar Lorenz equations for the evolution of the convection amplitudes with time. Two different formulations (named the “original” and the “modified” systems) are being used in an anticipation to obtaining a smooth transition in the form of an imperfect bifurcation from the “modified” system formulation. The results show that the transition remains sudden and the accuracy of the “modified” system results is being tested in comparison with the “original” system showing a sufficiently high degree of accuracy.

A continuum‐discrete hydromechanical analysis of granular deposit liquefaction

AbstractA coupled continuum‐discrete hydromechanical model was employed to analyse the liquefaction of a saturated loose deposit of cohesionless particles when subjected to a dynamic base excitation. The pore fluid flow was idealized using averaged Navier–Stokes equations and the discrete element method was employed to model the solid phase particles. A well established semi‐empirical relationship was utilized to quantify the fluid–particle interactions. The conducted simulations revealed a number of salient micro‐mechanical mechanisms and response patterns associated with the deposit liquefaction. Space and time variation of porosity was a major factor which affected the coupled response of the solid and fluid phases. Pore fluid flow was within Darcy's regime. The predicted response exhibited macroscopic patterns consistent with experimental results and case histories of the liquefaction of granular soil deposits. Copyright © 2004 John Wiley & Sons, Ltd.

Necessary and sufficient conditions of global nonlinear stability for rotating double-diffusive convection in a porous medium

The nonlinear stability of the conduction-diffusion solution of a fluid mixture heated and salted from below (and of a homogeneous fluid heated from below) and saturating a porous medium is studied with the Lyapunov direct method. Both Darcy and Brinkman models have been used. The porous medium is bounded by two horizontal parallel planes and is rotating about a vertical axis. Necessary and sufficient conditions of unconditional stability are proved (i.e., the critical linear and nonlinear stability Rayleigh numbers coincide). Our results generalize those given by Straughan [21] for a homogeneous fluid in the Darcy regime. In the case of a mixture two stabilizing effects act: that of the rotation and of the concentration of the solute.

Revisited Analysis of Pressure Drop in Flow through Crushed Rocks

Experimental data of pressure drop in flow through crushed rocks are reanalyzed with the capillary model in comparison with a model based on the square root of permeability as characteristic length. The capillary model is described by two parameters: the pore diameter and the tortuosity. The comparison leads to relationships between the structural parameters, Reynolds number, and friction factor of each model. The interest of the capillary model is that a single equation can predict all the experimental data expressed in terms of a pore friction factor as a function of pore Reynolds number. The equation, which is the sum of a viscous term and an inertial one, is valid for the whole Reynolds number domain. The equation can be used for the determination of the limits of the Darcy and turbulent flow regimes. According to the criterion used to neglect the viscous or the inertial term, the flow regimes limits can be expressed by a simple numerical value.

Pressure drop modelling in cellular metallic foams

A theoretical model is presented for the prediction of pressure drop in a Newtonian fluid flowing through highly porous, isotropic metallic foams. The model is based on a rigorous assumption of piece-wise plane Poiseuille flow and a simplistic geometrical model, and shows promise to accurately predict the hydrodynamic conditions in both the Darcy and Forchheimer regimes, without a priori knowledge of the flow behaviour of the particular metallic foam.

Modelling of power law liquid–solid mass transfer in packed beds at Darcy regime

A model of liquid-to-particle mass transfer in packed beds with single phase non-Newtonian creeping flow is derived using a coherent approach which associates the Lévêque solution with a capillary-type representation of the porous structure. The model is compared to experimental data obtained with spherical and anisotropic parallelepipedal particles, respectively, as well as to literature results. The model allows to predict mass transfer in fixed beds of spheres and anisotropic parallelepipedal particles with a mean relative error of 11%. It is also shown that a better prediction of mass transfer is obtained for parallelepipedal particles if the surface area really offered to fluid flow is considered. Dimensionless numbers based on the pore dimension are shown to be more suitable than those based on a particle diameter.

A comparative study on the performance of two time stepping schemes for convection in a fluid saturated porous medium

A comparative study has been carried out to investigate the performance of two different time stepping schemes for convective heat transfer and flow in a fluid saturated porous medium. Both the schemes are based on the velocity correction procedure. The first scheme is a semi‐implicit one in which the linear and non‐linear porous medium terms of the momentum equation are treated implicitly but solution of the simultaneous equation system is avoided by lumping the mass. The second procedure (quasi‐implicit) treats the porous medium and viscous terms implicitly and a simultaneous equation system is constructed to solve the equations of momentum conservation. Two numerical examples have been considered and both the schemes are tested for various parameters governing the flow and heat transfer in these problems. Results show that, at smaller Rayleigh numbers and on fine meshes, the quasi‐implicit scheme gives faster convergence to steady state in both Darcy and non‐Darcy regimes than that of the semi‐implicit scheme. At higher Rayleigh numbers, the semi‐implicit scheme is faster in the Darcy regime. Also, the semi‐implicit scheme is faster than that of the quasi‐implicit scheme on a coarse mesh used in this study. In general both the schemes predict transient cyclic developments well.

Transient Natural Convection in Differentially Heated Porous Enclosures: Corrected Abstract

The following abstract should replace the one that appeared with this article in the Journal of Porous Media, 3(2), 165 (2000).Transient natural convective flow and heal transfer in a differentially heated enclosure filled with a saturated porous medium is investigated numerically by adopting the general flow model that accounts for both the boundary and the inertial effects in porous media. The present study covers investigation of the fluid Rayleigh number (RA) between 1 × 105 and 1 × 109, the Darcy number (DA) of 1 × 10−6 to 1 × 10−2 for aspect ratio, and the inertial parameter and modified Prandtl number of unity. For DA > 1 × 10−6, it is found that if the modified Rayleigh number (Ram − Ra.Da) is greater than 100, significant differences exist between the general model and the Darcy model predictions, especially at the early transient stage. Both in the Darcy and non-Darcy regimes, the Brinkman extended model agreed well with the general model. Also, it is found that Forchheimer's term is significant in the transient stage, but it is of no importance at the steady state.

Nonlinear Study of Stratified Fluid Through Porous Media

An analysis is presented to describe the effects of Darcy resistance, fluid inertia, and horizontal density gradient on a heterogeneous Boussinesq fluid saturated porous medium in the presence of the vertical gravitational field using the Forchheimer-Lapwood extended Darcy (DLF) equation. The fluid is initially at rest, and sets in motion owing to two forms of the initial density gradients. One is the constant initial horizontal density gradient and the other is the piecewise constant density gradient. In the case of the uniform initial horizontal density gradient a purely horizontal motion develops satisfying the nonlinear Forchheimer extended Darcy (FD) equation. Analytical solution of this equation is obtained and limiting solutions valid for the Darcy regime and for a nonviscous fluid in the absence of porous media are derived. A measure of the stability of the flow is discussed briefly using the Richardson number. A comparison between the nature of solutions satisfying the nonlinear and linear initial value problems is made. We found that the vertical density gradient varies continuously both with space z and time t but the horizontal density gradient remains unchanged. In the case of a piecewise constant initial density gradient with continuous distribution of density, stream function formulation is used and the solutions are obtained using time-series analysis. In this case the solution shows crowding of the density profiles in the lower half of the channel, reflecting an increase in density gradient and incipient frontogenesis there, because of the increase in circulation of the flow due to the piecewise initial density gradient. Copyright © 2001 by Begell House, Inc.

Analytical solution for the effect of viscous dissipation on mixed convection in saturated porous media

A new analytical solution is introduced for the effect of viscous dissipation on mixed convection flow and heat transfer about an isothermal vertical wall embedded in Darcy and non-Darcy porous media with uniform free stream velocity. The effect of viscous dissipation on mixed convection in both regimes has been analyzed for both the aiding and opposing flows using Gebhart number, Gex = gβx/cp. The governing parameters are Re, Ra, Pe and Gex. The case of Re = 0 corresponds to Darcy mixed convection region and Re/Pe is identified as the mixed convection governing parameter, Ra = 0 leading to pure forced convection. A good agreement was found between the numerical and analytical solutions. It was found from the Nusselt number results that viscous dissipation lowers the heat transfer rate in both Darcy and Forchheimer flow regimes for aiding as well as opposing flows.

Forced Convection of a Power-Law Fluid in a Porous Channel—Integral Solutions

The integral method based on the boundary layer analysis is used to conduct a theoretical study of the primary hydrodynamic and heat transfer mechanisms for forced convection of a power-law fluid in a porous channel for both cases of uniform heat flux (UHF) and uniform wall temperature (UWT) boundary conditions. The flow in the porous medium is modeled using the modified Brinkman—Forchheimer-extended Darcy model for power-law fluids. The results indicate that for a high-permeability porous medium, the thickness of the momentum boundary layer depends on the Darcy number, inertia parameter, and power law index, but for a low-permeability porous medium it depends only on the Darcy number. Consequently in the non-Darcy regime, the effects of power law index on hydrodynamic and heat transfer behaviors in the porous channel are significant, whereas in the Darcy regime, the effects of Darcy number are predominant. Also the hydrodynamic behavior of shear thickening fluids (n > 1.0) is more sensitive to the Darcy number whereas the behavior of shear thinning fluids (n < 1.0) is more sensitive to the inertia parameter in the non-Darcy regime. Based on the fully developed Nusselt number solutions, the valid region for the applicability of the present integral method is illustrated graphically. The valid region of the integral solutions covers the Darcy regime for any value of the power law index within the range considered, while in the non-Darcy regime, the valid region for a shear thinning fluid becomes smaller than that for a shear thickening fluid as the inertia parameter decreases.

Experimental characterisation of flow regimes in various porous media—I: Limit of laminar flow regime

This work is part of a global study of flow regimes in porous media beyond Darcy regime. In view of determining the end of stable laminar regime in various porous media such as beds packed with spheres, stratified media and reticulated media, electrochemical micro-probes are inserted inside the medium and at the wall of test sections. Local instantaneous measurements of the limiting current intensity are implemented. Spectral analysis of the signal fluctuations enables an accurate determination of the end of stable laminar regime in the studied media. The comparison between internal measurements and those implemented at the wall of the cell shows that the latter are representative of the flow in the case of reticulated media but not in the case of packed beds of particles for which the use of internal probes is necessary. Results are expressed thanks to a pore Reynolds number Re p , based on a capillary representation of the porous media. Re p ≈180 characterises the end of the stable laminar regime in packed beds of particles presenting an isotropy in the plane perpendicular to the main direction of the fluid flow. A ‘laminarising effect’ is observed for synthetic foams. For each porous media tested, the percentage of viscous and inertial contributions in the pressure drop are given at the onset of fluctuations.

Forced convection of a power-law fluid in a porous channel - numerical solutions

A detailed numerical study of laminar forced convection in a porous channel which contains a fibrous medium saturated with a power-law fluid was performed. Hydrodynamic and heat transfer results are presented for a configuration that has uniform heat flux or uniform temperature heating at the walls. The flow in the porous medium was modeled using the modified Brinkman-Forchheimer-extended Darcy model for power law fluids in which the non-Darcy effects of inertia and boundary were considered. Parametric studies were conducted to examine the effects of Darcy number, power law index, inertia parameter and Prandtl number. The results indicate that when the power law index is decreased, the velocity gradient near the walls increases but these effects are reduced gradually as the Darcy number decreases until the Darcy regime (Da≤10−6) is reached in which case the effects of power law index become negligible. As the power law index is decreased, the thermal boundary layer thickness decreases significantly only in the non-Darcy regime. Consequently, as the power law index decreases, the fully developed Nusselt number increases considerably in the non-Darcy regime whereas in the Darcy regime the change in Nusselt number is very small. As the Prandtl number increases, the local Nusselt number increases and this effect is more significant for shear thinning fluids (n<1.0).

FLOW THROUGH A HIGHLY POROUS ANISOTROPIC MULTIFILAMENT KNIT

Experimental results for isothermal Newtonian flow through a high porosity anisotropic porous medium are provided. The porous medium is formed by a knitted stainless steel wire mesh rolled up to form a cylindrical plug. This kind of porous medium is generally used as a mist eliminator in the chemical and petroleum industries. Experimental results are compared with pressure drop predictions obtained with a theoretical pore-scale method. The model allows for anisotropy and is applied predictively over both the Darcy and non-Darcy regimes, requiring only that the porosity, dimensions of the wire and characteristic lengths of the plug be known. The predictions compare reasonably well with the experimental results.

Double-diffusive natural convection in a fluid saturated porous cavity with a freely convecting wall

Double — diffusive natural convection in fluid saturated porous medium has been investigated using a generalised porous medium model. One of the vertical walls of the porous cavity considered is subjected to convective heat and mass transfer conditions. The results show that the flow, heat and mass transfer become sensitive to applied mass transfer coefficient in both the Darcy and non-Darcy flow regimes. It is also observed that the Sherwood number approaches a constant value as the solutal Biot number increases.

Non-Darcy double-diffusive natural convection in axisymmetric fluid saturated porous cavities

Double-diffusive natural convection in a fluid saturated porous medium has been investigated using the finite element method. A generalised porous medium model is used to study both Darcy and non-Darcy flow regimes in an axisymmetric cavity. Results indicate that the Darcy number should be a separate parameter to understand flow characteristics in non-Darcy regime. The influence of porosity on heat and mass transfer is significant and the transport rates may differ by 25% or more, at higher Darcy and Rayleigh numbers. When compared with the Darcy and other specialised models of Brinkman and Forchheimer, the present generalised model predicts the least heat and mass transfer rates. It is also observed that an increase in radius ratio leads to higher Nusselt and Sherwood numbers along the inner wall.

Effect of viscous dissipation on a non-Darcy natural convection regime

The effect of viscous dissipation on non-Darcy natural convection flow along an isothermal vertical wall embedded in a saturated porous medium is initiated. The flow field is divided into non-Darcy G = 0, intermediate G = 0(1), and limit Darcy regimes G → ∞ using the inertia-buoyancy scales. Thermal dispersion effects are also taken into consideration. It is observed that the viscous dissipation effect reduces the heat transfer rate by about 10% in all three flow regimes. It is also found that the effect of viscous dissipation greater as the value of the parameter G increases. The effect of viscous dissipation in all three flow regimes increases with the dispersion parameter.

Heat transfer by forced convection in pipes packed with porous media whose matrices are composed of spheres

Empirical correlation equations for the average Nusselt number have been determined that accurately represent an extensive body of new experimental data for the geometry under consideration for fully developed thermal conditions. The equations cover the Darcy, Forchheimer and turbulent regimes of flow. The correlation equations are based upon a hypothesis that regards the flow in a porous medium to be the superposition of a ‘fine’ component upon a ‘coarse’ component and takes into account the wall effect and dispersion. The results show that packed tubes provide heat transfer rates from two to seven times those of unpacked tubes for laminar flow and two to two and a half times for turbulent flow for equal pumping power.

Pressure drop prediction for flow through high porosity metallic foams

Experimental results for isothermal Newtonian flow through metallic foams are analysed in comparison with a theoretical model and the results interpreted. In both the Darcy and non-Darcy regimes the results show promise for the accurate theoretical prediction of fluid dynamic phenomena in foamlike porous materials of very high porosity.